RFC 1750
Randomness Recommendations for Security
RFC 1750: Randomness Recommendations for Security ✨
okay bestie, let's get into it 💖 this is RFC 1750, published in December 1994 under the Network Working Group. it's Informational, which means it's giving "here's some info" energy, not "you must do this" 🫶
- Authors: D. Eastlake, 3rd (DEC), S. Crocker (Cybercash), and J. Schiller (MIT)
- Category: Informational
- Date: December 1994
- Obsoletes / Updates: the title block doesn't list anything it obsoletes or updates, so it's standing on its own 🦋
Status of this Memo 📋
this memo is info for the Internet community. it does not specify an Internet standard of any kind 🍓 and distribution is unlimited, so anyone can pass it around with no limits 💅
Abstract 🔮
okay so security systems today are built on super strong crypto algorithms that are really good at foiling pattern analysis 💪 BUT the whole thing depends on generating secret stuff like passwords and crypto keys 🔑. if you make those secrets with pseudo-random processes, you can end up with pseudo-security 🙈, which looks secure but isn't. a sophisticated attacker might not even need to crack the algorithm. they could just recreate the environment that made the secret and search the small set of possibilities it could produce, instead of hunting through the whole number space. lowkey the lazy route, and it works 💀
picking random stuff that can hold up against a resourceful, motivated adversary is surprisingly hard 😭. this paper calls out a bunch of traps people fall into with traditional pseudo-random number generation 🪤. it recommends truly random hardware techniques and points out that a lot of systems already have hardware that can do this 🎀. if you don't have that hardware, it offers ways to make things better anyway 🧰. it also gives examples of how big these quantities need to be for particular applications 📏
Acknowledgements 🫶
shoutout to everyone whose comments got incorporated into this doc. they're listed in alphabetical order 🌸
- David M. Balenson (TIS)
- Don Coppersmith (IBM)
- Don T. Davis (consultant)
- Carl Ellison (Stratus)
- Marc Horowitz (MIT)
- Christian Huitema (INRIA)
- Charlie Kaufman (IRIS)
- Steve Kent (BBN)
- Hal Murray (DEC)
- Neil Haller (Bellcore)
- Richard Pitkin (DEC)
- Tim Redmond (TIS)
- Doug Tygar (CMU)
Table of Contents 🗺️
1. Introduction........................................... 3
2. Requirements........................................... 4
3. Traditional Pseudo-Random Sequences.................... 5
4. Unpredictability....................................... 7
4.1 Problems with Clocks and Serial Numbers............... 7
4.2 Timing and Content of External Events................ 8
4.3 The Fallacy of Complex Manipulation.................. 8
4.4 The Fallacy of Selection from a Large Database....... 9
5. Hardware for Randomness............................... 10
5.1 Volume Required...................................... 10
5.2 Sensitivity to Skew.................................. 10
5.2.1 Using Stream Parity to De-Skew..................... 11
5.2.2 Using Transition Mappings to De-Skew............... 12
5.2.3 Using FFT to De-Skew............................... 13
5.2.4 Using Compression to De-Skew....................... 13
5.3 Existing Hardware Can Be Used For Randomness......... 14
5.3.1 Using Existing Sound/Video Input................... 14
5.3.2 Using Existing Disk Drives......................... 14
6. Recommended Non-Hardware Strategy..................... 14
6.1 Mixing Functions..................................... 15
6.1.1 A Trivial Mixing Function.......................... 15
6.1.2 Stronger Mixing Functions.......................... 16
6.1.3 Diff-Hellman as a Mixing Function.................. 17
6.1.4 Using a Mixing Function to Stretch Random Bits..... 17
6.1.5 Other Factors in Choosing a Mixing Function........ 18
6.2 Non-Hardware Sources of Randomness................... 19
6.3 Cryptographically Strong Sequences................... 19
6.3.1 Traditional Strong Sequences....................... 20
6.3.2 The Blum Blum Shub Sequence Generator.............. 21
7. Key Generation Standards.............................. 22
7.1 US DoD Recommendations for Password Generation....... 23
7.2 X9.17 Key Generation................................. 23
8. Examples of Randomness Required....................... 24
8.1 Password Generation................................. 24
8.2 A Very High Security Cryptographic Key............... 25
8.2.1 Effort per Key Trial............................... 25
8.2.2 Meet in the Middle Attacks......................... 26
8.2.3 Other Considerations............................... 26
9. Conclusion............................................ 27
10. Security Considerations.............................. 27
References............................................... 28
Authors' Addresses....................................... 301. Introduction 🌈
so software cryptography is going mainstream 📈. stuff like Kerberos, PEM, PGP, etc. is maturing and becoming part of the network landscape [PEM] ✨. these systems give you solid protection against snooping and spoofing 🛡️. BUT there's a potential flaw 👀. at the heart of every crypto system is generating secret, unguessable (i.e., random) numbers 🎲.
right now, there aren't generally available facilities for generating those unpredictable numbers, and honestly? it's an open wound in crypto software design 🩹. if you're a dev building a key or password generator that has to run on lots of different hardware, the only safe move so far has been forcing the local install to supply a random number routine. awkward, error-prone, and unpalatable, to put it mildly 🙈
keep this in mind, bestie 🔑: the goal is data that an adversary has a very low chance of guessing or figuring out. it will fail if your pseudo-random data only passes traditional statistical randomness tests, or if it's based on limited-range sources like clocks ⏰. those random-ish quantities are often determinable by an adversary searching through an embarrassingly small space of possibilities 💀
this informational doc suggests techniques for making random quantities that hold up against that kind of attack 🛡️. it recommends that future systems include hardware random number generation, or at least give access to existing hardware that can do the job 🔌. got no hardware? it suggests methods for that case too 🧰. and it gives some estimates of how many random bits you'd need for sample applications 🔢
2. Requirements 📝
probably the most common randomness requirement you'll run into is the user password 🔐, and it's usually just a simple character string. obviously, if someone can guess your password, it's not providing any security 🙃. (for reusable passwords, it's desirable that people can actually remember them, so it might be a good idea to go with pronounceable strings or phrases made of ordinary words. but that only changes the format of the password info, not the requirement that it be very hard to guess 🧠.)
lots of other requirements come from the crypto world 🔮. crypto techniques can provide a bunch of services, including confidentiality and authentication 🤫. those services rest on quantities, traditionally called "keys", that are unknown to and unguessable by an adversary 🗝️.
