All information contained in this work is provided "as is." All warranties, expressed, implied or statutory, concerning the accuracy of the information or the suitability for any particular use are hereby specifically disclaimed. While every effort has been taken to ensure the accuracy of the information contained in this work, the author(s) and contributor(s) assume(s) no responsibility for errors or omissions or for damages resulting from the use of the information contained herein. This work may be copied in any printed or electronic form for non-commercial, personal, or educational purposes if the work is not modified in any way, provided that the copyright notice, the notices of any other author included in this work, and this copyright agreement appear on all copies.
Sam Simpson also grants permission to distribute this work in electronic form over computer networks for other purposes, provided that, in addition to the terms and restrictions set forth above, Sam Simpson and/or other cited authors are notified and that no fees are charged for access to the information in excess of normal online charges that are required for such distribution.
This work may also be mentioned, cited, referred to or described (but not copied or distributed, except as authorised above) in printed publications, on-line services, other electronic communications media, and otherwise, provided that Sam Simpson receives appropriate attribution.
IDEA(tm) is a trademark of Ascom-Tech AG. PGP and Pretty Good Privacy are trademarks of Network Associates, Inc. All other products mentioned are registered trademarks or trademarks of their respective companies.
Comments about, suggestions about or corrections to this document are welcomed.
v1.5 - 20th September 1999
Added section "Has RIPEMD been broken?"
Added section "What are the implications of the NSA key in CryptoAPI?"
Added detail to "How big should my asymmetric key be?"
Added detail to "What about Elliptic Curves?"
Added detail to "Which is stronger, RSA or DH/DSS?"
Added detail to "What is IDEA?"
Added detail to "What is "government certification"?"
Added detail to "How does the cryptographic security of the OpenPGP & S/MIME standards compare?"
v1.4 - 18th August 1999
Minor grammar / spelling related changes
Updated references to FIPS186 to the revised FIPS186-1
USENET references now contain link to deja.com where available
Added section "What is the chance of running out of primes?"
Added section "What about Elliptic Curves?"
Added section "What is AES?"
Added section "What is Twofish?"
Added section "What are the implications of Shamir's TWINKLE device on PGP security?"
Added detail to "The huge key debate."
Added detail to "What is DH / ElGamal?"
Added detail to "So there are no intellectual property issues relating to PGP v5+?"
v1.3 - 16th March 1999
Minor changes to numerous sections
Added section "What are the implications of the attacks against MD5?"
Added section "Why doesn't PGP use a "provably secure" cipher?"
Added section "Program x offers 200-bit keys, is it better than PGP?"
Added section "So PGP is perfect?"
Added detail to "What is DH / ElGamal?"
Added detail to "Why does PGP use precomputed primes for DSS/DH?!?"
Added detail to "Why are DSS keys significantly smaller than DH keys?"
Added detail to "Which is stronger, RSA or DH/DSS? "
v1.2 - 3rd March 1999
Minor grammar / spelling related changes
Added section "Why are DSS keys significantly smaller than DH keys?"
Added section "Doesn't the DSS suffer from subliminal channels?"
Added section "Summary of PGP DH versus PGP RSA"
Added detail to "What is DH / ElGamal?"
Added detail to "Why does PGP use precomputed primes for DSS/DH?!?"
Added detail to "The huge key debate."
Added detail to "How does the cryptographic security of the OpenPGP & S/MIME standards compare?"
v1.1 - 27th Jan 1999
Added section "In respect of RSA, what are strong primes?"
Added section "The PGP implementation of DH is based on Galois Fields, aren't they broken?"
Added section "How 'computationally hard' are symmetric keys?"
Added section "How does multiple encryption affect security?"
Added section "Why perform signing before message encryption?"
Added section "Get the threat in perspective!"
Added detail to "What is DH / ElGamal?"
Added detail to "Which is stronger, RSA or DH/DSS? "
Added detail to "What is 3DES?"
v1.01 - 15th Jan 1999
Minor grammar / spelling related changes
Some technical changes - mainly to notes.
Primarily, sincere thanks to Kurt Mueller for providing the Key Revocation Certificate section and Robert Munyer who has scrutinised each release of this FAQ and continues to improve it substantially. Many thanks to Dan "the" Horne & Andy Jeffries for proof reading this document while it was rough-and-ready.
Thanks to John Young for providing the excellent Cryptome web site - without which this FAQ would have taken months longer to research.
Many thanks to Jaime Suarez for converting the FAQ to Spanish.
Thanks to the following additional people who have helped (unwittingly in some cases) in the production or revision of this FAQ:
"Documentation is like sex: when it is good, it is very, very good; and when it is bad, it is better than nothing."
-- Dick Brandon
This document aims to answer several of the most common PGP related questions posed in comp.security.pgp.discuss, alt.security.pgp, sci.crypt etc:
The move from PGP v2.x to v5+ has brought about several major advances. The primary change has to be the User Interface (UI) improvements. The other major change is the move from RSA to DH/DSS keys. This has been a contentious issue, but one that I subjectively believe has been made with good reason. Hopefully this document will help to explain the rationale for certain changes and put concerned users' minds at rest.
In fact, by the end of the FAQ it should be clear that PGP / NAI had to change the implementation of PGP in ways that would have necessarily retired existing RSA keys.
This FAQ tries to remain objective in its approach and offers copious references to material authored by the most eminent cryptographers. Some may argue that this FAQ exhibits an excessive number of references, but I believe this is necessary to allow users to substantiate the claims I have made. After all, why should you trust what I say ;-)
This FAQ assumes that you have previously read (and understood!) the PGP v6.02 User Manual, especially the section "Phil Zimmermann on PGP". I would also recommend that the reader downloads the RSA FAQ [RSA98], but be aware that RSA has a commercial interest in PK cryptography.
Sam Simpson is a Computer Science graduate of the University of Hertfordshire and has also attended postgraduate Information Security / Cryptography courses at the University of London.
He is heavily involved with the freeware ScramDisk project [Scr99], with improving the implementation efficiency of Serpent, an AES candidate [Gla99], and advising on and critically reviewing several cryptographic products.
He had the "honour" of being the first to distribute PGP v6.0 outside of the US, after the program was anonymously mailed to him [WIR98].
Sam is an ardent privacy advocate and, as such, considers himself "vendor neutral" towards this goal. He is currently employed as a Communications Analyst specialising in Internet security and has previously been an application / systems developer.
RSA was announced in 1978 [RSA78]. The security of the RSA system is based upon the RSA Problem (RSAP). This problem is conjectured (but not proven) to be equivalent to the Integer Factorisation Problem (IFP) [MOV96], [Sti95], [Len96].
RSA is patented in the United states by Massachusetts Institute of Technology (U.S. patent number 4,405,829), though this expires 20 September 2000.
From [MOV96] the RSAP is: "given a positive integer n that is a product of two distinct odd primes p and q, a positive integer e such that gcd(e,(p-1)(q-1))=1, and an integer c, find an m such that me is congruent to c (mod n)." Basically RSAP involves finding eth roots modulo a composite integer.
There is not a lot to be said about the RSA algorithm that hasn't been said in the PGP manual or in the RSA FAQ. [Bon98] summarises nicely the security of the RSA system after twenty years of use.
Recently, RSA has been diminished as the algorithm of choice in PGP. It is no longer supported in the freeware version of PGP, due to a number of issues (see later sections).
Note 2 contains a brief overview of how RSA is actually performed in PGP. An observant reader will note that PGP keeps more parameters in the private key than is strictly necessary, this is to speed up computation via the Chinese remainder theorem [Sch96a].
Diffie-Hellman (DH) was the first openly published public key system [DH76] (more correctly Diffie-Hellman is a key-exchange mechanism) and as such has received extensive analysis by eminent cryptographers.
Diffie-Hellman, along with derivatives such as ElGamal are covered by U.S. patent number 4,200,770, which expired 6th September 1997.
PGP actually implements ElGamal [ElG85], which is a public-key encryption variant of the Diffie-Hellman Problem (DHP). It has been proven that the ElGamal encryption scheme is equivalent to the DHP [MOV96], [TY98], [Sti95] - that is to say that if you can break either ElGamal or DH then you can also break the other (see Note 1).
The security of the DH system is based upon the DH Problem (DHP). This problem is conjectured (but not proven) to be equivalent to the Discrete Logarithm Problem (DLP) [MOV96], [Sti95], [Odl83].
DHP is equivalent to the DLP under the "Diffie-Hellman assumption" [Kob94]. This assumes that it is infeasible to compute gab knowing only ga and gb. To quote [Kob94] "...no one can imagine a way of passing from ga and gb to gab without first being able to determine a or b; but it is conceivable that such a way might exist". The above assumes that all values are computed over GF(p).
There are several downsides to using DH compared to using RSA:
The other side of the coin is that there are several benefits to using DH/DSS over RSA:
DH always requires a secure Random Number Generator (RNG), but so does RSA when used to encrypt random session keys (as in PGP, S/MIME etc). A cryptographically secure RNG is already available (See section "How secure is the RNG in PGP?"). A failure in the RNG under RSA would allow an adversary to recover individual messages, but a failure in the RNG whilst using DH/DSS would allow an adversary to recover both the message text and also the private key.
Note 3 contains a brief overview of how DH is actually performed in PGP.
[Odl99] contains an excellent overview on the current state of the art in respect of DH.
DSS stands for Digital Signature Standard and is formally defined in [FIPS186-1] & [ANSI930-1].
DSS employs the ElGamal and Schnorr PK systems to produce a fixed width signature (irrespective of the public/private key size). In contrast, RSA signature length is a function of the key length employed.
The DSS uses discrete exponentiation modulo a prime p, the exponents are computed modulo a prime q. A signature produced with DSS is likely to remain safe for at least 20 years [Odl95]. Time stamping and document trail mechanisms can be used with DSA if longer-term signature verification is required.
DSS is thought to be secure assuming that a good RNG is used. Serge Vaudeny has an interesting attack on DSS that allows a Birthday Attack in only 274 steps, rather than the expected 280 when using "weak keys" [Vau96]. This attack is of no practical concern, and the weak keys are easily detected.
Slim. Very slim indeed.
From [PGP97], the chances of the prime number generation routines in PGP v5.0 producing a composite instead of a prime (that is to say, a number that passes the probabilistic tests) for a 1024-bit key (e.g. a 512-bit candidate prime) are 10-44.
To put things in perspective, the chances of another "Dinosaur Killer asteroid" hitting the earth TODAY are 2-36 [PGP97].
There is no chance we will run out of primes.
For example, there are approximately 3.7 x 10151 512-bit primes. There are approximately 1.5 x 10298 1,000-bit primes - how many keys do we need? :-)
NOTE: According to current mathematical & cryptographic knowledge, this section applies approximately equally to DH & RSA keys. (If anything, DH keys are more secure, but the difference currently appears marginal).
When choosing an asymmetric key size, we are constrained by two principles:
For the majority of users, the main factor in the selection of PK size will be security. Speed is very rarely a determining factor, due to the following reasons:
The following table, from [Sch96a] lists the recommended public key length to protect against attack by a single corporation (keys should be substantially bigger to protect against intelligence agencies!):
|Year||Recommended Key Length|
Schneier has since commented on these figures [Sch98e]: "In PGP, for example, breaking the symmetric algorithm yields one message. Breaking the public-key algorithm yields all messages."
You really need to consider how important your messages are, the "lifetime" of the messages (e.g. how long will the data need to be protected for) and your likely adversary.
Key lengths of 10,000 bits or more can sometimes be necessary, for example for protecting key-signing keys owned by a CA, or for storing data that will remain sensitive for a very long period [Odl95]. It is of interest to note that Rivest (the 'R' in RSA) predicted that a 129-bit factorization would take 40-quadrillion years whereas in reality it took just 8 months using idle cycles on computers around the globe [Len96]. I'd say it pays to be cautious with keylengths.
We know that 512-bit keys have been insecure for some time now [Sch96a], [Odl95], [Odl99], [Len96]; a well-funded adversary could certainly break these size keys (even if it does take a month or two). In reality, an adversary wouldn't even need to be well funded - they would just need access to a large network of computers. The adversary could thus be a computer manufacturer, a large corporation (using idle time on computers) or a co-ordinated effort.
If doubt exists about the ability to factor a 512-bit key one only has to see that a 465-bit key was broken with just 2000 MIPS-years of effort [Paa99] and very recently a 512-bit key was broken with 8000 MIPS-years of effort [RSA99]. For comparison, breaking a 56-bit DES key is 50 times harder than breaking a 512-bit RSA key [Sch99c]. Even more worrying is the fact that this break was undertaken secretly (e.g. didn't involve thousands of machines working across the internet) - in reality people could be breaking e-commerce (e.g. Compaq's online store) keys NOW. Thank god for export controls on encryption, huh?A very important point from [Sch99c]: "If 512-bit keys are insecure today, they were just as insecure last month. Anyone implementing RSA should have moved to 1028-bit keys years ago, and should be thinking about 2048-bit keys today. It's tiring when people don't listen to cryptographers when they say that something is insecure, waiting instead for someone to actually demonstrate the insecurity.".
Don Johnson says it best [Joh99b]: "One should assume RSA 512 can be broken by a determined attacker."
