28 January 1999. Thanks to Sam Simpson.
Last-modified: Wednesday, 27th January 1999
PGP DH vs. RSA FAQ
Copyright © 1999 S.Simpson - All Rights Reserved
Sam Simpson - http://www.hertreg.ac.uk/ss/
All information contained in this work is provided "as is." All warranties, expressed, implied or statutory, concerning the accuracy of the information of 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 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.
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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.
v1.1 - 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.
2. Public Key Algorithms in PGP
3. Hash Function Algorithms in PGP
4. Conventional Encryption Algorithms in PGP
5. Key Revocation Certificate issues
6. What would it take to "break" PGP?
7. Other issues
9. Further Reading
Primarily, sincere thanks to Kurt Mueller for providing the Key Revocation Certificate section. Also, many thanks to Dan "the" Horne & Andy Jeffries for proof reading this document while it was rough-and-ready.
Thanks to the following 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 and also with improving the implementation efficiency of Serpent, an AES candidate. 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].
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 publicly published public key system [DH76] (more correctly Diffie-Hellman is a key-exchange mechanism) and as such has received extensive analysis by eminent cryptographers.
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 one can break either ElGamal or DH then you 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 Discrete Logarithm Problem (DLP) under the "Diffie-Hellman assumption" [Kob94]. This assumes that it is unfeasible 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".
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:
Some have complained that calling the implementation of ElGamal used in PGP Diffie-Hellman is somehow misleading, the facts certainly don't support this view. Recall that ElGamal is in fact a Diffie-Hellman variant and that the two systems are proven cryptographically & mathematically equivalent.
Note 3 contains a brief overview of how DH is actually performed in PGP.
DSS stands for Digital Signature Standard and is formally defined in [FIPS186] & [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 "paper 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.
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|
Table 1: Recommended assymetric key lengths
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].
We know that 512-bit keys have been insecure for some time now [Sch96a], [Odl95], [Rob95]; 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.
PGP allows only the creation of keys >= 768-bits, so the naïve user is afforded some protection from creating an excessively weak key. The question still remains however, what is a "reasonable" size key?
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 feasible [Sil97].
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 has 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]), 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].
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 justifiably 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] and thus should be called something else.
Users should be warned that the "official flavours" of PGP v5+ will not support DSS keys in excess of 1024-bits and DH keys in excess of 4096-bits.
One worry when PGP v5 was first released was the use of precomputed or "canned" primes for the values of the finite field and the generator of this field (p & g respectively) in DH, and p & q for DSS.
This fear is totally unfounded, it is quite acceptable to use public, precomputed values for these values [Sch96a], [FIPS186], [MOV96], [Kob94], [Sti95]. I would also recommend that the concerned user reads [Sch97a].
The truly paranoid user can choose to switch off the "Faster key generation" if desired (and be prepared to wait far longer for the production of keys). I am however, yet to find any literature that supports this paranoid view!
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...
I would currently recommend against the use of strong primes:
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].
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-Helmman 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].
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].
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].
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." Or, in English, all currently known algorithm 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, more or 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 section 7.4, which covers the compelled production of keys.
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"
Implications: Some classes of the DHP are not computationally weaker than the IFP.
"Breaking RSA may not be equivalent to factoring"
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"
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)
Table 2: Assymetric encryption speed comparison
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]:
Table 3: Assymetric signature speed comparison
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 unfeasible.
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 etc 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 3.7). 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.
RIPE-MD is another hash function derived from MD4. It is formally defined in [DBP96].
To date there have been no successful cryptanalysis of RIPE-MD. PGP implements RIPE-MD160, which produces an output of 160-bits.
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 [Dob96].
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 [Dob96], for example in the creation of digital signatures (the main application of a hash function in PGP). To quote Hans Dobbertin directly [Dob96]: "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 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 attack.
Anyway, it's nice to see that the faceless NSA cryptographers are only human too!
Certainly not MD5 (see section 3.7).
The choice would therefore appear to be between SHA-1 and RIPE-MD. 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], [Dob96]. 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 SHA-1.
Hash function timings (in Mbit/s on an unspecified platform) from [PBD97]:
Table 4: Hash function speed comparison
MD5 may be fast, but remember that it is relatively insecure.
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, invertible.
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:
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 superceded by the Triple-DES alogrithm, 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 EDE configuration
with independent keys. Encryption is executed as follows:
CipherText = DESk1(DES-1k2(DESk3(PlainText)))
And decipherments 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 attack are not useful in breaking practical implementations of 3DES.