in some cases, like symmetric encryption with one time pads [CRYPTO*] or the US Data Encryption Standard [DES], everyone who wants to talk confidentially and/or with authentication must all know the same secret key 🤝. in other cases, using asymmetric or "public key" crypto techniques, keys come in pairs 👯. one key is private and must be kept secret by one party, while the other is public and can be published to the whole world 🌍. it's computationally infeasible to work out the private key from the public key [ASYMMETRIC, CRYPTO*] 🧮
how often you need random stuff, and how much, varies a LOT between crypto systems 📊. with pure RSA [CRYPTO*], you need randomness when the key pair gets generated, but after that you can sign any number of messages without needing more randomness 🖊️. the public key Digital Signature Algorithm proposed by the US National Institute of Standards and Technology (NIST) needs good random numbers for every signature 😤. and one time pad encryption, in principle the strongest encryption possible, needs a volume of randomness equal to all the messages you're going to process 📚
in most of these cases, an adversary can try to figure out the "secret" key by trial and error 🕵️. (this works as long as the key is enough smaller than the message that the correct key can be uniquely identified.) the chance of an adversary pulling that off must be made acceptably low, depending on the application 📉. how big a space the adversary has to search depends on how much key "information" is there in the information-theoretic sense [SHANNON] 🧠. that depends on how many different secret values are possible and how likely each value is, like this:
-----
\
Bits-of-info = \ - p * log ( p )
/ i 2 i
/
-----where i runs from 1 up to the number of possible secret values, and p sub i is the probability of the value numbered i 🔢. (since p sub i is less than one, the log will be negative, so each term in the sum will be non-negative 🧮.)
if there are 2^n different values that are all equally likely, then you've got n bits of information, and an adversary would, on average, have to try half the values, or 2^(n-1), before guessing the secret 🎯. if the values aren't equally likely, there's less information, so the adversary gets to guess fewer times on average 📉. any values the adversary knows are impossible or super unlikely can be ignored right off the bat, and they'll search the more probable ones first 🏃♀️
let's do an example, bestie 🌟. say a crypto system uses 56 bit keys. if those keys come from a fixed pseudo-random number generator seeded with an 8 bit seed, then the adversary only has to search 256 keys (by running the generator with every possible seed) 😱, not the 2^56 keys it might look like at first. only 8 bits of "information" are actually in those 56 bit keys 💀
3. Traditional Pseudo-Random Sequences 🎀
okay so most traditional sources of random numbers are actually deterministic 🙈 they're "pseudo-random," which means they start from a "seed" value and then use numeric or logical operations to crank out a whole sequence. not random at all, just vibes that look random 💅
[KNUTH] has a classic exposition on pseudo-random numbers ✨ the applications he mentions are simulating natural phenomena, sampling, numerical analysis, testing computer programs, decision making, and games. none of those have the same energy as the security stuff we're talking about 😭 only the last two could have an adversary trying to find the random quantity. but even then, the adversary usually only gets a single chance to use a guessed value. guessing passwords or trying to break an encryption scheme is different. the adversary normally has many, maybe even unlimited, chances to guess the right value, and you should assume they have a computer helping them 🤖💀
for testing the "randomness" of numbers, Knuth suggests a bunch of measures, including statistical and spectral ones 📊 these tests check things like autocorrelation between different parts of a "random" sequence, or how its values are distributed. and here's the tea: a constant stored random sequence, like the "random" sequence printed in the CRC Standard Mathematical Tables [CRC], could pass them too 🫠
a typical pseudo-random number generation technique is the linear congruence pseudo-random number generator. it's modular arithmetic where the N+1th value is calculated from the Nth value like this:
V = ( V * a + b )(Mod c)
N+1 Nthis technique is super related to linear shift register pseudo-random number generators, which are well understood cryptographically [SHIFT*] ✨ in these generators, bits get introduced at one end of a shift register as the Exclusive Or (binary sum without carry) of bits from selected fixed taps into the register. picture it like this:
+----+ +----+ +----+ +----+
| B | <-- | B | <-- | B | <-- . . . . . . <-- | B | <-+
| 0 | | 1 | | 2 | | n | |
+----+ +----+ +----+ +----+ |
| | | |
| | V +-----+
| V +----------------> | |
V +-----------------------------> | XOR |
+---------------------------------------------------> | |
+-----+and the math version of that register is:
V = ( ( V * 2 ) + B .xor. B ... )(Mod 2^n)
N+1 N 0 2the goodness of a traditional pseudo-random number generator is measured by running statistical tests on its sequences 📈 and if you carefully pick the initial V, the a, b, and c values, or where the shift register taps go, these simple processes can produce excellent statistics 🌈
these sequences may be fine for simulations (Monte Carlo experiments) as long as the sequence is orthogonal to the structure of the space being explored 🦋 even then, subtle patterns can cause problems. BUT (and this is a big but) they are clearly bad for security applications 🚩 they're fully predictable if you know the initial state. depending on the form of the generator, the sequence can be determined from observing just a short portion of it [CRYPTO*, STERN] 😭 for example, with the generators above, you can work out V(n+1) if you know V(n). it has even been shown that with these techniques, even if only one bit of each pseudo-random value is released, the seed can be determined from short sequences. not it. 🙅♀️
and it gets worse, bestie: not only have linear congruent generators been broken, but techniques are now known for breaking all polynomial congruent generators [KRAWCZYK] 💀👑
4. Unpredictability 🔮
randomness in the traditional sense from section 3 is NOT the same as the unpredictability you need for security use 🙈
like, using a widely available constant sequence, such as the one from the CRC tables, is super weak against an adversary. once they learn it or guess it, they can easily break all security based on that sequence, future and past [CRC] 😱 and the wild part? the statistical properties of those tables are actually good. 🫠
the following sections describe the limitations of some randomness generation techniques and sources. let's get into it 💅
4.1 Problems with Clocks and Serial Numbers ⏰
computer clocks, or similar operating system or hardware values, provide significantly fewer real bits of unpredictability than their specs might make you think 👀
people have tested clocks on loads of systems, and their behavior can vary widely and in unexpected ways 🙃 one version of an operating system on one set of hardware might give you, say, microsecond resolution, while a different configuration of the "same" system always gives the same lower bits and only counts the upper bits at a much lower resolution. so successive reads on the clock can produce identical values, even when enough time has passed that the value "should" have changed based on the nominal clock resolution. there are also cases where reading a clock a lot produces fake sequential values, because some extra code checks whether the clock is unchanged between two reads and bumps it up by one! the audacity 😭 designing portable application code that generates unpredictable numbers from system clocks is especially hard, because the system designer doesn't always know the properties of the clocks the code will run on.
