# Critique of PGP Key Generation

### by C. H. Lindsey

Key generation is probably the weakest link in PGP, or indeed within any RSA system. Specifically, it is the one place where a supplier of a binary version of the system could insert a Trojan Horse that could not be detected by any feasible tests run on it.

For this reason, having just acquired a copy of PGP 2.6.3ia (that is 2.6.3 with Stale's patch) and being of a paranoid disposition (just as Phil recommends we should be), I decided to inspect the source code for the Key Generation part to see if it contained anything that might compromise the security of the keys that are generated. For the record, I found nothing of the sort. The code makes strenuous (even paranoid) attempts to be utterly fair. However, I did uncover a few features which, under a false pretence of adding extra security, actually made my task of finding the true security much harder. So I thought it might be useful to document the whole process, pointing out the (mis-)features that caused me concern.

## Overview of key generation

First, you have to generate two random numbers. So, for a 512-bit key, say, you generate two 256-bit randoms. Actually, it insists that the top two bits of the numbers are '1', so you actually generate numbers in the range (2**255 + 2**254) <= p <= (2**256 - 1), so there are 2**254 possible numbers obtained from 254 random bits.

The whole security of the system depends on the fact that the program must be capable of generating each and every one of these 2**254 numbers with more-or-less equal probability.

Next, from each random number, you generate a prime number. Starting from the given number, you look at each successive number to see if it might be prime. Well, obviously you only look at the odd ones, and in fact you only look at every 4th number because it tries to create "Blum" numbers for some reason or other. It uses a simple sieve to eliminate obvious cases, using a table of the first 1028 prime numbers. Anything that survives the sieve is subjected to Fermat's test (if ((x**(p-1)) mod p) != 1, then p is not prime) using xĀ=Ā2,3,5,7 (not a very original selection!). See the comments in genprime.c for why this is considered more than a safe test. It insists that the two prime numbers generated differ by at least 1 part in 128.

Next, you multiply your two (alleged) primes P and Q to give the modulus N, and from P, Q and N the RSA public and private keys can be derived in the standard manner. I estimate that for a 512-bit key there are 8*10**73 possible primes P and Q, giving 1.6*10**147 possibilities for N. For a 1024-bit key, these numbers are increased to 7*10*150 and 1.4*10**301. Which shows why a brute-force attack on an RSA encription is not feasible.

## Random number generation

But the whole system is only as good as its random number generator. It is no use using a pseudo-random number generator starting from some simple 32-bit seed (derived from the time of day, perhaps). That could generate, at most, 2**32 different keys (since pseudo-random sequences are deterministic and hence reproducible). You need 512 genuinely random bits to meet the criterion of being able to generate all possible 512-bit keys.

The system uses an array of (currently) 96 32-bit words (i.e. 384 bytes or 3072 bits) known as randPool, initially set to zero. The '96' is required for technical reasons to be a multiple of 4; otherwise I should much have preferred a prime number (such as 97). It proceeds by XOR-ing supposedly random bytes into successive positions; in practice always a multiple of 4 bytes at a time (this is a Bad Thing) derived from some 32-bit word. Some of these words are quite predictable, some are well correlated with other such words, and some are more-or-less independent. The crucial question is whether a sufficient number of them are truly independent, so as to ensure the required number of truly random bits.

A certain pattern of such words (examined in detail below) is XOR-ed in repeatedly (typically for 73 cycles when generating a 512-bit key). When randPool has been filled in it wraps around. But this could mean that the same positions in the pattern are being XOR-ed on top of themselves, which is why I would have preferred a prime number for the size of randPool, and why I would have preferred the pattern itself not to have been a multiple of 4 bytes. Now actually this problem should not matter, because the randPool is "stirred" between each cycle, and if the stirring is as effective as claimed it makes all the bits totally "forget" where they were. But I believe that this is a case where "not only should security be achieved, but security should be seen to be achieved". Stirring is described below. Typically, in a UNIX system, the randPool will be stirred and wrapped 11 times when generating a 512-bit key.

## Sources of randomness

At this stage, the system tells you to type at random on your keyboard, informing you that it is going to "measure the time intervals between your keystrokes". This is not quite true. The actual behaviour is somewhat complex but you do get, for each keystroke, a cycle of words XOR-ed into randPool, as described above. It would be a good idea to vary your rate of typing in a random manner - following the rhythm of a tune such as the William Tell Overture should generate plenty of randomness (but now I have said that, please choose your own tune in case some wily eavesdropper now does a statistical analysis of William Tell).

