This FAQ is a small side project I have decided to undertake. It was originally just going to be a rather lengthy spur-of-the moment post to alt.2600 in order to clear up some incorrect assumptions and perceptions people had about the security of PGP. It has grown well beyond that...
There are a great many misconceptions out there about how vulnerable Pretty Good Privacy is to attack. This FAQ is designed to shed some light on the subject. It is not an introduction to PGP or cryptography. If you are not at least conversationally versed in either topic, readers are directed to The Infinity Concept issue 1, and the sci.crypt FAQ. Both documents are available via ftp from infonexus.com. This document can be found there as well.
PGP is a hybrid cryptosystem. It is made up of 4 cryptographic elements: It contains a symmetric cipher (IDEA), an asymmetric cipher (RSA), a one-way hash (MD5), and a random number generator (Which is two-headed, actually: it samples entropy from the user and then uses that to seed a PRNG). Each is subject to a different form of attack.
IDEA, finalized in 1992 by Lai and Massey is a block cipher that operates on 64-bit blocks of data. There have be no advances in the cryptanalysis of standard IDEA that are publicly known. (I know nothing of what the NSA has done, nor does most anyone.) The only method of attack, therefore, is brute force.
As we all know the keyspace of IDEA is 128-bits. In base 10 notation that is:
To recover a particular key, one must, on average, search half the keyspace. That is 127 bits:
If you had 1,000,000,000 machines that could try 1,000,000,000 keys/sec, it would still take all these machines longer than the universe as we know it has existed and then some, to find the key. IDEA, as far as present technology is concerned, is not vulnerable to brute-force attack, pure and simple.
If we cannot crack the cipher, and we cannot brute force the key-space, what if we can find some weakness in the PRNG used by PGP to generate the pseudo-random IDEA session keys? This topic is covered in more detail in section 4.
RSA, the first full fledged public key cryptosystem was designed by Rivest, Shamir, and Adleman in 1977. RSA gets it's security from the apparent difficulty in factoring very large composites. However, nothing has been proven with RSA. It is not proved that factoring the public modulus is the only (best) way to break RSA. There may be an as yet undiscovered way to break it. It is also not proven that factoring *has* to be as hard as it is. There exists the possibility that an advance in number theory may lead to the discovery of a polynomial time factoring algorithm. But, none of these things has happened, and no current research points in that direction. However, 3 things that are happening and will continue to happen that take away from the security of RSA are: the advances in factoring technique, computing power and the decrease in the cost of computing hardware. These things, especially the first one, work against the security of RSA. However, as computing power increases, so does the ability to generate larger keys. It is *much* easier to multiply very large primes than it is to factor the resulting composite (given today's understanding of number theory).
To understand the attacks on RSA, it is important to understand how RSA works. Briefly:
Encryption is done by dividing the target message into blocks smaller than n and by doing modular exponentiation:
c=m^e mod n
Decryption is simply the inverse operation:
m=c^d mod n
An attacker has access to the public-key. In other words, the attacker has e and n. The attacker wants the private key. In other words the attacker wants d. To get d, n needs to be factored (which will yield p and q, which can then be used to calculate d). Factoring n is the best known attack against RSA to date. (Attacking RSA by trying to deduce (p-1)(q-1) is no easier than factoring n, and executing an exhaustive search for values of d is harder than factoring n.) Some of the algorithms used for factoring are as follows:
- Trial division: The oldest and least efficient. Exponential running time. Try all the prime numbers less than sqrt(n).
- Quadratic Sieve (QS): The fastest algorithm for numbers smaller than 110 digits.
- Multiple Polynomial Quadratic Sieve (MPQS): Faster version of QS.
- Double Large Prime Variation of the MPQS: Faster still.
- Number Field Sieve (NFS): Currently the fastest algorithm known for numbers larger than 110 digits. Was used to factor the ninth Fermat number.
These algorithms represent the state of the art in warfare against large composite numbers (therefore against RSA). The best algorithms have a super-polynomial (sub-exponential) running time, with the NFS having an asymptotic time estimate closest to polynomial behavior.