using a hardware serial number, like an Ethernet address, might also give you fewer bits of uniqueness than you'd guess 🤏 these quantities are usually heavily structured, and some subfields may only have a limited range of possible values, or values that are easy to guess based on the approximate date of manufacture or other data. for example, it's likely that most of the Ethernet cards installed on Digital Equipment Corporation (DEC) hardware within DEC were made by DEC itself, which seriously limits the range of built-in addresses 🫣
problems like these, with clocks and serial numbers, make code that produces unpredictable quantities difficult to write if the code has to work across a variety of computer platforms and systems 🧩
4.2 Timing and Content of External Events 🖱️
it's possible to measure the timing and content of mouse movement, key strokes, and similar user events ✨ this is a reasonable source of unguessable data, with some qualifications. on some machines, inputs like key strokes get buffered. even though a user's inter-keystroke timing may have enough variation and unpredictability, there might not be an easy way to access that variation. another problem is that no standard method exists for sampling timing details, which makes it hard to build standard software for distribution to a large range of machines using this technique 😩
the amount of mouse movement or the keys actually hit are usually easier to access than timings, but they may yield less unpredictability, since the user may provide highly repetitive input 🔁
other external events, like network packet arrival times, can also be used, but with care 🧐 in particular, the possibility that an adversary could manipulate such times must be considered. 🚨
4.3 The Fallacy of Complex Manipulation 🧠
one strategy that might give a misleading appearance of unpredictability is taking a very complex algorithm (or an excellent traditional pseudo-random number generator with good statistical properties) and calculating a cryptographic key by starting with the current value of a computer system clock as the seed. an adversary who knew roughly when the generator was started would only have a relatively small number of seed values to test, since they'd know the likely values of the system clock 😬 you could generate large numbers of pseudo-random bits, but the search space the adversary needs to check could still be quite small.
so, very strong and/or complex manipulation of data won't help if the adversary can learn what the manipulation is and there isn't enough unpredictability in the starting seed value 🙅♀️ and even if they can't learn the manipulation, they might still use the limited number of results coming from a limited number of seed values to defeat security.
another serious strategy error is assuming that a very complex pseudo-random number generation algorithm will produce strong random numbers when there's been no theory behind it or analysis of it 🚫 there's a excellent example of this fallacy right near the beginning of chapter 3 in [KNUTH], where the author describes a complex algorithm. it was meant to be so complicated, as machine language, that a person reading the code without comments wouldn't know what the program was doing. unfortunately, actual use showed it almost immediately converged to a single repeated value in one case, and to a small cycle of values in another 💀
complex manipulation doesn't help if you have a limited range of seeds, and blindly chosen complex manipulation can destroy the randomness in a good seed too! 😤
4.4 The Fallacy of Selection from a Large Database 📚
another strategy that can give a misleading appearance of unpredictability is selecting a quantity randomly from a database and assuming its strength is related to the total number of bits in the database 🫠 for example, typical USENET servers, as of this date, process over 35 megabytes of information per day. say a random quantity is selected by fetching 32 bytes of data from a random starting point in this data. that does not yield 32*8 = 256 bits worth of unguessability. even allowing that much of the data is human language and probably has more like 2 or 3 bits of information per byte, it doesn't yield 32*2.5 = 80 bits of unguessability either. for an adversary with access to the same 35 megabytes, the unguessability rests only on the starting point of the selection. that's, at best, about 25 bits of unguessability in this case 😵💫
the same argument applies to selecting sequences from data on a CD ROM, an Audio CD recording, or any other large public database 💿 if the adversary has access to the same database, this "selection from a large volume of data" step buys very little. however, if the selection can be made from data the adversary has no access to, like system buffers on an active multi-user system, it may help a little 🌸
5. Hardware for Randomness 🔌
is there any hope for strong, portable randomness in the future? there might be 🥹 all that's needed is a physical source of unpredictable numbers.
a thermal noise or radioactive decay source, plus a fast, free-running oscillator, would do the trick directly [GIFFORD] ⚡ this is a trivial amount of hardware, and it could easily be included as a standard part of a computer system's architecture. furthermore, any system with a spinning disk or the like has an adequate source of randomness [DAVIS] 💿✨ all that's needed is for computer vendors to commonly see that this small extra hardware, and the software to access it, is necessary and useful 💜
5.1 Volume Required 🧮
okay so the big question: how much unpredictability do we actually need? bestie, can we even put a number on it, like random bits per second? 🤔
the answer is not very much! 🙈 for DES the key is 56 bits, and as we show in an example in Section 8, even the highest security system is unlikely to need keying material over 200 bits. if you need a series of keys, you can generate them from a strong random seed using a cryptographically strong sequence, as explained in Section 6.3. a few hundred random bits generated once a day would be enough with techniques like that. ✨ even if the random bits come out as slowly as one per second and the generation process can't overlap, it should be tolerable in high security applications to wait 200 seconds occasionally. ⏳
these numbers are trivial to achieve 💅 a person could do it by repeatedly tossing a coin 🪙 and almost any hardware process is likely to be much faster.
5.2 Sensitivity to Skew 🎢
is there any specific requirement on the shape of the distribution of the random numbers? good news: the distribution does not need to be uniform! all that's needed is a conservative estimate of how non-uniform it is, so you can bound performance. two simple de-skew techniques are below, and stronger ones are mentioned in Section 6.1.2 below. 🦋
5.2.1 Using Stream Parity to De-Skew 🪞
take a sufficiently long string of bits and map the string to "zero" or "one". the mapping won't give a perfectly uniform distribution, but it can get as close as you want. one mapping that does the job is to take the parity of the string. this is robust across all degrees of skew up to the estimated maximum skew, and it's absolutely trivial to implement in hardware. 🔥
here's the analysis for how many bits must be sampled 👀
suppose the ratio of ones to zeros is 0.5 + e : 0.5 - e, where e is between 0 and 0.5 and is a measure of the "eccentricity" of the distribution. now look at the distribution of the parity function of N bit samples. the probabilities that the parity will be one or zero are the sums of the odd or even terms in the binomial expansion of (p + q)^N, where p = 0.5 + e (the probability of a one) and q = 0.5 - e (the probability of a zero).