Modern version of PGP allow only the creation of keys >= 768-bits, so the naive user is afforded some protection from creating an excessively weak key. It has been said that starting from v6.5.1, PGP will have a minimum key length of 1,024-bits [Pri99b]. The question still remains however, what is a "reasonable" size key?
[Odl99] states that "...keys have to be at least 1024 bit even for moderate security, and a least 2048 for anything that should remain secure for a decade...". ANSI X9.31 mandates a minimum RSA key size of 1024-bits [ANSI931].
Personally, I would suggest creating asymmetric encryption keys no smaller than 2048-bits. Why so large? Well, assuming "reasonable" advances in number theory and computing power, along with the growth of distributed computing via the Internet, a 1024-key will only offer guaranteed security (assuming the RSAP / DLP is indeed intractable) for a period of around 5-years [Odl95].
No demonstration of breaking a 768-bit key has been performed, but one must assume that it is now at least theoretically possible [Sil97], [Ley99]. RSA still recommend 768-bit as a minimum but it is expected that keys of this size will be vulnerable within the next couple of years [Sch99c] - it would appear that RSA's recommendations are far too optimistic.
In view of the fact that PGP uses 128-bit session keys, it would be prudent to use DH/RSA keys of around 3072-bits to ensure "equivalent security" [Cer99b].
Any major advance in factoring algorithms, computer speed, computational power available in a distributed effort etc would adversely affect the security both DH / RSA. Remember that the fastest algorithms for factoring and computing discrete logs are now sub-exponential - so any increase in computing power affects to a large degree the security afforded by public keys.
We would do well to remember one of the basic maxims of information security: "Cryptography is about making the cost of retrieving a message much higher than the value of the message itself."
For further details of recommended key sizes, refer to [Sch96a], [Sch96b]. For a great paper discussing the future prospects for integer factorisation see [Odl95].
Cyber-Knights Templar [Cyb99a] have produced an amended version of PGP v5.5 that supports the following enhancements:
There are some arguments about the use of these gigantic keys. Some argue (including Phil Zimmermann [Zim98b] & Will Price [Pri98] of NAI), that keys of this size are of absolutely no use.
I don't particularly agree with the day to day use of keys which are this large as they are tediously slow, but I do partially disagree with Phil's argument...If an adversary breaks a PGP symmetric key, then the adversary can read a single message. If the adversary breaks an asymmetric key, then they can read all messages, past, present and future (and forge signatures if an RSA key is being used).
There is therefore a reasonable argument for using asymmetric keys that offer security in excess of that provided by the underlying symmetric cipher [Sch98e], [Sim98].
A further fact which is often overlooked: there are now sub-exponential algorithms for factoring and computing discrete logs (i.e. increase in computing power greatly affects the feasibility of attacking the keys). Such a growth in computing power affects to a lesser degree the feasibility of breaking symmetric keys (i.e. the problem of brute forcing symmetric keys are exponential). Doesn't it therefore make sense to be more conservative (in terms of security) with respect to asymmetric keys, especially since key bits are "cheap"?
Still, if an adversary manages to break a > 3000-bit PGP key (either RSA or DH) today, then it is likely to have occurred either due to a mathematical breakthrough or through a broken implementation which would thus render any size asymmetric keys ineffectual.
Future versions of PGP will support the AES block cipher standard with a keysize of 256-bits - then users will be able to use larger keys. For now, I would suggest only using very large keys under extreme circumstances.
Users should also be reminded that DSS keys greater than 1024 are no longer compliant with the official DSS standard [FIPS186-1] and thus should be called something else.
The following table highlights versions of PGP that are compatible or incompatible with certain key-sizes [Cyb99b]:
|1|| Blue text indicates the maximum key size (in bits) that can be generated by this version. |
Red text indicates the maximum key size (in bits) that can be understood by this version.
|2||The international commercial version available from www.pgpinternational.com|
|3||DSA keys greater than 1024-bits should not technically be called "DSA" as they are no longer compliant with the standard. It is thought that these longer keys do not significantly increase the security offered by the signature scheme.|
|4||This is a "double-width" version of SHA-1, as per Hash Algorithm ID 4 in [RFC2440]. This hash function has not been shown to offer significantly more security than SHA-1.|
One worry when PGP v5 was first released was the use of precomputed or "canned" values of the finite field and the generator of this field (p & g respectively) in DH, and p & q for DSS.
It is quite acceptable to use public, precomputed values for these values [Sch96a], [FIPS186-1], [MOV96], [Kob94], [Sti95]. I would also recommend that the concerned user reads [Sch97a].
The concerned user can choose to switch off the "Faster key generation" if desired (and be prepared to wait far longer for the production of keys).
The problem with using canned primes is that a table of p values needs only to be computed once for the field. Breaking individual keys in this field is then a relatively fast operation. Of course, computing a database of factor base logarithms for reasonable parameters is still impossible, but it would seem prudent from a security perspective to not use canned primes for long term keys [MOV96].
Or in plain English (from [BB99]): "With ElGamal, for example, the expensive key component to generate is the public prime modulus. A group of keys can share a common public modulus with no negative security implications other than that the key presents a fatter target for pre-computation attacks."
My recommendation: turn off the Fast Key Gen option when creating long term keys...It may take longer, but potentially offers more long term security against a concerted attack.
Historically, it was desirable to select strong primes (p & q) for use in the RSA system. These strong primes helped defend against Pollard's p-1 factoring algorithm. More recently however faster factoring algorithms have been discovered that work just as well against strong primes, so there isn't any real advantage to using these strong primes.
PGP doesn't use "strong" primes, which is a decision I agree with for the following reasons:
It may be necessary in the future to once again rely on "strong" parameters when using the RSA system - but at the moment prime length is far more important than structure [Sil97], [Sch96a], [MOV96], [RSA98].
No. There are two general types of Galois Fields with cryptographic significance, GF(p) with p prime, and GF(2n).
When first introduced, GF(2n) was the preferred implementation, basically because it is easier to implement in hardware [Sch96a], [Odl83]. However, this was shown to be relatively insecure. The field GF(p) where p is around 2750 and is prime is thought to offer roughly the same security as GF(2n) where n is around 2000. Clearly, the Galois Field GF(p) offers better security for the same parameter size.
It is unfortunate that these two systems, though related, are both often discussed in the same breath - theory in one field isn't necessarily applicable in the other field.
Anyway, PGP implements Diffie-Hellman over GF(p) which, as we'll see later, is still secure.
If you are still interested in the relation between GF(p) and GF(2n) then I most highly recommend [Odl83].
Clearly, if DH keys can be up to 4,096 bits while DSS keys can only be 1,024-bits then there is a serious disparity between the strength offered by these two types of keys.
An initial thought was that DSS keys may offer more security by combining both ElGamal and Schnorr signature schemes but this is untrue however as breaking ElGamal clearly breaks DSS.
A 1,024-bit DSS key appears far easier to break than a DH key of greater length. This is indeed so; DH and DSS are based on the same underlying mathematical theory - a key of 1,024-bits is inherently easier to break than a 4,096-bit key.
So, why the contrast? Well, firstly, PGP simply implements the Digital Signature Standard as per [FIPS186-1]. DSS is the de facto standard for digital signatures, and PGP implements DSS to the maximum strength possible within the bounds of the standard (e.g. with p up to 1024-bits). An implementation of "DSS" with p greater than 1024-bits would no longer conform to the standard.
Secondly, let's look at the purpose of encryption & signature keys. Encryption is used to hide data - a compromise of the key would clearly make all messages readable and would thus make encryption entirely useless. This is a real problem; it will one day certainly be possible to read messages secured with 1,024-bit keys.
Signature keys are used to offer integrity, authentication and non-repudiation. The ability to "break" a DSS key would allow an adversary to forge digital signatures. DSS keys of 1,024-bits are secure for several years, then what? Well, time stamping and document trail mechanisms can be used to "prove" the genuineness of an old digital signature (which could have been forged using what will be current technology). This assumes that there are no totally unexpected breakthroughs in computing discrete logs.
So, one can see that breaking an encryption key is catastrophic, while breaking a signature key at some date in the future is nowhere near as disastrous.
Still, I would personally like to see the option of a "stronger" signature scheme; ElGamal signatures keys with the same length as the encryption keys would be a good choice. I note that ElGamal signatures keys are in the OpenPGP standard, just not currently implemented.
Interestingly though, making the signature keys longer doesn't necessarily make an overall stronger signature scheme. One notes that the only "generally accepted" hash functions MD5, SHA-1 & RIPEMD only offer security around 160-bits (128-bits for MD5). If one uses a 4096-bit RSA / ElGamal signature key with one of these hash functions, are we really increasing security significantly? I would suggest not; a birthday attack could be undertaken against the hash function in 280 operations (or, worse 264 for MD5) thus, against some attacks, throwing asymmetric key bits at the signature scheme may give the naive user a false impression of security in the signature scheme.
DSS is a "balanced" signature standard; the hash function falls to some attacks in around 280 operations, and the signature key length allows some attacks in around 280 operations. Compare this to deprecated v2.x keys, where the signature scheme falls to some attacks in 264 operations while the RSA key can offer security well in excess of 2128. Is this very wide discrepancy useful?
One can, quite rightly, make the point that asymmetric keys should afford more security than the strength provided by the underlying hash function; "brute-forcing" the hash function only allows an adversary to forge one message, while breaking the asymmetric key allows an adversary to fake many signatures, so this "equivalent security" argument is tenuous.
Will Price [Pri98] has pointed out that the cryptographic community need to decide upon (or create as necessary) stronger cryptographic primitives. The AES process is selecting a stronger block encryption algorithm and one assumes NIST/NSA will pull a stronger SHS out of the hat?
If you are interested in reading further on this point then I recommend A.M.Odlyzko, "The Future of Integer Factorization" [Odl95] - which also similarly covers key lengths for DH/ElGamal based systems.
The web of trust is based purely on the use of signature keys, the user signs another users public signature key with his private signature key to create a "certificate". The use of a weak signature scheme would make the web of trust in new versions of PGP untrustworthy [Bac99b]. Is this the case yet? No. Will it be the case? Yes, someday.
[Mun99] offers an interesting argument why signature keys can afford to be smaller than encryption keys: "If a highly secret agency acquired the ability to break 1,024-bit keys, long before the public crypto community believed it would be possible, then they could read lots of encrypted traffic, but only as long as the rest of the world believed 1,024-bit keys were still safe. They could use this technology to read encrypted messages routinely, and no one would ever know. But if they used it to forge signatures routinely, certainly this would be noticed. The word would soon get out that 1,024-bit keys aren't safe, and everyone would switch to 2,048-bit keys for encryption as well as for authentication. In other words, by using their ability to forge, they risk losing their ability to decrypt!"
A subliminal channel allows the leaking of information or key-data to an adversary. For example, an amended version of PGP could be distributed that allows a third party to collect details of your private key from digital signatures.
Farfetched? No! Such a system has been developed [YY96].
DSS can suffer from subliminal channels, but so can RSA, ElGamal, Diffie-Hellman and even symmetric ciphers [YY96]. If you believe that the software you are using could be exploiting subliminal channels then it is important to check the source code to ensure PK parameters are generated in an acceptable fashion. This is simple with PGP, as source code is available for inspection, but may be impossible with other programs.
One notes that Simmons commented that DSS "...provides the most hospitable setting for subliminal communications discovered to date" (also quoted in [Sch96a]). This has subsequently been disproved in [AVPN96]: "...the design of DSA does not maximise the covert utility of its signatures, but minimises them."
Elliptic Curve systems are specified in ANSI x9.62 & x9.63 & IEEE P1363.
Elliptic Curves first proposed for use in cryptographic applications in 1985 independently by V.S.Miller & N.Koblitz. Elliptic Curves themselves have been studied for many centuries and are [Kob94] "among the most richly structured and intensively studied objects in number theory".
Elliptic Curves are of interest because they have the following benefits over both DH and RSA:
A sub-exponential algorithm exists for solving one class of elliptic curves known as "supersingular curves", though these curves are easily avoided. For the great majority of curves, the only algorithms applicable run in exponential time. This means that EC keys can be significantly smaller than RSA or DH keys to provide similar expected levels of security.
Even though not all factoring methods can be used to break discrete logarithm systems, it is conceivable that a new factoring or DL algorithm could be produced that breaks both RSA & DH. This creates a need for diversification in the design of PK algorithms [Len96]. Or more simply, if RSA & DH are broken by a mathematical breakthrough, it is highly unlikely that elliptic curve based systems will also break.
OpenPGP [RFC2440] supports the implementation of EC technology. PK algorithm ID 18 is reserved for "Elliptic Curve", whilst ID 19 is reserved for "ECDSA" - which is an implementation of the DSA over elliptic curves.[Kob94] states that "It is likely that the DLP on elliptic curves will prove to be more intractable than DLP in finite fields". Of course, only time will tell.
It has been shown that ECDHP is fully exponential if ECDLP is. This is a major achievement as such an equivalence has never been shown for RSA / IFP or DHP / DLP [Joh99a].
Recently, NIST have specified a collection of elliptic curves for USG use - which is indicative that they trust elliptic curve crypto to some degree [NIST99a].