Still, factoring large numbers is hard. However, with the advances in number theory and computing power, it is getting easier. In 1977 Ron Rivest said that factoring a 125-digit number would take 40 quadrillion years. In 1994 RSA129 was factored using about 5000 MIPS-years of effort from idle CPU cycles on computers across the Internet for eight months. In 1995 the Blacknet key (116 digits) was factored using about 400 MIPS-years of effort (1 MIPS-year is a 1,000,000 instruction per second computer running for one year) from several dozen workstations and a MasPar for about three months. Given current trends the keysize that can be factored will only increase as time goes on. The table below estimates the effort required to factor some common PGP-based RSA public-key modulus lengths using the General Number Field Sieve:
KeySize MIPS-years required to factor ----------------------------------------------------------------- 512 30,000 768 200,000,000 1024 300,000,000,000 2048 300,000,000,000,000,000,000
The next chart shows some estimates for the equivalences in brute force key searches of symmetric keys and brute force factoring of asymmetric keys, using the NFS.
Symmetric Asymmetric ------------------------------------------------------------------ 56-bits 384-bits 64-bits 512-bits 80-bits 768-bits 112-bits 1792-bits 128-bits 2304-bits
It was said by the 4 men who factored the Blacknet key that "Organizations with 'more modest' resources can almost certainly break 512-bit keys in secret right now." This is not to say that such an organization would be interested in devoting so much computing power to break just anyone's messages. However, most people using cryptography do not rest comfortably knowing the system they trust their secrets to can be broken...
My advice as always is to use the largest key allowable by the implementation. If the implementation does not allow for large enough keys to satisfy your paranoia, do not use that implementation.
These attacks do not exhibit any profound weakness in RSA itself, just in certain implementations of the protocol. Most are not issues in PGP.
An attacker listens in on the insecure channel in which RSA messages are passed. The attacker collects an encrypted message c, from the target (destined for some other party). The attacker wants to be able to read this message without having to mount a serious factoring effort. In other words, she wants m=c^d.
To recover m, the attacker first chooses a random number, r < n. (The attacker has the public-key (e,n).) The attacker computes:
x=r^e mod n (She encrypts r with the target's public-key)
y=xc mod n (Multiplies the target ciphertext with the temp)
t=r^-1 mod n (Multiplicative inverse of r mod n)
The attacker counts on the fact that:
If x=r^e mod n, Then r=x^d mod n
The attacker then gets the target to sign y with her private-key, (which actually decrypts y) and sends u=y^d mod n to the attacker. The attacker simply computes:
tu mod n = (r^-1)(y^d) mod n = (r^-1)(x^d)(c^d) mod n = (c^d) mod n = m
To foil this attack do not sign some random document presented to you. Sign a one-way hash of the message instead.
As it turns out, e being a small number does not take away from the security of RSA. If the encryption exponent is small (common values are 3,17, and 65537) then public-key operations are significantly faster. The only problem in using small values for e as a public exponent is in encrypting small messages. If we have 3 as our e and we have an m smaller than the cubic root of n, then the message can be recovered simply by taking the cubic root of m because:
m [for m < 3rdroot(n)]^3 mod n will be equivalent to m^3
3rdroot(m^3) = m.
To defend against this attack, simply pad the message with a nonce before encryption, such that m^3 will always be reduced mod n.
PGP uses a small e for the encryption exponent, by default it tries to use 17. If it cannot compute d with e being 17, PGP will iterate e to 19, and try again... PGP also makes sure to pad m with a random value so m > n.
A very new area of attack publicly discovered by Paul Kocher deals with the fact that different cryptographic operations (in this case the modular exponentiation operations in RSA) take discretely different amounts of time to process. If the RSA computations are done without the Chinese Remainder theorem, the following applies:
An attacker can exploit slight timing differences in RSA computations
to, in many cases, recover d. The attack is a passive one where the attacker sits on a network and observes the RSA operations.
The attacker passively observes k operations measuring the time t it takes to compute each modular exponentiation operation: m=c^d mod n. The attacker also knows c and n. Pseudo code of the attack is:
Algorithm to compute m=c^d mod n: Let m0 = 1. Let c0 = x. For i=0 upto (bits in d-1): If (bit i of d) is 1 then Let mi+1 = (mi * ci) mod n. Else Let mi+1 = mi. Let di+1 = di^2 mod n. End.