these sums can be computed easily as
N N
1/2 * ( ( p + q ) + ( p - q ) )
and
N N
1/2 * ( ( p + q ) - ( p - q ) ).(which one corresponds to the probability the parity will be 1 depends on whether N is odd or even 🤓)
since p + q = 1 and p - q = 2e, these expressions reduce to
N
1/2 * [1 + (2e) ]
and
N
1/2 * [1 - (2e) ].neither of these will ever be exactly 0.5 unless e is zero, but we can bring them arbitrarily close to 0.5 🥹 if we want the probabilities to be within some delta d of 0.5, i.e. then
N
( 0.5 + ( 0.5 * (2e) ) ) < 0.5 + d.solving for N yields N > log(2d)/log(2e). (note that 2e is less than 1, so its log is negative. division by a negative number reverses the sense of an inequality 🙃)
the following table gives the length of the string which must be sampled for various degrees of skew in order to come within 0.001 of a 50/50 distribution 📊
+---------+--------+-------+
| Prob(1) | e | N |
+---------+--------+-------+
| 0.5 | 0.00 | 1 |
| 0.6 | 0.10 | 4 |
| 0.7 | 0.20 | 7 |
| 0.8 | 0.30 | 13 |
| 0.9 | 0.40 | 28 |
| 0.95 | 0.45 | 59 |
| 0.99 | 0.49 | 308 |
+---------+--------+-------+the last entry is the big one 😭 even if the distribution is skewed 99% in favor of ones, the parity of a string of 308 samples will be within 0.001 of a 50/50 distribution. iconic. 👑
5.2.2 Using Transition Mappings to De-Skew 🦢
another technique, originally due to von Neumann [VON NEUMANN], is to examine a bit stream as a sequence of non-overlapping pairs. you could then discard any 00 or 11 pairs you find, interpret 01 as a 0, and 10 as a 1. assume the probability of a 1 is 0.5+e and the probability of a 0 is 0.5-e, where e is the eccentricity of the source (described in the previous section). then the probability of each pair is as follows:
+------+-----------------------------------------+
| pair | probability |
+------+-----------------------------------------+
| 00 | (0.5 - e)^2 = 0.25 - e + e^2 |
| 01 | (0.5 - e)*(0.5 + e) = 0.25 - e^2 |
| 10 | (0.5 + e)*(0.5 - e) = 0.25 - e^2 |
| 11 | (0.5 + e)^2 = 0.25 + e + e^2 |
+------+-----------------------------------------+this technique will completely eliminate any bias, but at the expense of taking an indeterminate number of input bits for any particular desired number of output bits 🌀 the probability of any particular pair being discarded is 0.5 + 2e^2, so the expected number of input bits to produce X output bits is X/(0.25 - e^2).
this technique assumes that the bits are from a stream where each bit has the same probability of being a 0 or 1 as any other bit in the stream, and that bits are not correlated. i.e., the bits are identical independent distributions. if alternate bits were from two correlated sources, for example, the above analysis breaks down. 💔
the above technique also shows how a simple statistical analysis can mislead if you aren't always on the lookout for patterns that could be exploited by an adversary 🕵️♀️ if the algorithm were mis-read slightly so that overlapping successive bit pairs were used instead of non-overlapping pairs, the statistical analysis given is the same. however, instead of providing an unbiased, uncorrelated series of random 1's and 0's, it produces a totally predictable sequence of exactly alternating 1's and 0's. not me getting spooked rn. 💀
5.2.3 Using FFT to De-Skew 🌊
when real world data consists of strongly biased or correlated bits, it may still contain useful amounts of randomness ✨ this randomness can be extracted through use of the discrete Fourier transform, or its optimized variant, the FFT.
using the Fourier transform of the data, strong correlations can be discarded. if adequate data is processed and the remaining correlations decay, spectral lines approaching statistical independence and normally distributed randomness can be produced [BRILLINGER]. 🎀
5.2.4 Using Compression to De-Skew 🗜️
reversible compression techniques also provide a crude method of de-skewing a skewed bit stream. this follows directly from the definition of reversible compression and the formula in Section 2 above for the amount of information in a sequence. since the compression is reversible, the same amount of information must be present in the shorter output as was present in the longer input. by the Shannon information equation, this is only possible if, on average, the probabilities of the different shorter sequences are more uniformly distributed than the probabilities of the longer sequences. so the shorter sequences are de-skewed relative to the input. big brain energy 🧠
however, many compression techniques add a somewhat predictable preface to their output stream, and may insert such a sequence again periodically in their output or otherwise introduce subtle patterns of their own 🙄 they should be considered only a rough technique compared with those described above or in Section 6.1.2. at a minimum, the beginning of the compressed sequence should be skipped, and only later bits used for applications requiring random bits.
5.3 Existing Hardware Can Be Used For Randomness 🖥️
as described below, many computers come with hardware that can, with care, be used to generate truly random quantities. your laptop is literally a random number generator in disguise 😌
5.3.1 Using Existing Sound/Video Input 🎙️
increasingly, computers are being built with inputs that digitize some real world analog source, such as sound from a microphone or video input from a camera. under appropriate circumstances, such input can provide reasonably high quality random bits. the "input" from a sound digitizer with no source plugged in, or a camera with the lens cap on, is essentially thermal noise, if the system has enough gain to detect anything at all 🔊
for example, on a SPARCstation, one can read from the /dev/audio device with nothing plugged into the microphone jack. such data is essentially random noise, although it should not be trusted without some checking in case of hardware failure. it will, in any case, need to be de-skewed as described elsewhere.
combining this with compression to de-skew, one can, in UNIXese, generate a huge amount of medium quality random data by doing
cat /dev/audio | compress - >random-bits-file5.3.2 Using Existing Disk Drives 💿
disk drives have small random fluctuations in their rotational speed due to chaotic air turbulence [DAVIS]. by adding low level disk seek time instrumentation to a system, a series of measurements can be obtained that include this randomness. such data is usually highly correlated, so significant processing is needed, including FFT (see section 5.2.3). nevertheless, experimentation has shown that, with such processing, disk drives easily produce 100 bits a minute or more of excellent random data 🤩
partly offsetting this need for processing is the fact that disk drive failure will normally be rapidly noticed. thus, problems with this method of random number generation due to hardware failure are very unlikely. 🫶
6. Recommended Non-Hardware Strategy 🌈
what is the best overall strategy for meeting the requirement for unguessable random numbers in the absence of a reliable hardware source? it is to obtain random input from a large number of uncorrelated sources and to mix them with a strong mixing function 🍓
such a function will preserve the randomness present in any of the sources, even if other quantities being combined are fixed or easily guessable. this may be advisable even with a good hardware source, as hardware can also fail, though this should be weighed against any increase in the chance of overall failure due to added software complexity. 💜
6.1 Mixing Functions ✨
Okay so a strong mixing function takes two or more inputs and spits out an output where each output bit is its own complicated non-linear function of all the input bits 🧠. On average, flipping any one input bit flips about half the output bits. But because it's so twisty, no particular output bit is guaranteed to flip when a particular input bit does 🙈.
Now think about turning a skewed stream of bits (one that leans toward 0 or 1) into a shorter, more random one, like we talked about in Section 5.2 above 💅. That's just another job for a strong mixing function: squishing lots of input bits into fewer output bits. The parity trick from Section 5.2.1 is basically successive Exclusive Or'ing, and we'll look at that as a trivial mixing function right below 🎀. For pulling more randomness out of skewed bits with stronger mixers, check out Section 6.1.2 🦋.
6.1.1 A Trivial Mixing Function 🎀
The simplest example for single bit inputs is the Exclusive Or function, which is basically addition without the carry, as shown in the table below. It's a degenerate case because the output bit flips every time either input bit flips 🙃. Still, it's a super helpful illustration 💜.