Different systems behave differently at this point, but they all assume that some form of "tick" is generated by the operating system for use for timing processes, giving the time-of-day, etc. In UNIX systems, there are two ticks: one fine tick (possibly every microsecond) derived from the real-time quartz clock for time-of-day purposes and one coarse tick (typically every 10 milliseconds) for timing processes (each time this tick occurs there is an interrupt, and whatever process is running at that moment is credited with 10msec of CPU time - or something like that). Since the random number generation works by counting ticks, it is important to know how coarse the ticks are.

All systems start off the same way (see noise.c):

A.
A call of clock(). Gives (supposedly) the number of microseconds of CPU used in this process. But it usually relies on a coarse tick (in Solaris it is always a multiple 0f 10000). Moreover it increases monotonically with successive calls, and successive calls frequently return the same value (this part of the process is not CPU-intensive). NOT a good source of randomness.
B.
A call of time(). Gives the number of seconds since Jan. 1st 1970. Even if the wily eavesdropper is not online to know the exact time you generated your key, the date on which you generated it is indeed public knowledge. Again, it increases monotonically and slowly (this phase of the process should take no more than half a minute all told). NOT a good source of randomness.
Now we need to distinguish between different systems.
1. MSDOS
C.
A call of pctimer0(). Seemingly this gives an unsigned (hopefully 32 bits) taken from a clock ticking every .84 Ąsec (but you will not get .84 Ąsec resolution). But, apart from being monotonic, the values returned should be as random as your typing.
2. MAC
C.
A call of TMTicks(). Again an unsigned long and presumably as random as your typing.
3. Win32 with the Microsoft compiler
C.
A call of QueryPerformanceCounter(). An unsigned (hopefully 32 bits) and presumably as random as your typing.
4. VMS
C.
A call of SYS\$GETTIM(). A unsigned long divided by 100000 (supposedly the ticks per update). Presumably as random as your typing, but you will not get much randomness unless you type rather slowly.
5. AMIGA
C.
A call of either ReadEClock() (the low-order 32 bits) or am_GetSysTime() (32 bits of seconds - presumably since 1970 - and 32 bits of microseconds). With am_GetSysTime() the seconds since 1970 is pretty useless, since we already had those bits under 'A.'. I could believe the microseconds or the output of ReadEClock().
D.
16 bits of additional noise from the video beam position. Neat!
E.
The ExecBase dispatch count, whatever that is.
6. ATARI
C.
A system call giving the 32-bit output from a 200Hz counter and 32 bits from a 50/60/70Hz Vertical BLank counter. Under the Pure C compiler, only the second of these is taken (the first is said to give the same as the earlier call of clock()). Coarse as these ticks are, they can still give true randomness if called often enough.
7. UNIX
This is the system I have studied most intensively. It calls every system call it can think of (and would have called the kitchen sink too, had it been available). As I will show, this just creates FUD rather than randomness.
C.
A call of gettimeofday(). This returns two 32-bit words. The first is just the seconds since 1970 which we have already had, so it adds nothing new. The second is the number of microseconds since the last full second. This should be based on a fine tick size, and is therefore as random as your typing, but it is the only source of true randomness that I would trust from the UNIX systems.
D.
A call of times(). This gives four 32-bit words:
D1.
The number of coarse ticks in this process. Typically this is one byte increasing monotonically but slowly. Useless as a source of randomness.
D2.
The number of coarse system ticks on behalf of this process. Increases perhaps twice as fast as the previous, but still useless.
D3.
The number of coarse ticks by children of this process. Always zero, because PGP never forks.
D4.
The number of coarse system ticks by children of this process. Also zero.
E.
Also from the same call of times(), the number of elapsed coarse ticks since boot time. The bottom byte changes somewhat faster than in the previous case (it is measuring elapsed rather than process time), but it is totally correlated with the call of clock() already made, so it adds nothing new.
At the end of all this, in all the systems, it XOR-es in the actual key you typed. This is good, but you likely only typed all lower-case letters, and probably not many punctuation or control counters, but every little helps. The bad news is that it still adds 4 bytes to randPool, even though only one byte is provided (see trueRandEvent()). Surely it would have been better to add just one byte here, at least to overcome the deficiency arising from the size of the randPool not being a prime number. Alternatively, the procedure randPoolAddBytes could have been made to discount any zero bytes at its most significant end.