This is very new (the public announcement was made on 12/7/95) and intense scrutiny of the attack has not been possible. However, Ron Rivest had this to say about countering it:
From: Ron Rivest
Newsgroups: sci.crypt Subject: Re: Announce: Timing cryptanalysis of RSA, DH, DSS Date: 11 Dec 1995 20:17:01 GMT Organization: MIT Laboratory for Computer Science The simplest way to defeat Kocher's timing attack is to ensure that the cryptographic computations take an amount of time that does not depend on the data being operated on. For example, for RSA it suffices to ensure that a modular multiplication always takes the same amount of time, independent of the operands. A second way to defeat Kocher's attack is to use blinding: you "blind" the data beforehand, perform the cryptographic computation, and then unblind afterwards. For RSA, this is quite simple to do. (The blinding and unblinding operations still need to take a fixed amount of time.) This doesn't give a fixed overall computation time, but the computation time is then a random variable that is independent of the operands. ============================================================================== Ronald L. Rivest 617-253-5880 617-253-8682(Fax) firstname.lastname@example.org ==============================================================================
The blinding Rivest speaks of simply introduces a random value into the decryption process. So,
m = c^d mod n
m = r^-1(cr^e)^d mod n
r is the random value, and r^-1 is it's inverse.
PGP is not vulnerable to the timing attack as it uses the CRT to speed RSA operations. Also, since the timing attack requires an attacker to observe the cryptographic operations in real time (ie: snoop the decryption process from start to finish) and most people encrypt and decrypt off-line, it is further made impractical.
While the attack is definitely something to be wary of, it is theoretical in nature, and has not been done in practice as of yet.
There are other attacks against RSA, such as the common modulus attack in which several users share n, but have different values for e and d. Sharing a common modulus with several users, can enable an attacker to recover a message without factoring n. PGP does not share public-key modulus' among users.
If d is up to one quarter the size of n and e is less than n, d can be recovered without factoring. PGP does not choose small values for the decryption exponent. (If d were too small it might make a brute force sweep of d values feasible which is obviously a bad thing.)
It is pointless to make predictions for recommended keysizes. The breakneck speed at which technology is advancing makes it difficult and dangerous. Respected cryptographers will not make predictions past 10 years and I won't embarrass myself trying to make any. For today's secrets, a 1024-bit is probably safe and a 2048-bit key definitely is. I wouldn't trust these numbers past the end of the century. However, it is worth mentioning that RSA would not have latest this long if it was as fallible as some crackpots with middle initials would like you to believe.
MD5 is the one-way hash used to hash the passphrase into the IDEA key and to sign documents. Message Digest 5 was designed by Rivest as a successor to MD4 (which was found to be weakened with reduced rounds). It is slower but more secure. Like all one-way hash functions, MD5 takes an arbitrary-length input and generates a unique output.
The strength of any one-way hash is defined by how well it can randomize an arbitrary message and produce a unique output. There are two types of brute force attacks against a one-way hash function, pure brute force (my own terminology) and the birthday attack.
The output of MD5 is 128-bits. In a pure brute force attack, the attacker has access to the hash of message H(m). She wants to find another message m' such that: H(m) = H(m').
To find such message (assuming it exists) it would take a machine that could try 1,000,000,000 messages per second about 1.07E22 years. (To find m would require the same amount of time.)
Find two messages that hash to the same value is known as a collision and is exploited by the birthday attack.
The birthday attack is a statistical probability problem. Given n inputs and k possible outputs, (MD5 being the function to take n -> k) there are n(n-1)/2 pairs of inputs. For each pair, there is a probability of 1/k of both inputs producing the same output. So, if you take k/2 pairs, the probability will be 50% that a matching pair will be found. If n > sqrt(k), there is a good chance of finding a collision. In MD5's case, 2^64 messages need to be tried. This is not a feasible attack given today's technology. If you could try 1,000,000 messages per second, it would take 584,942 years to find a collision. (A machine that could try 1,000,000,000 messages per second would take 585 years, on average.)
For a successful account of the birthday against crypt(3), see: url: ftp://ftp.infonexus.com/pub/Philes/Cryptography/crypt3Collision.txt.gz
Differential cryptanalysis has proven to be effective against one round of MD5, but not against all 4 (differential cryptanalysis looks at ciphertext pairs whose plaintexts has specific differences and analyzes these differences as they propagate through the cipher).
There was successful attack at the compression function itself that produces collisions, but this attack has no practical impact the security. If your copy of PGP has had the MD5 code altered to cause these collisions, it would fail the message digest verification and you would reject it as altered... Right?
According to conventional information theory, the English language has about 1.3 bits of entropy (information) per 8-bit character. If the pass phrase entered is long enough, the resulting MD5 hash will be statistically random. For the 128-bit output of MD5, a pass phrase of about 98 characters will provide a random key:
(8/1.3) * (128/8) = (128/1.3) = 98.46 characters
How many people use a 98 character passphrase for their secret-key in PGP? Below are 98 characters...