B.Schneier had the following to say about elliptic curves [Sch99c]: "Certicom used the event to tout the benefits of elliptic curve public-key cryptography. Elliptic-curve algorithms, unlike algorithms like RSA, ElGamal, and DSA, are not vulnerable to the mathematical techniques that can factor these large numbers. Hence, they reason, elliptic curve algorithms are more secure than RSA and etc. There is some truth here, but only if you accept the premise that elliptic curve algorithms have fundamentally different mathematics. I wrote about this earlier; the short summary is that you should use elliptic curve cryptography if memory considerations demand it, but RSA with long keys is probably safer."
S.Schnell of RSA says [Sch97d] "the current limited expertise and understanding of elliptic curve cryptosystems (ECC) represents a significant risk for today's data security applications" - but lets not forget that he works as VP of Marketing for a direct competitor to EC crypto.
T.Elgamal says [Elg97]: "Elliptic curve technology is interesting, perhaps a little newer than factoring or discrete logs, but needs more study and analysis before it is mature." and L.M.Adleman says [Adl97]: "It is correct that I am suspicious of elliptic curve cryptosystems. I have never heard an argument which persuaded me that there were reasons in principle for believing that the discrete logarithm problem on elliptic curves is strictly exponential. I suspect that the lack of a sub-exponential algorithm is merely a matter of neglect and that intense scrutiny - which a commercial implementation of an elliptic curve cryptosystem might engender - could readily change the situation."
At the moment I see no compelling reason to move towards an elliptic curve asymmetric cipher, though one day there may be such a reason. From a security perspective I feel it is prudent to remain cautious in the use of elliptic curves. When used, it is recommended that keys sizes of at least 300-bits are used even for moderate security [Odl99].
Certicom has issued recommendations on EC vs RSA key sizes (of course, one is reminded that Certicom have a commercial interest in EC technology...) [Cer99b]:
|Block Cipher Keylength||RSA Key Length||EC Key Length|
I suggest those interested in elliptic curves consult the excellent text by Koblitz [Kob94] or the brief summary in [Odl99].
Both of these algorithms are based on supposedly intractable problems that have been subjected to scrutiny by the world's finest mathematicians and cryptographers. The asymptotics of RSA and DH based systems are similar but in practice RSA keys appear more vulnerable [Sch98f].
Bob Silverman, who is now Senior Research Scientist with RSA Labs, has said [Sil96a]: "I am smug enough to say that NSA can't break RSA or discrete logs."
It is, in fact, slightly harder to compute discrete logs modulo an appropriate prime than to factor a "hard" integer of the same size - so RSA would appear slightly weaker than DHP [Odl95], [Sch97c]. From [Sch99a]: "RSA users have to choose a larger key size those using than DH over GF(p), for equivalent security."
It is thought possible, though unlikely, that there is a polynomial way to generally factor large numbers or compute discrete logarithms. There is also the chance that the RSAP or DHP could be broken without having to solve the underlying "hard" problem (IFP / DLP respectively).
Discussing the DLP & IFP, [RSA96a] states: "This suggests that the degrees of difficulty of both problems are closely linked, and that any breakthrough, either positive or negative, will affect both problems equally."
And to quote [Sch96a]: "Computing discrete logarithms is closely related to factoring. If you can solve the DLP then you can factor."
Another relevant quote [Wie98]: "The most important factor in choosing a public-key technology is security. Based on the best attacks known, RSA at 1024 bits, DSA and Diffie-Hellman at 1024 bits, and elliptic curves at about 170 bits give comparable levels of security."
One notes the interesting statement in [Odl83]: "In general, while there are algorithms for factorization that do not generalize to give discrete logarithm algorithms (the Schnorr-Lenstra algorithm, for example), the converse is not the case. Therefore it seems fairly safe to say that discrete logarithms are at least as hard as factoring and likely to remain so." More recently [Len96] it has been found that ECM cannot be used against discrete logarithms [Len96]. In English, all currently known algorithms for solving the DLP can be applied to the IFP, whereas the reverse is not always the case.
IF DH is broken by solving the DLP then RSA will also fall, since, if you know how to take discrete logs, then you can factor (that is the basis of Shor's quantum factoring algorithm [Sho94]). Thus, DLP would seem stronger than the IFP, since factoring does not allow you to solve discrete logs [MOV96]. More rigorously; "the Diffie-Hellman problem in Z *n is at least as difficult as the problem of factoring n."
I am unaware of any literature that states or predicts that either DH or RSA are, in operation, significantly less secure than the other, given correct implementation and parameter selection.
From a security perspective, one good argument for using DH/DSS keys is the fact that the encryption and signature keys are now autonomous. Thus if someone manages to obtain your DH private key (for example by brute forcing the key or by court order) they will be able to read all mail sent to you. They will not be able forge signatures however. Compare this to RSA, where divulging a key allows someone to both read all mail and forge signatures. See the further discussion in the section "Compelled Production of encryption keys", which covers the compelled production of keys.
Comparing the "largest breaks" of each key type, we see that an 512-bit RSA key has been broken [RSA99] but only a 283-bit DH key has been broken [Web99]. The 512-bit RSA break used around 8,000-MIPS years and it has been predicted that the task is equivalent to a DL in a prime field with a characteristic of 365-bits [Web99].
PGP Version 6.0 provides the ability to create and revoke new encryption keys without sacrificing your master signing key and the signatures collected on it. This feature would not have been possible with the old style RSA keys.
In the words of Cylink [CYL98b]: "Diffie-Hellman based systems can be used in place of RSA in any application requiring public-key cryptography."
Number theory is a fast changing topic. Recently several developments and proofs have affected the problems that both DH & RSA keys are based upon.
Obviously, proof that either DH or RSA is computationally equivalent to the underlying problem (DLP and IFP respectively) drastically enhances the confidence that should be placed in the public key system.
I briefly highlight relevant papers and try to identify how they affect the security of the asymmetric algorithms utilised in PGP.
"Generalized Diffie-Hellman modulo a composite is not weaker than factoring" [BBR98]
Implications: Some classes of the DHP are not computationally weaker than the IFP.
"Breaking RSA may not be equivalent to factoring" [BV98]
Implications: Provides evidence that certain instances of RSA cannot be equivalent to the IFP. This is contrary to the belief by some that RSA and IFP are equivalent.
"On the Complexity of Breaking the Diffie-Hellman Protocol" [MW96]
Implications: Provides a proof that some classes of the DHP are equivalent to the underlying DLP.
Basically, there have been some significant steps to prove the security of DHP is equivalent to DLP, while no such proof may be available for showing the equivalence between RSA and IFP. This may have long term security ramifications for RSA based keys.
Encryption timings (in milliseconds on a Sparc II) from [Sch96a]:
(1024-bits, 160-bit exp)
Messages are enciphered once, but may be decrypted many more times - thus ElGamal is considered better in terms of performance.
Note that the asymmetric cipher is only used to encrypt the random session key - so performance hit is negligible. These figures may impact low cost (e.g. smart cards) or high throughput environments.
Digital signature timings (in milliseconds on a 200 MHz Pentium Pro) from [Wie98]:
One can see that producing digital signatures using the DSA scheme is much faster than under RSA, assuming identical key lengths. Signature verification is far faster under RSA.
Signatures are created once but may be verified many more times - thus RSA is considered better in terms of performance.
In real world use, the performance difference between these two systems will go unnoticed. It is more likely to be an issue in low cost (e.g. smart cards) or high throughput environments.
A cryptographic hash function takes an arbitrary length message as input and produces a fixed length output. The input is typically a file or a message. The output of the hash function is called a Message Digest, hash value or message fingerprint.
By their very nature, hash functions are many to one - that is to say there will certainly be more than one input message that produces a given hash value. The purpose of the hash function is to make the job of finding such "collisions" computationally infeasible.
Being able to produce collisions for a hash function in reasonable time renders a hash function ineffectual.
Most modern hash functions are built from a compression function, which is iterated on the input stream. Like a complete hash function, a compression function can suffer from collisions. The precise design of the hash function dictates how detrimental compression function collisions are to the security of the overall hash function - but a collision free compression function is necessary for an overall secure iterated hash function [MOV96].
A checksum or CRC function is a non-cryptographic mechanism for detecting transmission errors [MOV96], [RSA98].
PGP uses a checksum value to:
Note that the checksum function is only used in areas that don't require cryptographic strength. When cryptographic strength is required, PGP uses a hash function.
The hash function is responsible for two primary tasks in PGP:
It is therefore important that the hash function has the following two characteristics:
MD5 is a hash function by Ron Rivest [RFC1321]. It is an enhanced version of the MD4 hash function (MD4 has been totally broken [RSA98]).
MD5 has been included in PGP since inception, but has recently been deprecated due to several security concerns (see section "Has MD5 been broken?"). MD5 seems to have been a catalyst for the changes that we have seen post PGP v2.x.
MD5 is supported in S/MIME (v3) only for [IETF98]: "providing backward compatibility".
SHA-1 is the current hash function of choice as implemented in both the PGP and S/MIME standards. SHA-1 is formally defined in [FIPS180-1], [ANSI930-2] & [ISOIEC10118-3].
SHA-1 is based upon the MD4 algorithm, but was enhanced by NIST / NSA. The output of SHA-1 is 160-bits.
SHA-1 has been extensively analysed but to date there have been no successful attacks.
RIPEMD-160 is another hash function derived from MD4. It is formally defined in [DBP96].
To date there have been no successful cryptanalysis of RIPEMD-160 which produces an output of 160-bits.
There are versions of RIPEMD hash function offering hash lengths of greater than 160-bit security (256-bit and 320-bit to be precise) but these hash functions are, according to the authors [DBP99]: "RIPEMD-256 and RIPEMD-320 are optional extensions of, respectively, RIPEMD-128 and RIPEMD-160, and are intended for applications of hash functions that require a longer hash result without needing a larger security level."
Not totally. First, pseudo-collisions in the compression function were found by den Boer and Bosselaers [BB94], then collisions in the compression function were found by Dobbertin [Dob96a].
This doesn't mean that collisions have been found in the full hash function, but it does mean that MD5 shouldn't be used in situations where collision resistance is required [Dob96a], for example in the creation of digital signatures (the main application of a hash function in PGP). To quote Hans Dobbertin directly [Dob96a]: "Therefore we suggest that in the future MD5 should no longer be implemented in applications like signature schemes, where a collision-resistant hash function is required. According to our present knowledge, the best recommendations for alternatives to MD5 are SHA-1 and RIPEMD-160."
One should be reminded that the design goal of MD5 was to build a CRHF from a collision resistant compression function [Sch96a], [MOV96], this design goal has now been violated.
In the words of RSA Labs [RSA96b]: "With regards to existing applications which use MD2 and MD5, collisions for these hash functions have not yet been discovered but this advance should be expected...RSA Laboratories currently recommends that in general, the hash function SHA-1 be used instead but RIPEMD-160 would also be a good alternative."
[MOV96] says: "An iterated hash function is thus in this regard at most as strong as its compression function".
A further complaint about MD5 is the 128-bit output. This is insufficient for long term security [PBD97] & [Sch96a].
Overall, Schneier says [Sch96a] "I am wary of using MD5".
One also notes that MD5 is supported in S/MIME (v3) only for [IETF98]: "providing backward compatibility". It would seem foolish to continue to use technology that is so dubious when far superior unencumbered algorithms are available.
To summarise, there are three main concerns about the use of MD5:
An ignorant view is that "MD5 is secure until someone demonstrates a break" - this is just not true. For example, we knew that DES was ineffectual against a determined adversary even before the Internet, and later the EFF, broke the cipher. I think Schneier has the right idea on this subject [Sch96b]: "History has taught us: never underestimate the amount of money, time, and effort someone will expend to thwart a security system. It's always better to assume the worst. Assume your adversaries are better than they are. Assume science and technology will soon be able to do things they cannot yet. Give yourself a margin for error. Give yourself more security than you need today. When the unexpected happens, you'll be glad you did."
The MD5 attack was first detailed in [Dob96a].
The attack does not currently allow an adversary to create a slightly altered message given an arbitrary message that will match hash values. That is to say given a message A, an adversary cannot feasibly find a message B that hashes to the same hash M.
What it does potentially allow an adversary to do is create a message A with a related, but different message B, that hashes to the same M. Practically, you should sign message only when a third party does not have influence over the message being signed.
The long and the short of it: MD5 should no longer be used universally without forethought. Users have to be careful when considering which documents to notarise or sign with MD5. Compare this with SHA-1 & RIPEMD with which no such forethought is necessary (because no B can be found that hashes to the same M with these two alternative algorithms).
If you are interested then I recommend [Dob96b] for a (slightly outdated) description of the status of MD5. The paper says in conclusion "...might indicate that there is a serious security problem with MD5...but in future implementations better move away from MD5".
The only difference between SHA-1 and SHA is the inclusion of a one-bit rotation in the block expansion from 16 to 80 words - something to do with "linear error correction codes", apparently [MOV96]. The interested reader is referred to [CJ98] for an in-depth analysis of attacks against the original SHA.
To answer the question, it would appear that SHA was not broken, but may have been susceptible to an advanced, and as yet unknown, attack.
Anyway, it's nice to see that the faceless NSA cryptographers are only human too!
The original RIPEMD was released in 1992 but was subsequently found to have some significant weaknesses [Dob95], [MOV96]. Subsequently a modified version was created, called RIPEMD-160 which does not suffer from the deficiencies of the previous version. The new version of RIPEMD is approximately half as fast as the previous version - which gives some idea of the significant security improvements made between versions.
RIPEMD-160 as implemented in PGP has no known weaknesses.