It will be seen that the only source of true randomness in all the systems described above is the call labeled 'C.' in each case (also the actual key struck). Everything else is more-or-less predictable FUD. This call measures the instant at which each keystroke occurred (possibly modulo 1000000 microseconds) using as fine a tick size as the system affords. Where the call gives a time in microseconds rather than a tick count (systems AMIGA and UNIX) it endeavours to determine the ticksize by experiment. It usually overestimates it (which is safe) because the length of the measuring loop is longer than the ticksize (for example, on a SPARC1+ running Solaris2 it computes a ticksize of 11 microseconds, whereas the true value is believed to be 1 microsecond).

It now computes a time delta, modulo any computed ticksize, between this call and the previous one. Now this is a truly random estimate of the time between keystrokes but, oddly, this value is not incorporated into randPool (but it is represented by the absolute time of the keystroke which is so included). Since some systems (e.g. VMS and the ATARI) use an extremely coarse tick, this delta may contain only a few bits; so the bits in the delta are counted, and it carries on looking at keystrokes until the cumulative value of these bits is equal to the length of the key being generated (plus a few extra random bits needed for other purposes). But the maximum number of bits taken from any one keystroke is 8, so with fast systems it always demands more keystrokes than it truly needs.

I believe that when counting the bits in delta it counts one more than it should (the most significant bit should not be included in the count in my opinion, since it is by definition always '1', and only serves to tell how many bits there are). Note also that the value of delta will have a minimum (it is impossible for the time between two keystrokes to be absolutely zero), which is a further reason to discount one bit when counting the bits of delta (see trueRandEvent()).

## Stirring

Each time the randPool wraps around, and again at the end, it is "stirred" (see randPoolStir()). Note that stirring is a completely deterministic process, exactly repeatable on different systems (it even converts the entire randPool to big-endian format beforehand in order to ensure this, and afterwards rewrites each group of four bytes the other way round).

Its modus operandi is to encrypt the whole randPool using the MD5 algorithm (that's the one used for generating signatures). That is why the length of the randPool had to be a multiple of 128 bits. It starts from a zero key (I told you it was going to be deterministic). By the time it has got to the end of the randPool, the last bit deposited is a function of every preceding bit. It then encrypts it all again, so that every bit in the new randPool now potentially depends on every bit in the original randPool. If you start with an empty randPool, it will now appear (to the naked eye) as a complete random mess. If you start with a randPool containing just one bit, it will appear as a different random mess.

Finally, it takes the first 128 bits of the new randPool as the starting key for the next stirring.

Note that the purpose of stirring is not to generate extra randomness. It is merely to redistribute the randomness that is already there (derived from your keystrokes). So it matters not whether the randPool is completely filled by your keystroking, nor that the two sets of 256 bits that you extract from it in order to make your 512-bit key are not the actual bits that you randomly generated. Every bit that you take out is affected by every random bit that you put in (as well as by the many non-random, predictable bits that the UNIX systems in particular insert). So it is still possible to generate each and every one of the possible keys, even when starting from a known time of day and a known number of ticks since boot time.

## Conclusion

So is it safe? The answer seems to be Yes.

But I would have been much happier to see a more straightforward system. It took me three days and much work examining the operation of the program with a debugger to convince myself that most of the information being put into randPool was mere FUD, and to identify exactly where the true required number of random bits was actually coming from. In view of the excellent performance of randPoolStir(), I cannot see that any of the FUD serves any useful purpose. Well, I can see a little merit in including the time and date just once, in case any extremely regular typist manages to get exactly the same intervals between each keystroke during her interpretation of William Tell, or unless some other flaw is found in the system (which is also why I would like to see the wrap-around of the randPool a little less regular, as already mentioned).

So I have at last felt it safe to generate my PGP key. And as a final check against any wily eavesdropper who may have been compiling a complete set of all the popular 1024-bit keys (will the universe last long enough for him to complete his task?) I have set my key length to 820 bits. How's that for paranoia?