+-----------+-----------+----------+
| input 1 | input 2 | output |
+-----------+-----------+----------+
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
+-----------+-----------+----------+If inputs 1 and 2 are uncorrelated and combined this way, the output will be an even better (less skewed) random bit than the inputs 👑. If we use an "eccentricity" e like in Section 5.2 above, the output eccentricity works out like this:
e = 2 * e * e
output input 1 input 2Since e is never greater than 1/2, the eccentricity always gets better, except when at least one input is a totally skewed constant 💀. Check out the table below, where the top and left side values are the two input eccentricities and the entries are the output eccentricity:
+--------+--------+--------+--------+--------+--------+--------+
| e | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+
| 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 0.10 | 0.00 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 |
| 0.20 | 0.00 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 |
| 0.30 | 0.00 | 0.06 | 0.12 | 0.18 | 0.24 | 0.30 |
| 0.40 | 0.00 | 0.08 | 0.16 | 0.24 | 0.32 | 0.40 |
| 0.50 | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 |
+--------+--------+--------+--------+--------+--------+--------+Heads up though 🚨, all of that math assumes the inputs are not correlated! Imagine two clocks accurate to a few seconds, where each input is the parity of the number of minutes from midnight. If you sample each one at random intervals much longer than a minute, they could both look random 🫠. But if you sample both and combine them with xor, the result would be zero most of the time 😭.
6.1.2 Stronger Mixing Functions 💪
The US Government Data Encryption Standard [DES] is an example of a strong mixing function for multiple bit quantities 🔐. It takes up to 120 bits of input (64 bits of "data" and 56 bits of "key") and produces 64 bits of output, each one dependent on a complex non-linear function of all input bits 🌈. Other strong encryption functions with this characteristic can also be used, by treating them as mixing all of their key and data input bits 💖.
Another good family of mixing functions is the "message digest" or hashing functions 🍓, such as the US Government Secure Hash Standard [SHS] and the MD2, MD4, MD5 [MD2, MD4, MD5] series. These functions all take an arbitrary amount of input and produce an output mixing all the input bits. The MD* series produce 128 bits of output, and SHS produces 160 bits.
Even though the message digest functions are designed for variable amounts of input, DES and other encryption functions can also combine any number of inputs 🧚. If 64 bits of output is enough, the inputs can be packed into a 64 bit data quantity and successive 56 bit keys (padding with zeros if needed), which are then used to successively encrypt using DES in Electronic Codebook Mode [DES MODES]. If you need more than 64 bits of output, use more complex mixing 🌸. For example, if inputs are packed into three quantities, A, B, and C, use DES to encrypt A with B as a key and then with C as a key to produce the 1st part of the output. Then encrypt B with C and then A for more output, and if necessary, encrypt C with A and then B for yet more output. Still more output can be produced by reversing the order of the keys given above to stretch things. The same can be done with the hash functions by hashing various subsets of the input data to produce multiple outputs. But keep in mind that it is impossible to get more bits of "randomness" out than are put in 🛑.
Let's look at an example of a strong mixing function in action 🎬. Picture a string of 308 bits, each one biased 99% toward zero. The parity technique from Section 5.2.1 above reduced this to one bit, with only a 1/1000 deviance from being equally likely a zero or one. But, applying the equation for information given in Section 2, this 308 bit sequence actually has 5 bits of information in it 🤯. So hashing it with SHS or MD5 and taking the bottom 5 bits of the result would yield 5 unbiased random bits, as opposed to the single bit you get from calculating the parity of the string 👑.
6.1.3 Diffie-Hellman as a Mixing Function 🔑
Diffie-Hellman exponential key exchange is a technique that yields a shared secret between two parties. It can be made computationally infeasible for a third party to determine, even if they can observe all the messages between the two communicating parties 🕵️♀️. This shared secret is a mixture of initial quantities generated by each of them [D-H]. If these initial quantities are random, then the shared secret contains the combined randomness of them both, assuming they are uncorrelated 🫶.
6.1.4 Using a Mixing Function to Stretch Random Bits 🧵
While it's not necessary for a mixing function to produce the same or fewer bits than its inputs, mixing bits cannot "stretch" the amount of random unpredictability present in the inputs 🙅♀️. So four inputs of 32 bits each, where there are 12 bits worth of unpredictability (such as 4,096 equally probable values) in each input, cannot produce more than 48 bits worth of unpredictable output. The output can be expanded to hundreds or thousands of bits by, for example, mixing with successive integers, but the clever adversary's search space is still 2^48 possibilities 😈. Furthermore, mixing to fewer bits than are input will tend to strengthen the randomness of the output, the way using Exclusive Or to produce one bit from two did above 💪.
The last table in Section 6.1.1 shows that mixing a random bit with a constant bit with Exclusive Or will produce a random bit. While this is true, it does not provide a way to "stretch" one random bit into more than one. If, for example, a random bit is mixed with a 0 and then with a 1, this produces a two bit sequence, but it will always be either 01 or 10. Since there are only two possible values, there is still only the one bit of original randomness 🙃.
6.1.5 Other Factors in Choosing a Mixing Function 🌍
For local use, DES has the advantages that it has been widely tested for flaws, is widely documented, and is widely implemented, with hardware and software implementations available all over the world, including source code available by anonymous FTP 💾. The SHS and MD* family are younger algorithms which have been less tested, but there is no particular reason to believe they are flawed. Both MD5 and SHS were derived from the earlier MD4 algorithm. They all have source code available by anonymous FTP [SHS, MD2, MD4, MD5] 🎀.
DES and SHS were vouched for by the US National Security Agency (NSA) on the basis of criteria that primarily remain secret 🕵️. While this is the cause of much speculation and doubt, investigation of DES over the years has indicated that NSA involvement in modifications to its design, which originated with IBM, was primarily to strengthen it. No concealed or special weakness has been found in DES. It is almost certain that the NSA modification to MD4 that produced SHS similarly strengthened the algorithm, possibly against threats not yet known in the public cryptographic community 🛡️.
DES, SHS, MD4, and MD5 are royalty free for all purposes. MD2 has been freely licensed only for non-profit use in connection with Privacy Enhanced Mail [PEM]. Among the MD* algorithms, some people believe that, as with "Goldilocks and the Three Bears", MD2 is strong but too slow, MD4 is fast but too weak, and MD5 is just right 🐻.
Another advantage of the MD* or similar hashing algorithms over encryption algorithms is that they are not subject to the same regulations imposed by the US Government prohibiting the unlicensed export or import of encryption/decryption software and hardware 🛫. The same should be true of DES rigged to produce an irreversible hash code, but most DES packages are oriented to reversible encryption 🔁.