Certainly not MD5 (see section "Has MD5 been broken?").
The choice would therefore appear to be between SHA-1 and RIPEMD-160. Neither of these has succumbed to any known attacks and the finest cryptographers in the field produced both.
SHA-1 is the standard used in PGP v5+ and there is absolutely no reason to doubt this choice [Sch96a], [PGP98], [RSA96b], [MOV96], [Dob96a]. The PGP manual [PGP98] summarises the cryptographic communities feelings on SHA-1: "SHA has been published in the open literature and has been extensively peer-reviewed by most of the best cryptographers in the world who specialise in hash functions, and the unanimous opinion is that SHA is extremely well designed."
I am yet to find a single quote that casts doubt on the cryptographic security of either SHA-1 or RIPEMD.
Hash function timings (in Mbit/s on an unspecified platform) from [PBD97]:
MD5 may be fast, but remember that it is relatively insecure against certain attacks.
A conventional encryption algorithm is a function that maps an n-bit plaintext block to an n-bit ciphertext block where n is the blocksize. Typically, n is equal to 64 or 128-bits.
The function takes a parameter, the "key" which specifies which mapping between the plaintext and ciphertext is used.
Block ciphers are, given the same key, invertable.
An excellent (and free!) introduction to block ciphers is the paper [Mir98].
Block ciphers are used in numerous areas of PGP:
IDEA is a strong block cipher by Xuejia Lai and is formally defined in [Lai91].
IDEA is an 8-round cipher with a 64-bit block size and uses keys of 128-bits. The strength of the cipher is provided by "mixing operations from different algebraic groups". IDEA is resistant to both differential [LMM91] and linear cryptanalysis [Sch96a]. Currently, there is no known way of breaking IDEA short of brute force [Sch96a].
From [RSA96a]: "IDEA is generally considered secure and both the cipher development and its theoretical basis have been openly and widely discussed."
The best known attacks on IDEA are:
IDEA also has a large (251) class of weak keys, as outlined in [DGV93]. This is not a major concern as the chances of picking a weak key is 2-77 [MOV96].
None of these attacks are useful in breaking practical implementations of IDEA. Full IDEA is strong against differential, linear, and related- and chosen-key attacks, though there is an interesting side channel attack that can be undertaken on an implementation of IDEA that allows high-resolution timing [KSWH98b].
IDEA is no longer the default cipher of choice in PGP due to patents issues (IDEA requires a license for commercial use).
CAST is a family of block ciphers by Carlisle Adams and Stafford Tavares. PGP v5 implements a version of CAST known as CAST5, or CAST-128. This version of CAST is the most standard cipher that people usually mean when they say "CAST". CAST is formally specified in [RFC2144].
This version of CAST has a blocksize of 64-bits and a key length of 128-bits. It is resistant to both linear and differential cryptanalysis [Sch96a]. Currently, there is no known way of breaking CAST short of brute force [Sch96a].
There are no known attacks on CAST with reduced rounds- it looks incredibly secure.
CAST is now the default cipher in PGP.
DES is formally defined in [FIPS46-2] and Triple-DES (3DES) in [ANSI952].
Recently, NIST has suggested that FIPS46-2 should be superseded by the Triple-DES algorithm, which we expect to be formally approved as [FIPS46-3]. This is primarily due to the cracking of DES keys in under a day by the EFF.
3DES consists of three applications of the DES cipher in Encrypt-Decrypt-Encrypt (EDE) configuration with independent keys. Encryption is executed as follows:
CipherText = DESk1(DES-1k2(DESk3(PlainText)))And decipherment is:
PlainText = DES-1k3(DESk2(DES-1k1(CipherText)))
3DES has a block size of 64-bits and a key length of 168 (3*56). Because of the construction of 3DES, it is thought to offer strength equivalent to a 112-bit block cipher [BK98].
The best known attacks on 3DES are:
These attacks are not useful in breaking practical implementations of 3DES.
Advanced Encryption Standard (AES) is the US Governments search for a replacement to the ageing DES standard(or Data Encryption Standard).
The National Institute of Standards and Technology (NIST), first called for algorithms 12th Sept 1997 [NIST97]. Several very well known cryptographers (Rivest, Schneier, Knudsen, Biham, Rijmen, Coppersmith etc) developed AES candidate algorithms that met the criteria. Of the 15 initial algorithms meeting the criteria, 5 (Mars, RC6, Rijndael, Serpent & Twofish) have been selected to enter round 2. An eventual candidate is expected to be named as 'AES' sometime in year 2000.
Although this selection process only selects a cipher for USG use, it is believed that AES, when selected, will become the international standard for encryption into the next millennium. All AES candidates accept keys of 128, 196 or 256-bits and have a block size of 128-bits.
The OpenPGP standard [RFC2440] has allocated ID 7,8 & 9 for the 128, 192 & 256-bit keys respectively.
Anyone interested in reading more about AES is advised to see the NIST Round 1 report [NIST99b].
Twofish is an AES candidate produced by Schneier, Kelsey, Whiting, Wagner, Hall, Ferguson first presented in [SKWWHF98]. It is included as a separate section in this document because OpenPGP & NAI have decided that Twofish is such a promising cipher that it should be added to PGP prior to the outcome of the AES process [Koc98], [Pri99a].
Twofish is an evolution of the cipher Blowfish. Most changes seem to have been made to meet the AES criteria. Currently, there is no known way of breaking Twofish short of brute force, though this is to be expected considering the short time since publication.
Personally, I believe that it is a BAD idea to include Twofish as a cipher at this stage. My original comments on the choice [Sim99] were:
"It has previously been mentioned that Twofish would be added to PGP before completion of the AES process. I think everyone would agree that the move to a 128-bit block, 128/192/256-bit key cipher is a Good Thing, but the point of this message is to ask "why the choice of Twofish?"
Is strength (or "expected strength" at this stage) the criteria? If you're going to select a cipher from the x candidates (many of which, including Twofish, have only had 8 months peer-review) then surely you would go for the most ultra-conservative design? Even better, wouldn't you select a cipher which has been analysed for a longer period?
Is speed a selection criteria for block ciphers in PGP (if so why 3DES)?
Is heritage the criteria? If so, there are other candidates with a similar heritage: MARS (Coppersmith) , RC6 (Rivest), Serpent (Anderson, Biham & Knudsen), Rijndael (Daemen & Rijmen), DFC (Vaudenay), CAST-256 (Adams) etc.
Are patent issues part of the criteria? We would expect so (in view of IETF views on this matter and the OpenPGP standard). This potentially discounts RC6 & MARS and one or two others.
Are some of the more obscure (and of little practical importance in PGP?) criteria being considered? For example: key agility, resistance to timing / power analysis, smart card / hardware implementation, scalability to 64-bit architectures.
I am a great fan of Bruce Schneier (and Twofish actually) - but isn't it too soon jump on the bandwagon? One notes that Twofish has received some criticism in AES Conference II papers:
1) "Report on the AES Candidates" by Vaudenay et al. Points out that S-Boxes should "no longer be called key-dependant". Also says "consists of a collection of patches" & "we do not think this design comes from deep investigation". Of course, this paper is written by the authors of another AES candidate.
2) "An observation on the Key Schedule of Twofish" by Mirza & Murphy (RHBNC), points out several deficiencies with the key schedule. Implications unknown, but it does directly contradict implicit claims in the original Twofish paper.
None of the candidates escaped criticism from a "cryptographic security" point of view, but if one wanted to objectively select an AES candidate by any combination of the above criteria, would we logically select Twofish at the moment?
My main concern is that naive users will see "Twofish" & "Schneier" and create keys specifying this algorithm. Sure, the User Manual will no doubt state "Twofish is new blah blah blah" but, in my experience, users never RTFM anyway."
Basically I believe that Twofish is far to new to field in live cryptographic systems. Similar thoughts are expressed in [Pre99], [Knu99].
In my opinion, the saving grace is that recipients of messages in PGP have the choice of which symmetric ciphers they are willing to accept. For example, if the OpenPGP standard offered a simple XOR algorithm (totally insecure!), then recipients can simply disable this algorithm and they will never receive messages encrypted with this algorithm.
No. "Single" DES has been successfully brute forced by the EFF using a custom built cracking machine in just 56 hours [EFF98]. Brute force basically involves trying all possible 256 keys until the correct one is found. Brute force takes, on average, 2KeyLen-1 operations - so in the case of DES the machine has to try approximately 255 trial decryptions before being rewarded with the correct plaintext. One notes that brute force attacks are usually thought of as known-plaintext, but [WB94] outlines a chip capable of recognising correct decryption - this makes a brute force feasible even if a known-plaintext block is not known.
More recently, the third DES challenge posed by RSA was cracked in under a day by the EFF machine [EFF99]!
Triple-DES is at least 256 (approx. 1017) times harder to break than single DES [CYL98a] and as such can be considered very secure. An adversary would have to try on average 2111 keys in order to break a single 3DES message (and would need a large amount of storage space to keep intermediate values). It is important in multiple encryption schemes that the underlying cipher is not a group; fortunately Campbell & Wiener have shown that DES is not a group [CW93].
Properly implemented 3DES is still thought to be very strong [Wag94], [WB94], [Sch96a], [MOV96], [RSA96a], [RSA98], [Sch98g]. In fact, I have not been able to find a single cryptographer who has cast doubt upon the strength of 3DES.
In the words of [CYL98a]: "Since triple-DES is 1017 stronger than single-DES, we do a little arithmetic and find that this $300M computer can break a triple-DES key in about 4x1010 years, or about the age of the universe."
In the words of Bruce Schneier [Sch98g]: "Certainly triple-DES is a better choice than AES, in some applications. Triple-DES will probably be the algorithm of choice for many banking applications even after AES is approved as a standard."
Finally, in the words of Dorothy Denning [Den98]: "Triple DES has not been broken and its security has not been compromised."
As previously mentioned, CAST is a family of ciphers. Some of the other "CAST" ciphers have succumbed to advanced attack (Rijmen and Preneel have attacked some CAST designs and so have Kelsey, Schneier & Wagner [KSW97]). The same attacks have been tried against the implementation of CAST used in PGP and have, thus far, failed.
This is a contentious and subjective issue. None of these algorithms has either been broken or had any serious doubts cast upon their strength.
Some people don't trust 3DES because it is based upon DES, which is produced by the NSA. However, respected cryptographers hold 3DES in very high regard.
CAST5, as implemented in PGP, has not been subjected to any successful analysis, but Bruce Schneier has said the following [Sch98a] "I give a big yuk to CAST-128" and [Sch98c] "I don't buy the design process.".
Twofish is far to new to be considered stronger than either of the other 3 ciphers. In time it may be seen as secure, but for now it just hasn't been subjected to enough analysis. Some have claimed that because Blowfish is trusted, we should trust Twofish [Blu98] though this is simply not true. They are both BFN etc and the key schedule consists of iterations of a function used in encryption, but there are also many differences between the two ciphers. Small changes in block ciphers can move a cipher from "thought to be secure" to "trivially" broken.
IDEA is possibly the second most widely known cipher (after DES). This is mainly due two to influences:
IDEA appears to have been held back due to patent issues. Without these, I don't doubt that IDEA would now be the de facto standard. [RSA96a] says the following about IDEA: "IDEA is generally considered secure and both the cipher development and its theoretical basis have been openly and widely discussed."
More recently IDEA seems to have fallen out of favour (possibly due to the small block size compared to the AES candidates). Recently Bruce Schneier has said the following about IDEA [Sch98d]: "For the record, I am less enamoured of IDEA these days. It is still secure, in that there are no published attacks, but I like other algorithms a lot better."
When recently posed with the question "Which among the following is considered to be the most difficult algorithm to crack; RC6, IDEA, CAST, 3DES, Blowfish" Bruce replied [Sch98b] "Triple-DES. Without any doubt. Nothing else has had anywhere near the analysis." He has also said [Sch98a] "If you want to be conservative, use Triple-DES."
David Wagner has said [Wag99a]: "3DES is still my cipher of choice for security-critical applications."
The NSA objected to the 3DES algorithm becoming an ANSI X9 standard with the comment [Fro95]:
So, it can't be all bad!
I am personally reassured by the presence of three strong algorithms in PGP. If any of these algorithms were broken, even though this development appears unlikely at the moment, then PGP users could simply disable the broken algorithm and use one of the still secure algorithms. Compare this to users of standards (e.g. S/MIME) that are forsaken with only one secure cipher.
In summary, none of the algorithms implemented in PGP v5+ are broken, or anywhere near broken. PGP will certainly add the AES cipher once selected and this should then be adopted as the symmetric algorithm of choice. For now, it would appear that 3DES is the symmetric algorithm of choice for pessimists [Wag94], [Sch96a], [Sch98g].
An NAI employee has expressed the opinion that [Pri99a]: "I'll choose CAST5 over either of those algorithms [Blowfish & 3DES] any day -- although I do believe all three are 'secure' in a general sense." A view I (and many other cryptographers) disagree with....
To summarise, CAST is the default algorithm for new keys produced with PGP v5+, but 3DES is the MUST algorithm in the OpenPGP standard (e.g. all implementations of the standard must provide the 3DES cipher). Personally, I'd use 3DES as my default cipher if I were creating a new key.
To be honest, I wasn't looking forward to writing this section. Whatever I write is bound to be wrong one way or the other, and may be subject to serious misinterpretation.