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.
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], [Sch96a], [MOV96], [RSA96a], [RSA98], [Sch98f]. 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 [Sch98f]: "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.".
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."
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], [Sch98f].
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 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 5 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|
Table 5: Block cipher strength against a very well funded
|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)?
Symmetric cipher timings (in Mbit/s on a 200Mhz Pentium) from [Lip99]:
Table 6: Block cipher speed comparison
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]. 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 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 CAPI (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 7.10 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].
As a matter of personal interest, I abstracted the RNG functionality from PGP v6.02 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
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] states decisively "The Department of Commerce is not aware of any patents that would be infringed by this standard."
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.
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, RIPE-MD against test vectors. I have also tested the code used for DSS signature generation against the test vectors provided in [FIPS186] 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 7.7.
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 [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:
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.
I highly recommend the papers [Bea95] & [Sho94] as an overview of DNA and QC respectively, and in particular their practical application to factoring.
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
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"
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].
NAI have confirmed that they are in the process of obtaining FIPS140-1 certification for the primitives used within PGP. In fact, they have already received certification for DES & SHA-1.
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 then 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], 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!
This question can be interpreted in a number of ways:
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 any of the asymmetric or symmetric 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 unfeasible. But hang on - breaking a single level of military grade encryption is totally unfeasible 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. PGP compresses data before encryption - this may reduce the feasability of the plaintext attack against the outer layers.
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).
When a user requests PGP encrypt and 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)|
Table 7: S/MIME / OpenPGP comparison
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 (section 2.5) 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, allows 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 keys 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.
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 7.10. 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].
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 (3DES) do not interoperate 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:
An article written by a well known cryptographer [GW96] has claimed that "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." in respect of Netscape Navigator (a well known program that purports to implement the S/MIME standard).
"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] (also see Note 4), all of which have, by now, been extensively tested.
In summary, 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
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. Why would they bother?
The most significant threat to PGP comes from user slopiness. 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 - A comprehensive and definitive introduction to cryptography.
Handbook of Applied Cryptography by Menezes, van Oorschot & Vanstone - 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 z*m mod p, 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 97 posting to ietf-open-pgp mailing list