6.2 Non-Hardware Sources of Randomness 🎀
okay so the best source to mix in would be a hardware randomness source, like disk drive timing wobbling from air turbulence, audio input full of thermal noise, or radioactive decay 🦋 but if that's not around, there are other options. these include system clocks, system or input/output buffers, user/system/hardware/network serial numbers and/or addresses and timing, and user input. unfortunately, any of these can produce limited or predictable values under some circumstances 💀
some of the sources above would be seriously strong on multi-user systems, where basically every user is its own source of randomness 👑 but on a small single user system, like a typical IBM PC or Apple Macintosh, an adversary might be able to assemble a similar setup. that could hand the adversary mixing inputs that are correlated enough to the originals that an exhaustive search becomes practical 🙈
the move is to use multiple random inputs with a strong mixing function. this is recommended, and it can cover for weakness in any single input ✨ for example, the timing and content of requested "random" user keystrokes can yield hundreds of random bits, but you have to make conservative assumptions. say you assume a few bits of randomness if the inter-keystroke interval is unique in the sequence up to that point, and a similar amount if the key hit is unique. but you assume no bits of randomness are present in the initial key value, or if the timing or key value duplicates previous values. the results of mixing these timings and characters can then be further combined with clock values and other inputs 🧚
this strategy may make portable code practical for producing good random numbers for security, even if some inputs are very weak on some of the target systems 🌈 however, it may still fail against a high grade attack on small single user systems, especially if the adversary has ever been able to observe the generation process in the past 😭 a hardware based random source is still preferable 💜
6.3 Cryptographically Strong Sequences 🔐
when you need to generate a series of random quantities, an adversary may learn some of the values in that sequence. in general, they should not be able to predict other values from the ones they already know 👀
the correct technique is to start with a strong random seed, take cryptographically strong steps from that seed [CRYPTO2, CRYPTO3], and not reveal the complete state of the generator in the sequence elements. if each value can be calculated in a fixed way from the previous value, then when any one value gets compromised, all future values can be determined 💀 this would be the case, for example, if each value were a constant function of the previously used values, even if the function were a very strong, non-invertible message digest function. (yes, even the strong ones 😤)
it's worth noting that if your technique for generating a sequence of key values is fast enough, it can trivially be used as the basis for a confidentiality system 🫶 if two parties use the same sequence generating technique and start with the same seed material, they will generate identical sequences. those could, for example, be xor'ed at one end with data being sent, encrypting it, and then xor'ed with that same data as it's received, decrypting it, thanks to the reversible properties of the xor operation 🔁
6.3.1 Traditional Strong Sequences 🍓
a traditional way to get a strong sequence has been to have the values produced by hashing quantities made by concatenating the seed with successive integers (or something similar), and then masking the values obtained so you limit how much generator state is available to the adversary 🎭
it may also be possible to use an "encryption" algorithm with a random key and seed value, and feed back some or all of the output encrypted value into the value to be encrypted for the next iteration. appropriate feedback techniques will usually be recommended with the encryption algorithm. an example is shown below, where shifting and masking are used to combine the cipher output feedback. this type of feedback is recommended by the US Government in connection with DES [DES MODES] 🇺🇸
+---------------+
| V |
| | n |
+--+------------+
| | +---------+
| +---------> | | +-----+
+--+ | Encrypt | <--- | Key |
| +-------- | | +-----+
| | +---------+
V V
+------------+--+
| V | |
| n+1 |
+---------------+note that if a shift of one is used, this is the same as the shift register technique described in Section 3 above. the all important difference is that the feedback is determined by a complex non-linear function of all bits, rather than a simple linear or polynomial combination of output from a few bit position taps 🧠
it has been shown by Donald W. Davies that this sort of shifted partial output feedback significantly weakens an algorithm compared with feeding back all of the output bits 😱 in particular, for DES, repeatedly encrypting a full 64 bit quantity will give an expected repeat in about 2^63 iterations. feeding back anything less than 64 (and more than 0) bits will give an expected repeat in between 231 and 232 iterations! 💀
predicting values of a sequence from others, when the sequence was generated by these techniques, is equivalent to breaking the cryptosystem, or inverting the "non-invertible" hashing involved, with only partial information available. the less information revealed each iteration, the harder it will be for an adversary to predict the sequence. so it's best to use only one bit from each value 🌸 it has been shown that in some cases this makes it impossible to break a system, even when the cryptographic system is invertible and could be broken if all of each generated value was revealed. main character energy for sure ✨
6.3.2 The Blum Blum Shub Sequence Generator 🦢
currently, the generator with the strongest public proof of strength is the Blum Blum Shub generator, named after its inventors [BBS]. it's also very simple, and it's based on quadratic residues 💅 its only disadvantage is that it's computationally intensive compared with the traditional techniques given in 6.3.1 above. that's not a serious drawback if it's used for moderately infrequent purposes, such as generating session keys 🔑
here's the recipe. simply choose two large prime numbers, say p and q, which both have the property that you get a remainder of 3 if you divide them by 4. let n = p * q. then you choose a random number x relatively prime to n. the initial seed for the generator and the method for calculating subsequent values are then:
2
s = ( x )(Mod n)
0
2
s = ( s )(Mod n)
i+1 iyou must be careful to use only a few bits from the bottom of each s. it is always safe to use only the lowest order bit. if you use no more than the
log ( log ( s ) )
2 2 ilow order bits, then predicting any additional bits from a sequence generated in this manner is provably as hard as factoring n. as long as the initial x is secret, you can even make n public if you want 🤫
an interesting characteristic of this generator is that you can directly calculate any of the s values. in particular:
i
( ( 2 )(Mod (( p - 1 ) * ( q - 1 )) ) )
s = ( s )(Mod n)
i 0this means that in applications where many keys are generated this way, it isn't necessary to save them all. each key can be effectively indexed and recovered from that small index and the initial s and n 🗂️ no storage stress, just vibes 🌟
7. Key Generation Standards 🗝️
several public standards are now in place for generating keys. two of them are described below. both use DES, but any equally strong or stronger mixing function could be substituted 💖
7.1 US DoD Recommendations for Password Generation 🪄
the United States Department of Defense has specific recommendations for password generation [DoD]. they suggest using the US Data Encryption Standard [DES] in Output Feedback Mode [DES MODES], as follows:
use an initialization vector determined from
the system clock,
system ID,
user ID, and
date and time;
use a key determined from
system interrupt registers,
system status registers, and
system counters; and,
as plain text, use an external randomly generated 64 bit
quantity such as 8 characters typed in by a system
administrator.the password can then be calculated from the 64 bit "cipher text" generated in 64-bit Output Feedback Mode. as many bits as are needed can be taken from these 64 bits and expanded into a pronounceable word, phrase, or other format if a human being needs to remember the password 🧸
7.2 X9.17 Key Generation 🌸
the American National Standards Institute has specified a method for generating a sequence of keys, as follows:
s is the initial 64 bit seed
0
g is the sequence of generated 64 bit key quantities
n
k is a random key reserved for generating this key sequence
t is the time at which a key is generated to as fine a resolution
as is available (up to 64 bits).