The first thing to say is that all these figures assume that brute force is the best attack (or in the case of 3DES MITM). If an adversary manages to find a practical cryptanalytic method of bettering the workload required, then these figures can be reduced accordingly.
Secondly, if QCs or DNA computing become a reality, or if Moore's law is a gross underestimate of the growth in computer power, then again this section is useless.
So, back to the question... To brute force the symmetric ciphers now under a number of assumptions:
Then a single (3DES) key can be brute forced in an average of 1011 (more precisely 457,351,814,728) years. This assumption is from some perspectives very optimistic (from an adversaries perspective); can all computers be harnessed in one go and are they all running at the speed of a PII 450? No.
This figure is worked out by: 2(KeyLen-1) / (NumComputers) / NumOpsPerYear or: 2111 / 3*108 / 18,921,600,000,000
These figures are "a little" pessimistic from another angle:
To try and take all of the above factors into account is difficult. Table 7 tries to show how long the various types of keys remain secure for. A number of assumptions are made:
|Cipher||Effective Key Size||Years until break feasible with different HW1|
|500 Supercomputers2||10 Billion computers (PII 450)3||100,000 Deep Crack machines4|
|1||Calculated using 'LOG((2KeySize-1)/((KeysPerSecond*60*60*24*365)*NumberOfMachines*YearsSafetyMargin))*YearsToDoubleComputerSpeed)/LOG(2)' as published [Pet97]. Basically this column indicates the number of years until we will need to be concerned about a brute force attack using the assumptions mentioned above. More precisely, this figure shows how many years until a brute force attack is feasible with 10 years effort on the stated hardware.|
|2||It is assumed that each supercomputer can manage 10 Billion keys per second, regardless of the cipher employed.|
|3||Figures based on 10 processors per person [Odl95], CAST & 3DES implementation by A. Bosselaers ASM implementations, IDEA implementation by H. Lipmaa, timings from [Lip99].|
|4||Figures from [EFF98]. BIG assumption - each key test takes the same time as a DES test.|
I'll take a figure and explain exactly what it means - it isn't immediately obvious. If an adversary has 10 Billion "state of the art" consumer level processor (analogous to a PII 450) to use solely for cracking a 3DES key, then we would need to be looking to changing cipher system in 44 years - the adversary could break the cipher in 10 years from this point. Remember that this only breaks a single key!
Another example; if an adversary takes 500 state of the art supercomputers, each capable of cracking 1 billion keys a second (these are serious computers!) and to use solely for cracking a 3DES key, then we would need to be looking to changing cipher system in 61 years - the adversary could break the cipher in 10 years from this point. Again, this only breaks a single key.
Another example; if an adversary takes 100,000 "Deep Crack" machines, each capable of cracking 92,625,000,000 keys per second to use solely for cracking a 3DES key, then we would need to be looking to changing cipher system in 74 years - the adversary could break the cipher in 10 years from this point. Again, this only breaks a single key.
In all honestly, this section was "produced under duress" - lots of people requested it. I'm not sure it helps prove a single thing other than that 3DES, IDEA & CAST keys are safe for 50 years as long as the best way to break the ciphers is by brute force.
I personally think that the asymmetric cipher will prove to be the "weak link" in the chain - e.g. factoring / computing discrete logs will become more feasible whilst attacking symmetric keys remains less feasible. Even worse, either the DLP or IFP could turn out to be "computationally easy".
If there is a moral of this story, it's "The sooner you start computing, the longer it will take to finish." [Sil98].
One has to ask one's self whether an intelligence agency would really bother to spend billions of pounds worth of effort to read one message...If the message were that important, wouldn't they just kidnap you (and murder you as appropriate)?
Because there are no practical ciphers which offer a general proof of security!
Some ciphers offer provable security against certain specific classes of attacks (e.g. DFC [GGH+98]), and others offer a more general proof of security based upon some currently unproven underlying assumptions (e.g. Bear [AB96]). But none of these algorithms have undergone a significant amount of analysis.
The Twofish paper [SKWWHF98] states that "Any reasonable proof of general security for a block cipher would also prove P != NP". Similarly, [Bih99] states that "The search for provable security is still in progress, but it is not expected that a full proof of security of any cipher will be found in the next decades, as this proof is closely related to proving lower bounds of computational complexity, and to the problem of the relation of the classes P and NP."
[MOV96] says about unconditional security: "...it requires as many bits of secret key as plaintext and cannot be provided by a block cipher used to encrypt more than one block". Not very useful!
So, without using a One-time pad (also known as a Vernam cipher), which is impractical, PGP has to do the "next best thing" - using a well trusted block cipher offering heuristic security. Of all the block ciphers in the public literature, 3DES appears to be the closest we have to an "ideal cipher" in terms of strength.
The best we can hope to do is use our complete arsenal of analysis tools to try and prove that a cipher is insecure. If it fails to succumb to these tools then it is not proven to be secure, but it indicates that a degree of faith can be placed in the cipher.
Symmetric cipher timings (in Mbit/s on a 200Mhz Pentium) from [Lip99]:
NOTE: These figures are not from the PGP implementation of the ciphers, but are indicative of the underlying cipher performance.
All too often, someone posts to a *.pgp news groups with a question resembling "I have lost my private key, how can I revoke my key?". All too often, the members of those news groups have to let those people know that they cannot revoke their key.
You need two things to revoke your key: your private key and your passphrase. What if the passphrase you so easily remember now you forget some day? Or what if you accidentally delete your private key ring? Both happen, and either way, except for disabling your key or hacking the hex of your key, there is no way to signify that your key cannot be used.
Also, both of the ways just mentioned do not prevent people from using your key, while a KRC does. Bottom line, it is just plain wise to generate a KRC immediately after your make your key - it may be impossible to create the KRC when you actually need it.
If someone gets a hold of your KRC, they can revoke your key. Take care to store the KRC securely enough that no one who is not supposed to can get to it. Also make sure that your KRC isn't so securely stored that you can't get to it!
In PGP v6.0+ and Business versions of v5.x, in the preferences, you can set PGP to automatically synchronise with the Key Servers upon certain operations. Make sure that the preference to automatically synchronise upon revoking a key is disabled or your KRC will be sent to the servers!!!
Also, by creating your KRC, you have made a backup copy of your private key. You may want to wipe that file. Also note that since Windows uses a swap file, and your private key information may have wound up in there. Heck, your private key may have even been burned on your RAM or something. Take whatever precautions you believe to be prudent to protect your private key info.
The following procedure covers the creation of a KRC under recent (e.g. Business version of v5.x and all versions of v6.x) version of PGP:
This is really a quick answer to the above question. For a more detailed explanation (relating to RSA versions only though :-() see the excellent "The PGP attack FAQ" [Inf96].
Any of the following would affect the security afforded by PGP:
The NSA may have done any of the above. It's a funny world.
The Private keyring (called "Secring.skr" by default) contains your private key(s). If someone obtains access to your private keyring then they can:
Basically, if someone obtains your private keys then the entire security of PGP is lost. It is therefore important to protect PGP private key rings stringently. PGP provides some protection; it encrypts the keyring with a passphrase - without the passphrase the private key(s) are inaccessible.
If you provide a good passphrase (see [Sch96a]), then your private keyring should be safe from any adversary. Still, you would be well advised to take some simple steps to protect the secret keyring file:
Combining the above two procedures (e.g. storing the files on a removable disk, which is also encrypted) will certainly improve security.
PGP has recently been submitted as an IETF standard. It is very unlikely that OpenPGP would be accepted as a standard if it mandated encumbered algorithms [IMC98], [WIR97b], [CNET97], [Bac99b]. This explains why RSA & IDEA have been deprecated in the latest versions of PGP.
The standard has now been accepted by the IETF and can be found in [RFC2440].
One can see a similar transition in cognate standards (e.g. S/MIME).
PGP / NAI were in an interesting position when deciding whether to support RSA in the Freeware version of PGP v5+. The most recent version of the RSA Reference library, which has to be used to prevent patent infringement, has some interesting stipulations [RSA94]:
PGP would also have been in the position of having to use a library that is inferior in terms of speed to the unencumbered MPILIB or BNLIB library.
Of course, PGP / NAI could have continued to use the legacy version of RSAREF (e.g. RSAREF v1), but this also has an onerous clause [RSA93]:
"...incorporate the Program into other computer programs for your own personal or internal use, provided that you provide RSA with a copy of any such modification or Application Program by electronic mail, and grant RSA a perpetual, royalty-free license to use and distribute such modifications and Application Programs on the terms set forth in this Agreement."
Basically, PGP/NAI would have to grant RSA a "royalty-free license to use and distribute PGP". This is certainly counter to NAI/PGP's commercial interests, and as such would appear to be prohibitive to distributing a free version of PGP that incorporates RSAREF. Which is the only current way of providing RSA support in a manner complicit with patent law.
Users, of course, are free to use the legacy versions of PGP if they absolutely must have RSA support. One also notes that International versions of PGP v5+ support RSA as standard (as there are no patent issues regarding the support of RSA outside of the United States).
It can be seen that, in the domestic version of PGP v6.02, NAI have made an honest attempt to provide maximum RSA functionality available under "reasonable terms" - this version of PGP uses the RSA functionality provided by CryptoAPI (licensed by Microsoft).
Maybe someone can see a commercially viable way of NAI/PGP to continue to support RSA in the freeware version of PGP? I, unfortunately, can't.
Note: This is a "grey area" in legal terms, there is very little legal precedent and as such users are warned to consult expert legal advice before acting upon comments in this section.
It is thought possible (at least in some countries) that Courts have the ability to compel the production of keys necessary to decrypt communications. In terms of PGP this means that a Court could compel the user to provide their passphrase that enables the Court to access the private key used for decryption.It is generally accepted [EC98] that "a key escrow system is not intended for access private keys used only for integrity functions." Compelling production or, even worse, the escrow of signature only keys would not allow for non-repudiation or "assured" authentication. To my knowledge, no countries are entertaining the idea of either compelling production of or escrowing signature keys.
In previous versions of PGP (e.g. v2.x), divulging the RSA key meant that:
Obviously, these two points are disastrous so PGP now offers solutions to both of these issues:
These facilities just weren't available using RSA keys. True, PGP could have built a similar feature for RSA keys but these new keys would necessarily be incompatible with existing keys.
From a cryptographic design viewpoint, it is also sound practice to avoid using the same key for multiple purposes [MOV96].
An early and unfounded complaint about PGP DH was that the new keys "break the web of trust". People thought that all their existing signatures, which construct the web of trust, would now be defunct.
A simple way to sustain the existing "RSA web of trust" is to simply sign new DH/DSS keys with legacy RSA keys.
There are three answers to this frequently asked question:
Dozens, if not hundreds of trained cryptographers and programmers have reviewed the various of implementations of PGP (I know at least 5 personally!). There is no doubt that breaking an implementation of PGP would make a cryptographer (recall that Matt Blaze was "made" by breaking a popular cryptographic standard). See section the "Does anybody really bother checking the PGP source code?" for further details.
The security of the PGP system relies quite heavily on the Random Number Generator (RNG).
The RNG is used in the following situations:
Fortunately, PGP v5+ implements a RNG according to ANSI X9.17, which is in conformance to the standard outlined in [FIPS186-1].
As a matter of personal interest, I abstracted the RNG functionality from PGP v5.0i and produced 50x 30Mb files of "random" data which were then tested with DieHard [Mar98], a popular program for testing data for non-randomness. According to DieHard the output of the RNG used in PGP exhibits no bias, correlation or other obvious statistical weakness. A couple of the tests failed, but this is to be expected [Rit98].
The PGP RNG also passes the statistical tests specified in [FIPS140-1].
NOTE: the RNG cannot be declared "secure" just upon my empirical testing.
If you are interested in testing the RNG in PGP then I recommend the excellent book by Knuth [Knu98], or the paper by Kelsey, Schneier, Wagner, Hall [KSWH98a]. Further particularly good sources information on RNG tests are available in [MOV96].
Yes. The new versions of PGP are based on a much more rigorous treatment of trust by Ueli Maurer [Mau96].
Not exactly. There are two possible patent issues relating to algorithms implemented in PGP (and in dozens of other pieces of software!):
[FIPS186-1] states decisively "The Department of Commerce is not aware of any patents that would be infringed by this standard." & in [NIST98] "NIST is not aware of any U.S. patent which might be infringed by the practice of the invention claimed in its patent on DSA....NIST is not aware of any U.S. patent which might be infringed by the use of SHA or SHA-1."
All programs that implement DSS or SHA may potentially infringe on these two claims. One would assume that according to US patent law, which states that if you hold a patent and fail to prosecute an infringement you may lose the patent [Sch96a], any valid claims would have been made by now.
RSA is still patented in the U.S. In Sept 2000 the patent will expire. Amusingly, RSALabs have indicated [SD99] that they are to trademark the term "RSA", despite previously assuring IEEE to the contrary [SD98]. It would appear that in Sept 2000 users will be allowed to use the RSA scheme, but vendors will be force to call it something else!
NOTE: I would suggest consulting a patent lawyer if the above causes concern!
Yes! I have personally tested (in PGP v5.0i) the implementation of DES, CAST, IDEA, MD5, SHA-1, RIPEMD-160 against test vectors. I have also tested the code used for DSS signature generation against the test vectors provided in [FIPS186-1] which testifies that the Big Number library code is working correctly.