1.3 comes from the fact that an arbitrary readable English sentence is usually going to consist of certain letters, (e,r,s, and t are statistically very common) thereby reducing it's entropy. If any of the 26 letters in the Latin alphabet were equally possible and likely (which is seldom the case) the entropy increases. The so-called absolute rate would, in this case, be:
log(26) / log(2) = 4.7 bits
In this case of increased entropy, a password with a truly random sequence of English characters will only need to be:
(8/4.7) * (128/8) = (128/4.7) = 27.23 characters
For more info on passphrase length, see the PGP passphrase FAQ
PGP employs 2 PRNG's to generate and manipulate (pseudo) random data. The ANSI X9.17 generator and a function which measures the entropy from the latency in a user's keystrokes. The random pool (which is the randseed.bin file) is used to seed the ANSI X9.17 PRNG (which uses IDEA, not 3DES). Randseed.bin is initially generated from trueRand which is the keystroke timer. The X9.17 generator is pre-washed with an MD5 hash of the plaintext and postwashed with some random data which is used to generate the next randseed.bin file. The process is broken up and discussed below.
The ANSI X9.17 is the method of key generation PGP uses. It is officially specified using 3DES, but was easily converted to IDEA. X9.17 requires 24 bytes of random data from randseed.bin. (PGP keeps an extra 384 bytes of state information for other uses...) When cryptRand starts, the randseed.bin file is washed (see below) and the first 24-bytes are used to initialize X9.17. It works as follows:
E() = an IDEA encryption, with a reusable key used for key generation
T = timestamp (data from randseed.bin used in place of timestamp)
V = Initialization Vector, from randseed.bin
R = random session key to be generated
R = E[E(T) XOR V]
the next V is generated thusly:
V = E[E(T) XOR R]
The trueRand generator attempts to measure entropy from the latency of a user's keystrokes every time the user types on the keyboard. It is used to generate the initial randseed.bin which is in turn used to seed to X9.17 generator. The quality of the output of trueRand is dependent upon it's input. If the input has a low amount of entropy, the output will not be as random as possible. In order to maximize the entropy, the keypresses should be spaced as randomly as possible.
In most situations, the attacker does not know the content of the plaintext being encrypted by PGP. So, in most cases, washing the X9.17 generator with an MD5 hash of the plaintext, simply adds to security. This is based on the assumption that this added unknown information will add to the entropy of the generator. If, in the event that the attacker has some information about the plaintext (perhaps the attacker knows which file was encrypted, and wishes to prove this fact) the attacker may be able to execute a known-plaintext attack against X9.17. However, it is not likely that, with all the other precautions taken, that this would weaken the generator.
The randseed is washed before and after each use. In PGP's case a wash is an IDEA encryption in cipher-feedback mode. Since IDEA is considered secure (see section 1), it should be just as hard to determine the 128-bit IDEA key as it is to glean any information from the wash. The IDEA key used is the MD5 hash of the plaintext and an initialization vector of zero. The IDEA session key is then generated as is an IV. The postwash is considered more secure. More random bytes are generated to reinitialize randseed.bin. These are encrypted with the same key as the PGP encrypted message. The reason for this is that if the attacker knows the session key, she can decrypt the PGP message directly and would have no need to attack the randseed.bin. (A note, the attacker might be more interested in the state of the randseed.bin, if they were attacking all messages, or the message that the user is expected to send next).
Most of the attacks outlined above are either not possible or not feasible by the average adversary. So, what can the average cracker do to subvert the otherwise stalwart security of PGP? As it turns out, there are several "doable" attacks that can be launched by the typical cracker. They do not attack the cryptosystem protocols themselves, (which have shown to be secure) but rather system specific implementations of PGP.
These attacks do not do need to do anything proactive and can easily go undetected.