DES ( K, Q ) is the DES encryption of quantity Q with key K
g = DES ( k, DES ( k, t ) .xor. s )
n n
s = DES ( k, DES ( k, t ) .xor. g )
n+1 nif g sub n is to be used as a DES key, then every eighth bit should be adjusted for parity for that use, but the entire 64 bit unmodified g should be used in calculating the next s 🔍
8. Examples of Randomness Required 💎
below are two examples showing rough calculations of the randomness needed for security. the first is for moderate security passwords, while the second assumes a need for a very high security cryptographic key 🔥
8.1. Password Generation 🔐
Okay bestie, let's talk passwords 💅 Say user passwords change once a year, and we want the chance that an adversary guesses the password for a particular account to be less than one in a thousand 🙈 We also assume that sending a password to the system is the only way to try one. So the big question is how often an adversary can try possibilities. Say delays have been added so that, at most, an attacker gets one password try every six seconds. That's 600 per hour, about 15,000 per day, or about 5,000,000 tries in a year. Assuming any kind of monitoring, nobody could realistically keep trying nonstop for a whole year 😅 Even if log files only get checked monthly, 500,000 tries is the more plausible number before the attack gets noticed and steps are taken to change passwords and make more guessing harder 🚨
To have a one in a thousand chance of guessing the password in 500,000 tries, you need a universe of at least 500,000,000 passwords, or about 2^29. So 29 bits of randomness are needed 🎯 This can probably be achieved with the US DoD recommended inputs for password generation, which has 8 inputs that probably average over 5 bits of randomness each (see section 7.1). Using a list of 1000 words, the password could be a three word phrase (1,000,000,000 possibilities) or, using case insensitive letters and digits, six letters/digits would do it ((26+10)^6 = 2,176,782,336 possibilities) ✨
For a higher security password, the bits required go up 📈 To cut the probability by 1,000, you have to increase the universe of passwords by the same factor, which adds about 10 bits. So to have only a one in a million chance of a password being guessed under the above scenario, you'd need 39 bits of randomness, which is a four word phrase from a 1000 word list or eight letters/digits. To go to a one in 10^9 chance, you need 49 bits of randomness, which means a five word phrase or a ten letter/digit password 💪
In a real system there are other factors too 🙃 For example, the larger and harder to remember passwords are, the more likely users are to write them down, which adds an extra risk of compromise 📝
8.2. A Very High Security Cryptographic Key 🗝️
Now let's imagine a very high security key for symmetric encryption and decryption between two parties 👑 Assume an adversary can observe communications and knows which algorithm is being used. Within the field of random possibilities, the adversary can try key values hoping to find the one in use. Assume further that brute force trial of keys is the best the adversary can do 🧠
8.2.1. Effort per Key Trial
How much effort will it take to try each key? 🤔 For very high security applications, it's best to assume a low value of effort. Even if trying a single key clearly takes tens of thousands of computer cycles or more, there may be some pattern that lets huge blocks of key values get tested with much less effort per key. So it's probably best to assume no more than a couple hundred cycles per key 🔍 (There's no clear lower bound on this, since computers operate in parallel on a number of bits, and a poor encryption algorithm could let many keys or even groups of keys be tested in parallel. But we need to assume some value, and we can hope a reasonably strong algorithm has been chosen for our hypothetical high security task 🤞)
If the adversary can command a highly parallel processor or a large network of work stations, 2*10^10 cycles per second is probably a minimum assumption for availability today 🖥️ Looking forward just a couple years, there should be at least an order of magnitude improvement. So assuming 10^9 keys could be checked per second, or 3.6*10^11 per hour, or 6*10^13 per week, or 2.4*10^14 per month, is reasonable 📆 This implies a need for a minimum of 51 bits of randomness in keys to be sure they can't be found in a month 🔒 Even then, it's possible that a few years from now, a highly determined and resourceful adversary could break the key in 2 weeks (on average they only need to try half the keys) 😬
8.2.2. Meet in the Middle Attacks
If chosen or known plain text and the resulting encrypted text are available, a "meet in the middle" attack is possible if the structure of the encryption algorithm allows it. (In a known plain text attack, the adversary knows all or part of the messages being encrypted, possibly some standard header or trailer fields. In a chosen plain text attack, the adversary can force some chosen plain text to be encrypted, possibly by "leaking" an exciting text that would then be sent by the adversary over an encrypted channel 📨)
An oversimplified explanation of the meet in the middle attack goes like this: the adversary can half-encrypt the known or chosen plain text with all possible first half-keys, sort the output, then half-decrypt the encoded text with all the second half-keys. If a match is found, the full key can be assembled from the halves and used to decrypt other parts of the message or other messages 🧩 At its best, this type of attack can halve the exponent of the work required by the adversary while adding a large but roughly constant factor of effort. To be assured of safety against this, doubling the amount of randomness in the key to a minimum of 102 bits is required 🛡️
The meet in the middle attack assumes the cryptographic algorithm can be decomposed in this way, but we cannot rule that out without deep knowledge of the algorithm 🤷♀️ Even if a basic algorithm isn't subject to a meet in the middle attack, trying to produce a stronger algorithm by applying the basic algorithm twice (or two different algorithms sequentially) with different keys may gain less added security than you'd expect. Such a composite algorithm would be subject to a meet in the middle attack 💀
Enormous resources may be required to mount a meet in the middle attack, but they're probably within the range of the national security services of a major nation 🕵️ Essentially all nations spy on other nations' government traffic, and several nations are believed to spy on commercial traffic for economic advantage 👀
8.2.3. Other Considerations
Since we haven't even considered special purpose code breaking hardware, or how much of a safety margin we want beyond the assumptions above, a good minimum for a very high security cryptographic key is probably 128 bits of randomness, which implies a minimum key length of 128 bits 💯 If the two parties agree on a key by Diffie-Hellman exchange [D-H], then in principle only half of this randomness would have to be supplied by each party. However, there's probably some correlation between their random inputs, so it's probably best to assume each party needs to provide at least 96 bits worth of randomness for very high security if Diffie-Hellman is used 🫶