I have tested the output from the RNG used within PGP as detailed in section "How secure is the RNG in PGP?"
I would, of course, conduct the above tests on other similar security packages (S/MIME implementations for example) - but it just isn't possible.
From personal experience, more people compile, inspect and test the source code than one might naïvely believe. From my involvement with ScramDisk, I note that out of the user base (which we estimate to be around 20,000), I have received e-mails from in excess of 40 individuals asking sometimes very technical questions about the source code. PGP is used by many more people than ScramDisk, so one can predict that the number who inspect the source code of PGP is correspondingly higher.
Before encryption, messages are compressed using a standard compression algorithm [RFC1951]. Compressing data prior to encryption helps remove redundancy from the plaintext. Redundancy in the plaintext may aid cryptanalysis [BW94], [Sch96a], [MOV96], [Pfl97], [Bau97].
Interesting question! Actually, this is two separate questions as DNA and Quantum computing affect the security of PGP in different ways.
First, Quantum Computers (QC). If (and it's a big "if" [Sch96a], [RSA96a]) QCs become practical then it is likely to have several implications in cryptography:
Professor Peter Knight, a lead physicist researching into QCs has commented that [Bro99] "It looks a long way of...if in four years we have worked out how to build a machine that uses quantum mechanics to factories 15, everyone will be hugely pleased."
Secondly, DNA computing, which appears to be more practical than QC. DNA computers can used to solve any problem that reduces to the Hamiltonian path problem [RSA96a], [Bea95], [Sch96a]. Essentially, DNA computing is like classical computing, just highly parallelized. DNA computing can be thwarted by using keys of sufficient size (e.g. 4096-bit asymmetric keys are certainly secure against DNA computers. It is not yet known whether a 128-bit symmetric key is safe against attacks by DNA computers - a pessimists would certainly consider moving to 256-bit keys (e.g. AES) when practicable. [Bea95] notes that, according to current DNA computing theory, 10120 litres of DNA would be necessary to factor a single 1000-bit RSA key.
Currently, supercomputers can manage approximately 1012 operations per second. It is thought plausible that a DNA based computer could exceed 1012 operations per second [MOV96]. The DNA based computer would also require more modest energy & cooling requirements.
I highly recommend the papers [Bea95] & [Sho94] as an overview of DNA and QC respectively, and in particular their practical application to factoring. [Gro99] is also a good non-technical paper covering QC.
All of the comments in this section reflect on both PGP and other similar programs that exploit public key cryptography.
Totally untrue. PGP has recently been exported via the following method:
You lucky Americans have a great constitution! Try living in the
dark ages UK sometime...
Absolutely not. A reasonably determined adversary can simply reverse engineer the machine code that comprises the program and analyse this [GW96]. University students are capable of undertaking this task, so it is extremely naïve to believe that the intelligence agencies can't.
As an example, Netscape and Microsoft refuse to release the source code of their security related software for peer review [GW96]. As a result of this lack of peer review, two of the most popular implementations of SSL were totally insecure against a determined adversary. One notes that even the "secure" versions of the browsers (e.g. the domestic US versions of the software) suffered from this security hole.
Quoting directly from the paper: "Peer review is essential to the development of any secure software. Netscape did not encourage outside auditing or peer review of its software - and that goes against everything the security industry has learned from past mistakes. By extension, without peer review and intense outside scrutiny of Netscape's software at the source-code level, there is simply no way consumers can know where there will be future security problems with Netscape's products."
"We are concerned that companies are hiding information about their security modules and shunning peer review"
Without source code review, how do you know that the cryptographic system isn't leaking key information with every signature or encryption performed? Signatures employing subliminal channels are indistinguishable from normal signatures - only the adversary (who has a secret key that allows them to unlock the subliminal data) is even aware that the subliminal channel exists.
It is a standard assumption in information security (referred to as the Kerckhoff principle, also paraphrased by Shannon "The enemy knows the system being used") that an adversary has access to the methods used in the process of encryption [Sch96a], [MOV96], [Sti95], [Bau97] - the strength must thus lie in the secrecy of the key. Trying to obtain "security by obscurity" seems imprudent in light of the Goldberg and Wagner paper.
The government has a certification program for cryptographic modules [FIPS140-1].
Very recently NIST have announced that the DES, DSA & SHA-1 algorithms from the PGP Cryptographic SDK, version 1.5 have now been validated under FIPS140-1 [NIST99c].
The process of certification involves the submittal of the source code to a NIST approved testing lab and may require changes to the primitive used in the software. As such, it can be considered an iterative process. One of the problems with certification is that it only relates to the certified version. Thus the next release of the software needs to be resubmitted for certification.
In practice, the certification means that PGP can be purchased and used by government departments. It also provides a "base-line" level of security in the algorithms used.
For completeness, I note that some of the cryptographic primitives used within Netscape (DES, DSA, SHA-1) have already been certified to level 2 when run on "Sun Sparc 5 w/ Sun Solaris version 2.4SE" and level 1 when run on Windows NT Workstation.
The observant reader will note that the implementation of the public key algorithm (RSA / DH or whatever) is not checked as part of the standards evaluation.
One problem with the certification process is that it can give naïve users a false sense of security in the evaluated product. A program's implementation of cryptographic primitives could, for example, be evaluated and certified in accordance with Federal standards, but be used in a setting that contains serious security flaws that affect the overall security of the package. These flaws may be neither easy to find or self-evident. Only a review of the complete package can "prove" the security of the package.
It is more than the core cryptographic code that needs checking, the code around the periphery of the implementation also needs to be checked, which FIPS140-1 doesn't provide. For further information read any good textbook on information security or applied cryptography, for example [Sch96a] or [Pfl97].
Some have complained about the delay in the production of v5 of PGP. Remember that Phil Zimmermann was under a three-year criminal investigation, which ended early in 1996. I would suggest that PRZ had limited time and financial backing to conduct much development work during this period.
Also, one notes the large "feature-jump" between version 2.x and 5.x. PGP now has very strong hash functions, two additional ciphers, a new PK encryption scheme, a signature scheme that conforms to the USG standard [FIPS186-1], integration with key servers etc.
"PGP is used all over the world by human rights groups, human rights activists who are documenting the atrocities of death squads, interviewing witnesses and using that to keep track of human rights abuses, and they encrypt that stuff with PGP, and they tell me that if the government there could get their hands on it they would round up all the witnesses and kill them, after torturing them first.
That's in Central America, and I talked to somebody working down there on it. The resistance groups in Burma are using it. Burma has a really horrible government, and there's resistance groups using PGP in jungle training camps. They're being trained to use it on portable computers. Then they are taking them to other jungle training camps and teaching them.
They've said that it's helped morale there because before PGP was introduced there, captured documents would lead to the arrest, torture, and execution of entire families. The government in exile in Tibet uses PGP. There's several other examples of third world countries where brutal dictatorships are, where human rights activists are using PGP."
It would appear people readily trust trade secrets, client confidentiality and their lives with PGP. This certainly puts my use of it to encrypt personal communication in perspective!
I'll answer both of these questions in order.
Encrypting a message to multiple recipients is as easy to break as the weakest asymmetric key. For example, if a message was encrypted to a 768-bit and a 1024-bit key then an adversary would obviously be able to read the message if they broke the smaller key.
If a message is encrypted to recipients with different asymmetric or symmetric key types - e.g. to both RSA and DH keys or with both 3DES and IDEA, then reading the message is clearly as hard as breaking the symmetric cipher or any of the asymmetric ciphers employed.
This necessarily introduces a theoretical weakness, as an adversary can attack multiple cipher schemes in order to get at the protected message. All of the ciphers in PGP are thought to be secure, so this theoretical weakness doesn't transfer to a practical weakness.
The ultra-paranoid could thus make a tenuous argument for only using one key type (both asymmetric and symmetric). For example, a user could refuse to accept RSA encrypted traffic (by not creating an RSA key I would suggest) and refuse to accept or send messages using any cipher other than CAST. This would minimise the theoretical weakness but I would suggest is totally unnecessary.
On to the second question...Encrypting an already encrypted message again is certainly practically more secure than a single encryption. In order to read the plaintext message an adversary would have to decrypt two levels of military grade encryption - which is totally infeasible. But hang on - breaking a single level of military grade encryption is totally infeasible anyway, so we haven't gained anything practically!
Actually, there is a subtle theoretical weakness when nesting PGP messages; PGP encrypted messages have a standard heading, so breaking the outer layers of encryption may be equivalent to a known plaintext attack - which is far easier than just performing a brute force without knowing about the underlying message content. Note: this does not affect the "inner layer" - this is as hard to break as any normal single PGP message.
If you use chained remailers and nest encryption to these remailers, then be assured that the message is at least as hard to break as the plaintext encrypted once - and may be practically far more secure depending on the resources of the adversary.
[Sch96a] contains a nice description of multiple encryption schemes.
One worry when nesting encryption schemes (especially multiple applications of the same underlying primitives) is that the multiple applications will be a "group" - e.g. encrypting a message with any two keys Key1 and Key2 is equivalent to encrypting the message once with a single Key3. I can see no way that a applying PGP encryption multiple times could form a group - but I also haven't seen evidence that it couldn't happen (compressing the ASCII armoured text before encryption should help to "disfigure" any structure).
"TWINKLE" is a theoretical device thought up by Adi Shamir early in '99 [Sha99]. It stands for The Weizmann Institute Key Locating Engine.
TWINKLE is a sieving machine that can be used to break RSA keys and, to a lesser extent, DH/DSS keys. To my knowledge, no TWINKLE machines have yet been implemented, but they are thought to be practical. Rather than inventing a new method of factoring, TWINKLE simply speeds up the implementation of an existing algorithm (NFS).
NFS consists of two steps:
Where feasible, the TWINKLE device dramatically speeds up the sieving portion of NFS. A conventional computer is still then required to solve the resulting matrix.
Recently, a team broke a 465-bit number using NFS. The sieving portion took 200 computers 4 weeks and solving the matrix took a CRAY supercomputer 4 days and 810Mb of RAM. In contrast [Sil99a], 6 of Adi Shamir's TWINKLE machines would take about 6 days to undertake (plus 4 days on the supercomputer to solve the matrix).
Unfortunately, it doesn't appear likely that TWINKLE can scale to 1,024-bit (or even 768-bit) keys. It is projected that 45,000 TWINKLE devices would take 500,000 years to sieve a 1,024-bit number and solving the matrix would take at least 5 terabytes of RAM & in excess of 5 million years [Sil99b]. Realistically, it appears that TWINKLE is only a threat to keys in the 512-bit to 600-bit range.
TWINKLE can attack DH/DSS keys in the same way as RSA keys. BUT (and it's a big but...), solving the matrix with DH keys is much harder as one has to [Sil99b] "solve the matrix modulo the order of the group or field" rather than mod 2.
It is also interesting to note that ECC security estimates are unaffected by TWINKLE [Sil99b], [Cer99a].
Really, TWINKLE confirms previous thoughts:
The bottom line: PGP (since v5.5) only allows the creation of keys >=768-bit - so no recently produced PGP keys are at risk by this development.
If you are interested in reading about TWINKLE then I'd recommend the original paper by Adi Shamir [Sha99] and the analysis by Robert Silverman [Sil99a].
CryptoAPI is present in all modern version of Windows, including NTv4, Windows 2000, Windows 95 & Windows 98 and offers cryptographic primitives for use in high level applications (such as the SSL implementation in Internet Explorer and the S/MIME implementation in Outlook).
Recently it has been disclosed that CryptoAPI (Microsoft's Cryptographic API) has a "secret" key that enables the holder of the key to sign CryptoAPI modules (these modules are known as CSPs, or Cryptographic Service Providers) - these modules can then be silently installed onto a users machine. The real big news is that this key is call _NSAKEY internally by windows. This has led to speculation that this key is really a NSA owned key that allows them to remotely install weakened crypto onto users machines etc.
The _NSAKEY variable name was only discovered by accident - MS forgot to strip the symbolic debugging information from NT v4 SP5 which allowed A.Fernandes of Cryptonym to detect the name of the variable and work outs it's use in signing CSP's.
Microsoft have denied these accusations claiming that "We do not share them [the keys] with any third party, including the National Security Agency or any other government agency." and that the second key is required as a backup and to comply with export regulations. They also unconvincingly claim that the variable name _NSAKEY is "...simply an unfortunate name." [Mic99].
There is no doubt that this issue has been over-hyped to some degree, but personally I am not convinced by Microsoft's flimsy "rebuttal". I simply don't believe that we know the true reason that the second keys are in every copy of Windows.
The CryptoAPI matter is a concern to some PGP users who operate under Win32 (e.g. Windows 95, Windows 98, Windows NT v4, Windows 2000) as some versions of PGP rely on the CryptoAPI for RSA support.
From [Sal99], the following table details how RSA support is implemented in various versions of PGP:
|PGP Version||RSA Support Via:|
|v5.x Commercial (with RSA support)||PGPInc Code|
|v5.x Commercial (without RSA support)||None|
|v6.x Commercial (with RSA support)||BSAFE2|
|v6.0.2 Commercial (without RSA support)||CryptoAPI3|
|v6.5.1 Commercial (with RSA support)||BSAFE2|
|1||RSAREF = special non-commercial RSA library provided by RSADSI|
|2||BSAFE = standard commercial RSA library purchased from RSADSI|
|3||CryptoAPI = MS CryptoAPI including RSA code licensed by MS from RSADSI. RSA support via CryptoAPI is only provided on machines with "domestic" (e.g. 128-bit) browsers.|
It is important to note that, apart from RSA support in the highlighted versions, PGP does not rely on CryptoAPI (or any other 3rd party libraries...) for any other cryptographic primitive. Hash functions, symmetric ciphers, DSS etc are all provided by in house code.