Still a very effective method of attack, keypress snooping can subvert the security of the strongest cryptosystem. If an attacker can install a keylogger, and capture the passphrase of an unwary target, then no cryptanalysis whatsoever is necessary. The attacker has the passphrase to unlock the RSA private key. The system is completely compromised. The methods vary from system to system, but I would say DOS-based PGP would be the most vulnerable. DOS is the easiest OS to subvert, and has the most key-press snooping tools that I am aware of. All an attacker would have to do would be gain access to the machine for under 5 minutes on two separate occasions and the attack would be complete. The first time to install the snooping software, the second time, to remove it, and recover the goods. (If the machine is on a network, this can all be done *remotely* and the ease of the attack increases greatly.) Even if the target boots clean, not loading any TSR's, a boot sector virus could still do the job, transparently. Just recently, the author has discovered a key logging utility for Windows, which expands this attack to work under Windows-based PGP shells (this logger is available from the infonexus via ftp, BTW). ftp://ftp.infonexus.com/pub/ToolsOfTheTrade/DOS/KeyLoggers/
Keypress snooping under Unix is a bit more complicated, as root access is needed, unless the target is entering her passphrase from an X-Windows GUI. There are numerous key loggers available to passively observe keypresses from an X-Windows session. Check: ftp://ftp.infonexus.com/pub/SourceAndShell/Xwindows/
The original invisible threat. Below is a clip from a posting by noted information warfare guru Winn Schwartau describing a Van Eck attack:
Van Eck Radiation Helps Catch Spies "Winn Schwartau" < email@example.com > Thu, 24 Feb 94 14:13:19 -0500 Van Eck in Action Over the last several years, I have discussed in great detail how the electromagnetic emissions from personal computers (and electronic gear in general) can be remotely detected without a hard connection and the information on the computers reconstructed. Electromagnetic eavesdropping is about insidious as you can get: the victim doesn't and can't know that anyone is 'listening' to his computer. To the eavesdropper, this provides an ideal means of surveillance: he can place his eavesdropping equipment a fair distance away to avoid detection and get a clear representation of what is being processed on the computer in question. (Please see previous issues of Security Insider Report for complete technical descriptions of the techniques.) The problem, though, is that too many so called security experts, (some prominent ones who really should know better) pooh-pooh the whole concept, maintaining they've never seen it work. Well, I'm sorry that none of them came to my demonstrations over the years, but Van Eck radiation IS real and does work. In fact, the recent headline grabbing spy case illuminates the point. Exploitation of Van Eck radiation appears to be responsible, at least in part, for the arrest of senior CIA intelligence officer Aldrich Hazen Ames on charges of being a Soviet/Russian mole. According to the Affidavit in support of Arrest Warrant, the FBI used "electronic surveillance of Ames' personal computer and software within his residence," in their search for evidence against him. On October 9, 1993, the FBI "placed an electronic monitor in his (Ames') computer," suggesting that a Van Eck receiver and transmitter was used to gather information on a real-time basis. Obviously, then, this is an ideal tool for criminal investigation - one that apparently works quite well. (From the Affidavit and from David Johnston, "Tailed Cars and Tapped Telephones: How US Drew Net on Spy Suspects," New York Times, February 24, 1994.) From what we can gather at this point, the FBI black-bagged Ames' house and installed a number of surveillance devices. We have a high confidence factor that one of them was a small Van Eck detector which captured either CRT signals or keyboard strokes or both. The device would work like this: A small receiver operating in the 22MHz range (pixel frequency) would detect the video signals minus the horizontal and vertical sync signals. Since the device would be inside the computer itself, the signal strength would be more than adequate to provide a quality source. The little device would then retransmit the collected data in real-time to a remote surveillance vehicle or site where the video/keyboard data was stored on a video or digital storage medium. At a forensic laboratory, technicians would recreate the original screens and data that Mr. Ames entered into his computer. The technicians would add a vertical sync signal of about 59.94 Hz, and a horizontal sync signal of about 27KHz. This would stabilize the roll of the picture. In addition, the captured data would be subject to "cleansing" - meaning that the spurious noise in the signal would be stripped using Fast Fourier Transform techniques in either hardware or software. It is likely, though, that the FBI's device contained within it an FFT chip designed by the NSA a couple of years ago to make the laboratory process even easier. I spoke to the FBI and US Attorney's Office about the technology used for this, and none of them would confirm or deny the technology used "on an active case." Of course it is possible that the FBI did not place a monitoring device within the computer itself, but merely focused an external antenna at Mr. Ames' residence to "listen" to his computer from afar, but this presents additional complexities for law enforcement. 1. The farther from the source the detection equipment sits means that the detected information is "noisier" and requires additional forensic analysis to derive usable information. 2. Depending upon the electromagnetic sewage content of the immediate area around Mr. Ames' neighborhood, the FBI surveillance team would be limited as to what distances this technique would still be viable. Distance squared attenuation holds true. 3. The closer the surveillance team sits to the target, the more likely it is that their activities will be discovered. In either case, the technology is real and was apparently used in this investigation. But now, a few questions arise. 1. Does a court surveillance order include the right to remotely eavesdrop upon the unintentional emanations from a suspect's electronic equipment? Did the warrants specify this technique or were they shrouded under a more general surveillance authorization? Interesting question for the defense. 2. Is the information garnered in this manner admissible in court? I have read papers that claim defending against this method is illegal in the United States, but I have been unable to substantiate that supposition. 3. If this case goes to court, it would seem that the investigators would have to admit HOW they intercepted signals, and a smart lawyer (contradictory allegory :-) would attempt to pry out the relevant details. This is important because the techniques are generally classified within the intelligence community even though they are well understood and explained in open source materials. How will the veil of national security be dropped here? To the best of my knowledge, this is the first time that the Government had admitted the use of Van Eck (Tempest Busting etc.) in public. If anyone knows of any others, I would love to know about it.