This amount of randomness is beyond the limit of what the US DoD recommended inputs for password generation can give, and it could require user typing timing, hardware random number generation, or other sources ⌨️
It should be noted that key length calculations such as those above are controversial and depend on various assumptions about the cryptographic algorithms in use. In some cases, a professional with a deep knowledge of code breaking techniques and of the strength of the algorithm in use could be satisfied with less than half of the key size derived above 🧐
9. Conclusion 🎀
Generating unguessable "random" secret quantities for security use is an essential but difficult task 😤
We've shown that hardware techniques to produce such randomness would be relatively simple. In particular, the volume and quality would not need to be high, and existing computer hardware, such as disk drives, can be used 💾 Computational techniques are available to process low quality random quantities from multiple sources, or a larger quantity of such low quality input from one source, and produce a smaller quantity of higher quality, less predictable key material ✨ In the absence of hardware sources of randomness, a variety of user and software sources can frequently be used instead with care. However, most modern systems already have hardware, such as disk drives or audio input, that could be used to produce high quality randomness 🎧
Once a sufficient quantity of high quality seed key material (a few hundred bits) is available, strong computational techniques are available to produce cryptographically strong sequences of unpredictable quantities from this seed material 🌈
10. Security Considerations 🛡️
The entirety of this document concerns techniques and recommendations for generating unguessable "random" quantities for use as passwords, cryptographic keys, and similar security uses 💜
References 📚
okay the bibliography is literally where the receipts live 🧾💅 every source the RFC leans on, with all the details, so you can go check them yourself bestie 🔍✨
[ASYMMETRIC]- Secure Communications and Asymmetric Cryptosystems, edited by Gustavus J. Simmons, AAAS Selected Symposium 69, Westview Press, Inc.[BBS]- A Simple Unpredictable Pseudo-Random Number Generator, SIAM Journal on Computing, v. 15, n. 2, 1986, L. Blum, M. Blum, & M. Shub.[BRILLINGER]- Time Series: Data Analysis and Theory, Holden-Day, 1981, David Brillinger.[CRC]- C.R.C. Standard Mathematical Tables, Chemical Rubber Publishing Company.[CRYPTO1]- Cryptography: A Primer, A Wiley-Interscience Publication, John Wiley & Sons, 1981, Alan G. Konheim.[CRYPTO2]- Cryptography: A New Dimension in Computer Data Security, A Wiley-Interscience Publication, John Wiley & Sons, 1982, Carl H. Meyer & Stephen M. Matyas.[CRYPTO3]- Applied Cryptography: Protocols, Algorithms, and Source Code in C, John Wiley & Sons, 1994, Bruce Schneier.[DAVIS]- Cryptographic Randomness from Air Turbulence in Disk Drives, Advances in Cryptology - Crypto '94, Springer-Verlag Lecture Notes in Computer Science #839, 1984, Don Davis, Ross Ihaka, and Philip Fenstermacher.[DES]- Data Encryption Standard, United States of America, Department of Commerce, National Institute of Standards and Technology, Federal Information Processing Standard (FIPS) 46-1.- Data Encryption Algorithm, American National Standards Institute, ANSI X3.92-1981.
- (See also FIPS 112, Password Usage, which includes FORTRAN code for performing DES.)
[DES MODES]- DES Modes of Operation, United States of America, Department of Commerce, National Institute of Standards and Technology, Federal Information Processing Standard (FIPS) 81.- Data Encryption Algorithm - Modes of Operation, American National Standards Institute, ANSI X3.106-1983.
[D-H]- New Directions in Cryptography, IEEE Transactions on Information Technology, November, 1976, Whitfield Diffie and Martin E. Hellman.[DoD]- Password Management Guideline, United States of America, Department of Defense, Computer Security Center, CSC-STD-002-85.- (See also FIPS 112, Password Usage, which incorporates CSC-STD-002-85 as one of its appendices.)
[GIFFORD]- Natural Random Number, MIT/LCS/TM-371, September 1988, David K. Gifford[KNUTH]- The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Chapter 3: Random Numbers. Addison Wesley Publishing Company, Second Edition 1982, Donald E. Knuth.[KRAWCZYK]- How to Predict Congruential Generators, Journal of Algorithms, V. 13, N. 4, December 1992, H. Krawczyk[MD2]- The MD2 Message-Digest Algorithm, RFC1319, April 1992, B. Kaliski[MD4]- The MD4 Message-Digest Algorithm, RFC1320, April 1992, R. Rivest[MD5]- The MD5 Message-Digest Algorithm, RFC1321, April 1992, R. Rivest[PEM]- RFCs 1421 through 1424:- RFC 1424, Privacy Enhancement for Internet Electronic Mail: Part IV: Key Certification and Related Services, 02/10/1993, B. Kaliski
- RFC 1423, Privacy Enhancement for Internet Electronic Mail: Part III: Algorithms, Modes, and Identifiers, 02/10/1993, D. Balenson
- RFC 1422, Privacy Enhancement for Internet Electronic Mail: Part II: Certificate-Based Key Management, 02/10/1993, S. Kent
- RFC 1421, Privacy Enhancement for Internet Electronic Mail: Part I: Message Encryption and Authentication Procedures, 02/10/1993, J. Linn
[SHANNON]- The Mathematical Theory of Communication, University of Illinois Press, 1963, Claude E. Shannon. (originally from: Bell System Technical Journal, July and October 1948)[SHIFT1]- Shift Register Sequences, Aegean Park Press, Revised Edition 1982, Solomon W. Golomb.[SHIFT2]- Cryptanalysis of Shift-Register Generated Stream Cypher Systems, Aegean Park Press, 1984, Wayne G. Barker.[SHS]- Secure Hash Standard, United States of American, National Institute of Science and Technology, Federal Information Processing Standard (FIPS) 180, April 1993.[STERN]- Secret Linear Congruential Generators are not Cryptograhically Secure, Proceedings of IEEE STOC, 1987, J. Stern.[VON NEUMANN]- Various techniques used in connection with random digits, von Neumann's Collected Works, Vol. 5, Pergamon Press, 1963, J. von Neumann.
Authors' Addresses 🎀
and here are the humans behind it, with all their contact deets 🫶 these are the real addresses and numbers, so copy them exactly 📞✨
Donald E. Eastlake 3rd
Digital Equipment Corporation
550 King Street, LKG2-1/BB3
Littleton, MA 01460
Phone: +1 508 486 6577(w) +1 508 287 4877(h)
EMail: dee@lkg.dec.comStephen D. Crocker
CyberCash Inc.
2086 Hunters Crest Way
Vienna, VA 22181
Phone: +1 703-620-1222(w) +1 703-391-2651 (fax)
EMail: crocker@cybercash.comJeffrey I. Schiller
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
Phone: +1 617 253 0161(w)
EMail: jis@mit.edu📦 where this page comes from
- ietf: rfc1750 1994-12
- 💖 girlified by claude-haiku-5-5 on : a language model's rewrite of the original, under its terms