I can confirm that International builds by:
all use the PGPInc home grown code rather than RSAREF, CryptoAPI or BSAFE.
From a security perspective, I would certainly say that it makes senses at the moment to avoid any versions of PGP (e.g. v6.0.2) that rely on CryptoAPI if you need RSA support. Maybe if Microsoft come up with a reasonable explanation then I'll recommend RSA support provided by CryptoAPI - but somehow I doubt it.
Anyone interested in reading more about this saga should firstly consult the original Cryptonym press release [Cry99] then secondly read the totally unconvincing Microsoft press release [Mic99]. I'd also recommend reading the considered comments by B.Schneier [Sch99b], M.Blaze [Bla99], D.Wagner [Wag99b] & M.Kuhn [Kuh99] all of whom disagree that this is some major conspiracy involving the NSA & Microsoft.
When a user requests that PGP encrypt & signs a message, it (and all other PK systems that I am aware of) first signs a message and then encrypts the concatenation of the signature and the data.
It is possible to implement this another way; first encrypt the data and then sign the cipher text. Transmit both the encrypted data and the signature.
Two reasons for not applying the signature algorithm (either RSA or DSS) after the encryption:
I can imagine obscure situations whereby you would like message authentication / verification outside of encryption. To implement this, simply encrypt (and optionally sign) a message with PGP and then sign the resulting encrypted message. Simple.
The following table identifies the algorithms mandated (as "MUST" algorithms) in S/MIME (v3) [IETF98] and OpenPGP [RFC2440]:
|OpenPGP v1||S/MIME v3|
|Symmetric Algorithm||3DES (CFB)||3DES (CBC) & RC2/40|
|PK Algorithm (encryption)||ElGamal (4096)||Diffie-Hellman (1024)|
Of course, it is the implementation of the standard that dictates the security, not the "theoretical" standard. (Note that ElGamal & DH are equivalent - see Note 1). Readers who are confused by the significance of "MUST" in IETF documents are referred to [RFC2119].
Asymmetric key size is of great importance in PGP and S/MIME. Table 1 in section "How big should my asymmetric key be?" lists the recommended public key length to protect against attack by a single corporation (keys should be substantially bigger to protect against intelligence agencies!). So...The S/MIME standard specifies a maximum public key length of 1024-bits, which according to the table above doesn't offer sufficient long-term protection against a determined corporation! The OpenPGP standard, and the NAI implementation of the standard, allow public keys of up to 4096 bits, which should protect data for the foreseeable future.
ANSI X9F1 mandates 1024-bit minimum key-length for RSA and DH, but S/MIME only supports key sizes of up to 1024-bits.
Some would argue that future versions of the S/MIME standard could possibly mandate keys greater than 1024-bits. That means that a determined and well-funded adversary could read your current private communications within 5 years or so.
One of the S/MIME periphery documents [PKIX98] describes a feature of S/MIME called "Proof of Possession of Private Key (PoP)". This is a mechanism whereby end user's private keys may be deposited with the CA when certification is requested. This is a very worrying inclusion and makes the implementation of mandatory key escrow a trivial matter [Gut99].
The PGP draft standard contains no such references to key recovery technology but, interestingly, PGP v5.x + include a feature called Additional Decryption Key (ADK) - sometimes also referred to as Corporate Message Recovery. As previously mentioned, this feature does not form part of the OpenPGP standard. Users interested in reading about this feature are pointed towards [Bar99a].
It is believed that the inclusion of PoP was added to the S/MIME standard at the request of the USG. Personally I believe that there should be a clear statement on whether the S/MIME standard requires unconditionally or functionally trusted TTPs (as per [MOV96]) - not some fuzzy standard that is subject to political pressure.
For completeness, I note that PGP specifies the CFB chaining mode whilst S/MIME mandates CBC mode. Both of these modes are equivalent in terms of strength [Sch96a], [MOV96] assuming that a unique IV (or random data to attach to the front of the message) can be provided in CFB mode. PGP already implements a random number generator (RNG); so supplying this random value is not a concern.
Of course, one can apply simple statistical theory to the implementations of PGP and S/MIME. Independent programmers and cryptographers have subjected PGP source code to literally thousands of hours of analysis. No such analysis can be made of S/MIME implementations since the source code is not distributed.
Since we can't prove there are no weaknesses in either implementation, the probability of there being such weakness is a straightforward function of the expert man-hours spent searching for them. One doesn't have to assert there ARE such weaknesses to make this argument. Thus the risk in using PGP is less than the risk in using S/MIME implementations that are not available with source. It's straightforward applied statistical decision theory.
How many expert man-hours have been spent searching for bugs in PGP? See section "Does anybody really bother checking the PGP source code?". How many expert man-hours have been spent searching for bugs in S/MIME? Who can tell? One could possibly argue that the "core" S/MIME code has been checked extensively by those implementing the system (but note that different implementations of S/MIME use different cryptographic libraries), but remember that, in the context of security, code on the periphery (e.g. not just the cryptographic core) can have a direct impact on security. So we revert back to the "lack of peer review" issue, as presented in [GW96].
A more recent paper discussing the merits of open source security components is available at [Sch99c]. My favourite quote of the paper is: "As a cryptography and computer security expert, I have never understood the current fuss about the open source software movement. In the cryptography world, we consider open source necessary for good security; we have for decades."
A version of S/MIME could be released (or could already be deployed?) that implements "Kleptography" - e.g. implementing key-generation routines in DH or RSA that produce keys which appear normal, but which allow a third party (who has knowledge of an "unlocking key") to retrieve the private key. This is an interesting development on the concept of subliminal channels, first documented by Gus Simmons. For further information, the reader is directed towards the paper [YY96]. No such covert channel exists in PGP v5.0i - a quick review of the source code would reveal the offending code.
Importantly, non US S/MIME users are provided with the RC2/40 symmetric cipher which has a key length of 40-bits and asymmetric ciphers <=512-bits which is clearly insecure. From [IETF98]: "40-bit encryption is considered weak by most cryptographers. Using weak cryptography in S/MIME offers little actual security over sending plaintext." Bruce Schneier offers a screen saver which brute forces 40-bit keys using the idle time on a standard PC.
It's worse than that actually...Many implementations of stronger ciphers do not inter-operate above 40-bit RC2 [Sch97b], [WIR97a]. This is bad news for end users.
PGP uses symmetric keys of 128-bits and asymmetric keys of up to 4096-bits irrespective of the locale. Thus users of PGP can communicate securely globally whereas S/MIME users currently can't.
From a security perspective, PGP can rightfully be considered more secure than S/MIME for the following reasons:
Once again, to quote Schneier [Sch98i]: "The S/MIME 2 e-mail standard took a relatively strong design and implemented it with a weak cryptographic algorithm"..."Functionality does not equal quality, and no amount of beta testing will reveal every security flaw".
An article written by a well known cryptographer [GW96] has claimed about Netscape Navigator (a well known program that purports to implement the S/MIME standard) "Until they learn their lesson and open their security programming to public evaluation, many security experts will remain justifiably sceptical of the company's security claims".
This is a question often asked by new users, and it raises a number of interesting points:
Unfortunately, cryptography isn't like any other science; we (usually) rely on empirical evidence to judge the strength of algorithms employed and the system implementations. Bad crypto looks and smells just like good crypto.
Sometimes it's useful to think of encryption as a combination lock on a safe. Is it better to have a lock with a greater number of possible combinations? All other things being equal, the answer is yes. But those "other things" are usually not equal. Some locks can be "picked" without trying all the possible combinations. You wouldn't want your household valuables to be protected by a lock that has trillions of possible combinations, but can be opened quickly by a person with a stethoscope.
Short keys guarantee weakness, but long keys don't guarantee strength.
If you have read the above (and I'd suggest the Snake Oil FAQ), and still think that the benefits outweigh the disadvantages, then help yourself!
PGP is an evolving standard and as such is constantly improving. The following list highlights common gripes about cryptographic elements of OpenPGP:
"No one shall be subjected to arbitrary interference with his privacy, family, home or correspondence, nor to attacks upon his honour and reputation. Everyone has the right to the protection of the law against such interference or attacks."
-- Article 12 Universal Declaration of Human Rights
"Takes a man to suffer ignorance and smile."
It would appear that there are several reasons for the changes seen between v2.x and v5+, primarily:
PGP had to change the implementation of PGP in ways that would have made previous keys invalid, it is therefore not unreasonable that they also change the asymmetric cipher used in PGP. Users can still use RSA keys (though this may mean using an International version of PGP or paying for the RSA version) if they have a requirement to continue using old keys. In view of the MD5 issue however, this would appear imprudent.
One argument has been that ElGamal as implemented in PGP could be flawed - that is to say there could be a subtle weakness in the implementation. The user should be reminded that ElGamal relies on the same mathematical operations and functions as RSA, namely: general large integer arithmetic, modulo exponentiation, inverses modulo, GCD, extended Euclidean algorithm, Chinese remainder theorem, reciprocal, etc. [Dif98], [Sil96b] (also see Note 4), all of which have, by now, been extensively tested.
PGP v5+ is certainly cryptographically stronger than previous versions of PGP. There also seem to be other influences, primarily the list above, that have necessitated the changes seen between these versions.
The only valid reason I can find for continuing to use RSA keys in preference to the new DH/DSS keys is to communicate with a recipient who only has the deprecated keys, or if a new version of PGP is not available for your operating platform.
In summary, DH PGP keys can be considered stronger than PGP RSA keys for the following reasons (I have removed references from this section for clarity - refer to previous sections of the document for justification of these claims):
The only arguments I can make for not using DH/DSS keys is that RSA signature keys can be significantly longer than DSS keys, however this point is easily countered:
Please let me know if you can think of other reasons why either key type is more or less strong than the other.
The NSA (probably!) aren't specifically interested in you. They aren't going to break into your house to install bugs, or monitor your screen from a block away. They will however collect all of your messages sent over public networks.
PGP protects you from one form of monitoring - Echelon or other passive network sniffing. When your messages are captured by this global monitoring system, along with millions of other messages a day, the NSA can possibly decide to try and decode your message.
The most significant threat to PGP comes from user sloppiness. It is far easier to install a keylogger on your computer, install a trojan version of PGP, or bruteforce your passphrase than to break any of the cryptographic mechanisms employed by PGP.
If you are seriously worried about Intelligence Agencies actively monitoring you, then the last thing you should be worried about is them cryptographically attacking your PGP crypto implementation!
If, having read all of the above, you are still not convinced about the benefits of DH over RSA keys then you still have the option of:
Why should I offer my subjective opinion on PGP as a summary? I won't. Instead I shall use 3 quotes (I like quotes!):
[Sch96a] on PGP:
"Assuming you trust IDEA, PGP is the closest you're likely to get to military-grade encryption".
[PGP98] sums up the security of PGP v5+ perfectly:
"If your situation justifies worrying about very formidable attacks of this caliber, then perhaps you should contact a data security consultant for some customized data security approaches tailored to your special needs."
The closing words of this FAQ have to be from Whitfield Diffie, distinguished progenitor of Public Key cryptography [DL98]:
"In writing PGP, Phil Zimmermann did something for cryptography that no technical paper could do: he gave people who were concerned with privacy but were not cryptographers (and not necessarily even programmers) a tool they could use to protect their communications".
I'm only going to make three recommendations:
Applied Cryptography (2nd ed.) by Schneier, 1996 - A comprehensive and definitive introduction to cryptography.
Handbook of Applied Cryptography by Menezes, van Oorschot & Vanstone, 1996 - a very rigorous, "landmark" book.
Privacy on the Line by Diffie and Landau, 1998 - the father of PK reviews the politics surrounding the crypto debate.
(1) To break DH, one needs to be able to find gab knowing only ga and gb. To break ElGamal, one needs to determine P, knowing gk and Pgak, with ga being public also. But then breaking ElGamal is equivalent to finding gak, knowing only ga and gk --the same problem needed to be solved to crack DH. (All of the above is of course over a 'large' finite field, the size of which is public.) So cracking one cracks the other, whether you solve the DLP or not. However, it's conjectured that there is no way to solve the above problem without essentially solving the DLP.
(2) From the source code of PGP v5.0:
Public key components:
n The public modulus
e The public exponent
Private key components:
d Decryption exponent (which is equal to e-1 mod ((p-1)(q-1)))
p The smaller factor of n
q The larger factor of n
u 1/p (mod q)
Encryption is (pseudo-code!):
EncMsg = DecMsge mod n
Decryption is (pseudo-code!):
DecMsg = EncMsgd mod n
(3) Taken directly from the source code of PGP v5.0:
Public key components:
p Public prime
g Public generator
y Public key, gx mod p
Private key component:
x Secret key, discrete log of y
ElGamal encryption is a simple variant on non-interactive Diffie-Hellman. The recipient publishes g, p, and y = gx mod p. The sender picks a random xx, computes yy = gxx mod p, and the shared secret z = yxx mod p = yyx mod p = gx*xx mod p, then sends yy and z*m mod p to the recipient, where m is the message.