|AES||Advanced Encryption Standard|
|DES||Data Encryption Standard|
|CRHF||Collision Resistant Hash Function|
|DH||Diffie-Hellman. The PK system, named after the creators|
|DLP||Discrete Logarithm Problem|
|DSA||Digital Signature Algorithm|
|DSS||Digital Signature Standard|
|EES||Escrowed Encryption Standard|
|IETF||Internet Engineering Task Force|
|IESG||Internet Engineering Steering Group|
|IFP||Integer Factorisation Problem|
|FIPS||Federal Information Processing Standards|
|KRC||Key Revocation Certificate|
|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|
|SHA||Secure Hash Algorithm|
|TTP||Trusted Third Party|
|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
|[And93]||R.Anderson, "Why Cryptosystems Fail", Cambridge University, 1993.|
|[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.|
|[ANSI952]||ANSI X9.52, "Triple data encryption modes of operation", draft, 1996.|
|[Bal98]||R.W.Baldwin, "Preliminary Analysis of the BSAFE 3.x Pseudorandom Number Generators", RSA Labs Bulletin - Number 8, Sept 1998.|
|[Bau97]||F.L.Bauer, "Decrypted Secrets; Methods and Maxims of Cryptology", Springer, 1997.|
|[BB94]||B.den Boer and A.Bosselaers. "Collisions for the compression function of MD5", Advances in Cryptology Eurocrypt '93, Springer-Verlag, 1994.|
|[BBR98]||E.Biham, D.Boneh, O.Reingold "Generalized Diffie-Hellman modulo a composite is not weaker than factoring", 1998.|
|[Bea95]||D.Beaver, "Computing With DNA", Journal of Computational Biology, Spring 1995.|
|[BK98]||E.Biham, L.R.Knudsen, "DES, Triple-DES and AES", RSA CryptoBytes - Volume4, Number1, Summer 98.|
|[Bon98]||D.Boneh, "Twenty years of Attacks on the RSA Cryptosystem", Stanford University, 1998.|
|[Bor96]||J.Borst, "Differential-Linear Cryptanalysis of IDEA", ESAT-COSIC Technical Report 96-2, 1996.|
|[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.|
|[CJ98]||F.Chabaud, A.Joux, "Differential Collisions in SHA-0", Crypto '98 Proceedings, 1998.|
|[CW93]||K.W.Campbell, M.J.Wiener, "DES is not a group", Crypto '92, 1993.|
|[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.|
|[Den98]||D.Denning, message beginning "Eli Biham and Lars Knudsen have exposed a...", 3rd Apr 98.|
|[Dif98]||W.Diffie, "Whitfield Diffie interview", Lem Bingley interview with W.Diffie, Ziff-Davis, Nov 1998.|
|[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.|
|[Dob96]||H.Dobbertin, "The Status of MD5 After a Recent Attack", RSA CryptoBytes - Volume2, Number2, Summer 96.|
|[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.|
|[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]||"DIGITAL SIGNATURE STANDARD (DSS)", NIST Federal Information Processing Standards Publication 186, 1994.|
|[Fro95]||A.M.Froomkin, "THE METAPHOR IS THE KEY: CRYPTOGRAPHY, THE CLIPPER CHIP, AND THE CONSTITUTION", 1995.|
|[Gut99]||P.Gutmann, "Re: Opinions on S/MIME" sci.crypt USENET posting, 1st Jan 1999.|
|[GW96]||I.Goldberg, D.Wagner, "Randomness and the Netscape Browser - How secure is the World Wide Web?", Dr Dobbs Journal, 1996.|
|[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.|
|[Knu98]||D.Knuth, "The Art of Computer Programming Volume 2: Seminumerical Algorithms, 3rd Ed.", 1998.|
|[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, "Cryptanalytic Attacks on Pseudorandom Number Generators", 1998.|
|[Kob94]||N.Koblitz, "Graduate Texts in Mathematics: A course in number theory and cryptography", Springer, 1994.|
|[Lai91]||X.Lai, "Detailed Description and a Software Implementation of the IPES Cipher", Institute for Signal and Information Processing, ETH-Zentrum, Zurich, Switzerland, 1991.|
|[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.|
|[Luc98]||S.Lucks, "Attacking Triple Encryption", Fast Software Encryption, 1998.|
|[Mar98]||G.Marsaglia. Diehard Statistical Tests. http://stat.fsu.edu/~geo/ , 1998.|
|[Mau96]||U.Maurer, "Modelling a Public-Key Infrastructure", Proceedings 1996 European Symposium on Research in Computer Security (ESORICS '96), E.Bertino (Ed.), Lecture Notes in Computer Science, Berlin: Springer-Verlag, Rome, Italy, Sept 1996.|
|[MOV96]||A.J.Menezes, P.C.van Oorschot, S.A.Vanstone "Handbook of Applied Cryptography", CRC Press, 1996.|
|[MW96]||U.M.Maurer, S.Wolf, "On the Complexity of Breaking the Diffie-Hellman Protocol", Institute of Theoretical Computer Science, Switzerland, Apr 1996.|
|[CNET97]||"RSA under fire from IETF", CNet News.com Article, 3rd Nov 1997.|
|[Odl83]||A.M.Odlyzko, "Discrete Logarithms in finite fields and their cryptographic significance", 1993.|