The relevance to PGP is obvious, and the threat is real. Snooping the passphrase from the keyboard, and even whole messages from the screen are viable attacks. This attack, however exotic it may seem, is not beyond the capability of anyone with some technical know-how and the desire to read PGP encrypted files.
In a multi-user system such as Unix, the physical memory of the machine can be examined by anyone with the proper privileges (usually root). In comparison with factoring a huge composite number, opening up the virtual memory of the system (/dev/kmem) and seeking to a user's page and directly reading it, is trivial.
In multitasking environments such as Windows, the OS has a nasty habit of paging the contents of memory to disk, usually transparently to the user, whenever it feels the need to free up some RAM. This information can sit, in the clear, in the swapfile for varying lengths of time, just waiting for some one to come along and recover it. Again, in a networked environment where machine access can be done with relative impunity, this file can be stolen without the owner's consent or knowledge.
If you use PGP on a host which you access remotely, you can be vulnerable to this attack. Unless you use some sort of session encrypting utility, such as SSH, DESlogin, or some sort of network protocol stack encryption (end to end or link by link) you are sending your passphrase, and messages across in the clear. A packet sniffer sitting at a intermediate point between your terminal can capture all this information quietly and efficiently. Packet sniffers are available at the infonexus: ftp://ftp.infonexus.com/pub/SourceAndShell/Sniffers/
These attacks are more proactive in nature and tend to be a bit more difficult to wage.
The age old trojan horse attack is still a very effective means of compromise. The concept of a trojan horse should not be foreign to anyone. An apparently harmless program that in reality is evil and does potentially malicious things to your computer. How does this sound...: Some 31it3 coder has come up with a k3wl new Windows front-end to PGP. All the newbies run out and ftp a copy. It works great, with a host of buttons and scrollbars, and it even comes with a bunch of *.wav files and support for a SB AWE 32 so you can have the 16-bit CD quality sound of a safe locking when you encrypt your files. It runs in a tiny amount of memory, coded such that nothing leaks, it intercepts OS calls that would otherwise have it's contents paged to disk and makes sure all the info stays in volatile memory. It works great (the first Windows app that does). Trouble is, this program actually has a few lines of malevolent code that record your secret-key passphrase, and if it finds a modem (who doesn't have a modem these days?) it 'atm0's the modem and dials up a hard coded number to some compromised computer or modem bank and sends the info through.
Possible? Yes. Likely? No.
The code to PGP is publicly available. Therefore it is easy to modify. If someone were to modify the source code to PGP inserting a sneaky backdoor and leave it at some distribution point, it could be disastrous. However, it is also very easy to detect. Simply verify the checksums. Patching the MD5 module to report a false checksum is also possible, so verify using a known good copy. A more devious attack would be to modify the code, compile it and surreptitiously plant it in the target system. In a networked environment this can be done without ever having physical access to the machine.
I have presented factual data, statistical data, and projected data. Form your own conclusions. Perhaps the NSA has found a polynomial-time (read: *fast*) factoring algorithm. But we cannot dismiss an otherwise secure cryptosystem due to paranoia. Of course, on the same token, we cannot trust cryptosystems on hearsay or assumptions of security. Bottom line is this: in the field of computer security, it pays to be cautious. But it doesn't pay to be un-informed or needlessly paranoid. Know the facts.
PRZ, Collin Plumb, Paul Kocher, Bruce Schneier, Paul Rubin, Stephen McCluskey, Adam Back, Bill Unruh, Ben Cantrick and the readers of sci.crypt and the comp.security.* groups,
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