(4) From the source code of 2.6.3i MPILIB.H:
"These routines implement all of the multi-precision arithmetic necessary for number-theoretic cryptographic algorithms such as ElGamal, Diffie-Hellman, Rabin, or factoring studies for large composite numbers, as well as Rivest-Shamir-Adleman (RSA) public key cryptography."
(5) "While quantum computation can make problems such as factoring and discrete logarithms (which public-key cryptography are based on) easy, they can only reduce the complexity of any arbitrary computation by a square root. So, for example, if a 128-bit key length was long enough pre quantum computation, then a 256-bit key will be long enough post quantum computation."
-- B.Schneier, 10th Oct 1998 posting to soc.history.what-if USENET group
(6) "How can we with good conscience allow users to generate keys which we don't feel meet our security standards? We can't. Case closed. If you're unfamiliar with why RSA keys are not as secure as we'd like, you should check archives of the newsgroups for the past few years.
The weaknesses of MD5 and the KeyID attacks were the two primary security issues we felt absolutely had to be addressed in 5.0. The development team couldn't have cared less about RSA licensing issues.
The only issue was security. Fixing those required a new key format. As long as we were changing the key format, we decided to switch to unencumbered algorithms at the same time since the hit was the same either way -- everyone would need new keys. If a particular user doesn't mind the security issues with RSA keys, they should feel free to continue using them although the number of versions supporting those keys available from us will undoubtedly continue to dwindle, and at the same time the number of versions and platforms supporting DH/DSS keys will continue to grow dramatically."
-- W.Price, Nov 1997 posting to ietf-open-pgp mailing list
|ADK||Additional Decryption Key|
|AES||Advanced Encryption Standard|
|API||Application Programming Interface|
|BFN||Balanced Fiestel Network|
|DES||Data Encryption Standard|
|CMR||Corporate Message Recovery|
|CRHF||Collision Resistant Hash Function|
|CSP||Collision Cryptographic Service Provider|
|DH||Diffie-Hellman. The PK system, named after the creators|
|DLP||Discrete Logarithm Problem|
|DSA||Digital Signature Algorithm|
|DSS||Digital Signature Standard|
|ECM||Elliptic Curve Method|
|ECDH||Elliptic Curve Diffie-Hellman|
|ECDSA||Elliptic Curve Digital Signature Algorithm|
|EES||Escrowed Encryption Standard|
|GAK||Government Access to Keys|
|IETF||Internet Engineering Task Force|
|IESG||Internet Engineering Steering Group|
|IFP||Integer Factorisation Problem|
|FIPS||Federal Information Processing Standards|
|KRC||Key Revocation Certificate|
|MIPS||Millions of Instructions Per Second|
|MITM||Meet In The Middle|
|NIST||National Institute of Science and Technology|
|NSA||National Security Agency|
|OWHF||One Way Hash Function|
|PRNG||Pseudo-Random Number Generator|
|PRZ||Philip R Zimmermann, original author of PGP|
|RNG||Random Number Generator|
|RSA||Rivest, Shamir, Adleman. The PK system, named after the creators|
|RTFM||Read the F*cking Manual|
|SHA||Secure Hash Algorithm|
|SHS||Secure Hash Standard|
|TTP||Trusted Third Party|
|TWINKLE||The Weizmann Institute Key Locating Engine.|
|USG||United States Government|
"I must've seen it in a USENET posting; that's sort of like hearsay evidence from Richard Nixon..."
-- Blair Houghton
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|[ANSI930-1]||ANSI X9.30 (Part 1), "...Part 1: The Digital Signature Algorithm (DSA)", ASC X9 Secretariat - American Bankers Association, 1995.|
|[ANSI930-2]||ANSI X9.30 (Part 2), "...Part 2: The Secure Hash Algorithm (SHA)", ASC X9 Secretariat - American Bankers Association, 1993.|
|[ANSI931]||ANSI X9.31, "Digital Signatures using reversible public key cryptography for the financial services industry (rDSA)", 1998|
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|[Bar99b]||L.Baranyi, personal communication, 22nd Feb 1999.|
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|[Bea95]||D.Beaver, "Computing With DNA", Journal of Computational Biology, Spring 1995.|
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|[Bla99]||M.Blaze, "Re: NSA key in MSFT Crypto API", [email protected] mailing list, 3rd Sept 1999.|
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|[Bro99]||M.Brooks, "Quantum Leap in computing",Engineering and Physical Sciences Research Council (EPSRC), 1999.|
|[BV98]||D.Boneh, R.Venkatesan, "Breaking RSA may not be equivalent to factoring", Eurocrypt '98, Lecture Notes in Computer Science, Vol. 1233, Springer-Verlag, 1998.|
|[BW94]||D.Wagner, S.Bellovin, "A Programmable Plaintext Recognizer", 1994.|
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|[Cyb99a]||Cyber-Knights Templar homepage http://members.tripod.com/cyberkt/, 1999.|
|[Cyb99b]||Cyber-Knights Templar, readckt.txt, as distributed with PGP v6.0.2ckt Build 5, 1999.|
|[CNET97]||"RSA under fire from IETF", CNet News.com Article, 3rd Nov 1997.|
|[Cry99]||"Microsoft, the NSA, and You", Press release, Cryptonym, 31st Aug 1999.|
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|[CYL98a]||"What's all of the fuss about triple-DES? How strong is it anyway?", Cylink Corporation, 1998.|
|[CYL98b]||"Alternatives To RSA Using Diffie-Hellman With DSS", Cylink Corporation, Aug 1998.|
|[DBP96]||H.Dobbertin, A.Bosselaers, B.Preneel. "RIPEMD-160: A strengthened version of RIPEMD." Fast Software Encryption, 1996.|
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|[Den98]||D.Denning, message beginning "Eli Biham and Lars Knudsen have exposed a...", 3rd Apr 1998.|
|[Dif98]||W.Diffie, "Whitfield Diffie interview", Lem Bingley interview with W.Diffie, Ziff-Davis, Nov 1998.|
|[DGV93]||J.Daemen, R.Govaerts, J.Vandewalle, "Weak Keys for IDEA", 1993.|
|[DH76]||W.Diffie, M.E.Hellman, "New Directions in Cryptography", IEEE Transactions on Information Theory", Nov 1976.|
|[DL98]||W.Diffie, S.Landau "Privacy on the Line", MIT Press, 1998.|
|[Dob95]||H.Dobbertin, "RIPEMD with Two-Round Compress Function is Not Collision-Free", Journal of Cryptology, Volume 10, Number 1, 1997.|
|[Dob96a]||H.Dobbertin, "The Status of MD5 After a Recent Attack", RSA CryptoBytes - Volume2, Number2, Summer 1996.|
|[Dob96b]||H.Dobbertin, "MD5 Discussion" sci.crypt USENET posting, 11th Jun 1996.|
|[EC98]||"Legal and Regulatory Issues for the European Trusted Services Infrastructure" Final Report, ISTEV, European Commission.|
|[EFF98]||"Cracking DES", EFF, 1998.|
|[EFF99]||"RSA Code-Breaking Contest Again Won by Distributed.Net and Electronic Frontier Foundation (EFF) - DES Challenge III Broken in Record 22 Hours", Press Release, EFF, 19th Jan 1999.|
|[ElG85]||T.ElGamal, "A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms," IEEE Transactions on Information Theory, v.IT-31, n. 4, 1985.|
|[Elg97]||T.ElGamal, comments to NIST on proposed revision to FIPS 186 as quoted by [Sch97d], 1997.|
|[FIPS46-2]||"DATA ENCRYPTION STANDARD (DES)", NIST Federal Information Processing Standards Publication 46-2, 1993.|
|[FIPS46-3]||"DATA ENCRYPTION STANDARD (DES)", NIST Federal Information Processing Standards Publication 46-3, Draft, 1999.|
|[FIPS140-1]||"SECURITY REQUIREMENTS FOR CRYPTOGRAPHIC MODULES", NIST Federal Information Processing Standards Publication 140-1, 1994.|
|[FIPS180-1]||"SECURE HASH STANDARD", NIST Federal Information Processing Standards Publication 180-1, 1995.|
|[FIPS186-1]||"DIGITAL SIGNATURE STANDARD (DSS)", NIST Federal Information Processing Standards Publication 186-1, Dec 1998.|
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|[GGH+98]||H.Gilbert, M.Girault, P.Hoogvorst, F.Noilhan, T.Pornin, G.Poupard, J.Stern, S.Vaudenay "Decorrelated Fast Cipher: an AES Candidate", NIST AES Proposal, 1998|
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|[GW96]||I.Goldberg, D.Wagner, "Randomness and the Netscape Browser - How secure is the World Wide Web?", Dr Dobbs Journal, 1996.|
|[Hit98]||K.Ogawa, letter begining "We are pleased that IEEE P1363" sent to B.Kaliski, Chair, IEEE P1363, 3rd August 1998.|
|[Hof94]||L.J.Hoffman, "Building in Big Brother", Springer, 1994.|
|[IETF98]||B.Ramsdell, "S/MIME Version 3 Message Specification", Aug 1998.|
|[IMC98]||Internet Mail Consortium, "S/MIME and OpenPGP", http://www.imc.org/, 1998.|
|[INF96]||"infiNity", "The PGP attack FAQ", http://axion.physics.ubc.ca/pgp-attack.html|
|[ISOIEC10118-3]||ISO/IEC 10118-3, "Information Technology - Security Techniques - Hash Functions - Part3: Dedicated hash functions", draft, 1996.|
|[Joh99a]||D.Johnson, "RE: compare RSA and D-Hellman" sci.crypt USENET posting, 24th Mar 1999.|
|[Joh99b]||D.Johnson, "RE: RSA-512" sci.crypt USENET posting, 27th July 1999.|
|[Knu98]||D.Knuth, "The Art of Computer Programming Volume 2: Seminumerical Algorithms, 3rd Ed.", 1998.|
|[Knu99]||L.R.Knudsen, "Some thoughts on the AES process", 15th April 1999.|
|[KR97]||L.R.Knudsen, V.Rijmen, "Truncated Differentials of IDEA", 1997.|
|[KSW96]||J.Kelsey, B.Schneier, D.Wagner, "Key-Schedule Cryptanalysis of IDEA, G-DES, GOST, SAFER, Triple-DES", Advances in Cryptology - CRYPT'96, 1996.|
|[KSW97]||J.Kelsey, B.Schneier, D.Wagner, "Related-Key Cryptanalysis of 3-WAY, Biham-DES, CAST, DES-X, NewDES, RC2, and TEA", 1997.|
|[KSWH98a]||J.Kelsey, B.Schneier, D.Wagner, C.Hall, "Cryptanalytic Attacks on Pseudorandom Number Generators", 1998.|
|[KSWH98b]||J.Kelsey, B.Schneier, D.Wagner, C.Hall, "Side Channel Cryptanalysis of Product Ciphers", 1998.|
|[Kob94]||N.Koblitz, "Graduate Texts in Mathematics: A course in number theory and cryptography", Springer, 1994.|
|[Koc98]||W.Koch, "Twofish", IETF OpenPGP mailing list, 23rd Dec 1998.|
|[Kuh99]||M.Kuhn, "RE: NSA key in MSFT Crypto API", [email protected] mailing list, 4th Sept 1999.|
|[Lai91]||X.Lai, "Detailed Description and a Software Implementation of the IPES Cipher", Institute for Signal and Information Processing, ETH-Zentrum, Zurich, Switzerland, 1991.|
|[Len96]||A.K.Lenstra, "Securing the net - the fruits of incompetence", 1996.|
|[Ley99]||P.Leyland, "RE: Asymmetric Key sizes", UK-Crypto mailing list, 14th Feb 1999.|
|[Lip99]||H.Lipmaa, "AES Ciphers: speed" - available at http://home.cyber.ee/helger/aes/, 1999.|
|[LMM91]||X.Lai, J.L.Massey, S.Murphy, "Markov Ciphers and Differential Cryptanalysis", Advances in Cryptology - EUROCRYPT'91.|
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|[Mic99]||"There is no 'back door' in Windows", Press release, Microsoft, 7th Sept 1999.|
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|[Mun99]||R.Munyer, personal communication, 7th Mar 1999.|
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|[NIST98]||M.Smid, e-mail entitled "Re: P1363 patent update", NIST, 25th June 1998.|
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|[PGP97]||PGP v5.00i, source code, 1997.|
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|[Pri99b]||W.Price, "Re: [PGP]: Shamir's factoring device" pgp-users mailing list posting, 5th May 1999.|
|[RFC1321]||R.Rivest, "The MD5 Message Digest Algorithm", RFC 1321, April 1992.|
|[RFC1951]||P.Deutsch, "DEFLATE Compressed Data Format Specification version 1.3.", RFC 1951, May 1996.|
|[RFC2119]||S.Bradner, "Key words for use in RFCs to Indicate Requirement Levels", RFC 2119, Mar 1997.|
|[RFC2144]||C.Adams, "The CAST-128 Encryption Algorithm", May 1997.|
|[RFC2440]||J.Callas, L.Donnerhacke, H.Finney, R.Thayer, "OpenPGP Message Format", RFC 2440, Nov 1998.|
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