|[Odl87]||A.M.Odlyzko, "On the Complexity of Computing Discrete Logarithms and Factoring Integers", Bell Laboratories, 1998.|
|[Odl95]||A.M.Odlyzko, "The Future of Integer Factorization", RSA CryptoBytes, Volume 1, Number 2, Summer 1995.|
|[PBD97]||B.Preneel, A.Bosselaers, H.Dobbertin, "The Cryptographic Hash Function RIPEMD-160", RSA CryptoBytes - Volume2, Number2, Autumn 97.|
|[Pet97]||S.Peterson, "Re: Why is IDEA only 128 bits " sci.crypt USENET posting, 18 Jul 1997.|
|[Pfl97]||C.P.Pfleeger, "Security in Computing", 1997.|
|[PGP96]||S.Schumacher, "Pretty Good Privacy version 2.6.3i - READ ME FIRST", 1996.|
|[PGP97]||PGP v5.00i, source code, 1997.|
|[PGP98]||NAI, "An Introduction to Cryptography", (as distributed with PGP v6.02), 1998.|
|[PKIX98]||M.Myers, C.Adams, D.Solo, D.Kemp, "Internet X.509 Certificate Request Message Format", May 1998.|
|[RFC1321]||R.Rivest, "The MD5 Message Digest Algorithm", RFC 1321, April 92.|
|[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.|
|[Rit98]||T.Ritter, "Re: Need help evaluating output from a PRNG" sci.crypt USENET posting, 10 Jan 1998.|
|[Rob95]||M.J.B.Robshaw, "Security Estimates for 512-bit RSA", RSA Labs, June 29.|
|[RSA78]||R.L.Rivest, A.Shamir, L.M.Adleman, "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems", Communications of the ACM, Feb 1978.|
|[RSA93]||"RSAREF License from RSA Data Security", RSA LABORATORIES, 5th January 1993.|
|[RSA94]||"RSAREF(TM) 2.0: A Free Cryptographic Toolkit General Information", Apr 1994.|
|[RSA96a]||RSA FAQ v3, 1996.|
|[RSA96b]||Bulletin 4, RSA Labs, Nov 1996.|
|[RSA98]||RSA FAQ v4, 1998. Available at http://www.rsa.com/|
|[Sch96a]||B.Schneier, "Applied Cryptography, Second Edition", Wiley, 1996.|
|[Sch96b]||B.Schneier, "Why Cryptography Is Harder Than It Looks", 1996.|
|[Sch97a]||J.Schiller, "Re: Question about the 'Freeware Version'" alt.security.pgp USENET posting, 19th June 1997.|
|[Sch97b]||B.Schneier, "S/MIME Crack? Beware press bearing incomplete stories more options" sci.crypt USENET posting, 26th Sept 1997.|
|[Sch98a]||B.Schneier, "Re: linux kernel loopback encryption", ailab.coderpunks mailing list, 16th July 1998.|
|[Sch98b]||B.Schneier, "Re: Candidates for AES" sci.crypt USENET posting, 26th October 1998.|
|[Sch98c]||B.Schneier, "CAST" ailab.coderpunks mailing list, 16th July 1998.|
|[Sch98d]||B.Schneier, "Re: Why CAST as default in PGP?" sci.crypt USENET posting, 20th October 1998.|
|[Sch98e]||B.Schneier, "Re: AES and Symmetric vs Asymmetric key length more options" sci.crypt USENET posting, 11th November 1998.|
|[Sch98f]||R.Schlafly, "Re: Opinions on S/MIME" sci.crypt USENET posting, 30th December 1998.|
|[Sch98g]||B.Schneier, "Re: DES better than AES?" sci.crypt USENET posting, 26th September 1998.|
|[Sch98h]||B.Schneier, "Re: DES better than AES?" sci.crypt USENET posting, 26th September 1998.|
|[SH97]||B.Schneier, C.Hall "An Improved E-Mail Security Protocol", 1997.|
|[Sho94]||P.W.Shor, "Algorithms for Quantum Computation: Discrete Logarithms and Factoring", 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1994.|
|[Sil97]||B.Silverman, "Fast Generation of Random, Strong RSA Primes", RSA Labs, 17th May 1997.|
|[Sil98]||B.Silverman, "Re: Primes" sci.crypt USENET posting, 2nd October 1998.|
|[Sim98]||S.Simpson, "Re: To All Morons Using Greater That 3000 Bit RSA Keys!!!! READ!!!", alt.security.pgp USENET posting, 29th November 1998.|
|[Sti95]||D.R.Stinson, "Cryptography - Theory and Practice", CRC Press, 1995.|
|[Tuc79]||W.Tuchman, "Hellman Presents No Shortcut Solutions To DES,", IEEE Spectrum, v. 16, n. 7, July 1979.|
|[TY98]||Y.Tsiounis, M.Yung, "On the Security of ElGamal based Encryption", 1998.|
|[Vau96]||S.Vaudenay "Hidden Collisions on DSS", June 1996.|
|[Wag94]||D.Wagner, "Re: Algorithms" sci.crypt USENET posting, 15th November 1994.|
|[Wie98]||M.J.Wiener, "Performance Comparison of Public-Key Cryptosystems", RSA CryptoBytes, Volume 4, Number 1, Summer 1998.|
|[WIR97a]||"S/MIME Cracked by a Screensaver", Wired News Article, 26th Sept 1997.|
|[WIR97b]||"RSA Blows Standards Smoke", Wired News Article, 31st Oct 1997.|
|[WIR98]||"PGP's 6.0: Cat Out of the Bag", Wired News Article, 3rd Sept 1998.|
|[Zim96]||"Interview with author of PGP (Pretty Good Privacy)", R.D.Hoffman interview with P.R.Zimmermann, Feb 1998.|
|[Zim98a]||"Letters to Phil", NAI Web site, http://www.pgp.com/, Dec 1998.|
|[Zim98b]||P.R.Zimmermann, message beginning "There is no advantage for using the keys larger than about 3000 bits.", 31st July 1998.|