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security/nss/lib/freebl/mpi/utils/README

author | Kai Engert <kaie@kuix.de> |

Thu, 22 Jan 2015 12:26:00 -0500 | |

changeset 251103 | 6b4103d8c3f769d4d97937a9d5a3a9cba8884247 |

parent 142468 | c6f5c1bbcf761369c0d51f85ba1bb110f2f40fd8 |

permissions | -rw-r--r-- |

Bug 1107731 - Upgrade Mozilla 37 to use NSS 3.17.4. a=sledru

This Source Code Form is subject to the terms of the Mozilla Public License, v. 2.0. If a copy of the MPL was not distributed with this file, You can obtain one at http://mozilla.org/MPL/2.0/. Additional MPI utilities ------------------------ The files 'mpprime.h' and 'mpprime.c' define some useful extensions to the MPI library for dealing with prime numbers (in particular, testing for divisbility, and the Rabin-Miller probabilistic primality test). The files 'mplogic.h' and 'mplogic.c' define extensions to the MPI library for doing bitwise logical operations and shifting. This document assumes you have read the help file for the MPI library and understand its conventions. Divisibility (mpprime.h) ------------ To test a number for divisibility by another number: mpp_divis(a, b) - test if b|a mpp_divis_d(a, d) - test if d|a Each of these functions returns MP_YES if its initial argument is divisible by its second, or MP_NO if it is not. Other errors may be returned as appropriate (such as MP_RANGE if you try to test for divisibility by zero). Randomness (mpprime.h) ---------- To generate random data: mpp_random(a) - fill a with random data mpp_random_size(a, p) - fill a with p digits of random data The mpp_random_size() function increases the precision of a to at least p, then fills all those digits randomly. The mp_random() function fills a to its current precision (as determined by the number of significant digits, USED(a)) Note that these functions simply use the C library's rand() function to fill a with random digits up to its precision. This should be adequate for primality testing, but should not be used for cryptographic applications where truly random values are required for security. You should call srand() in your driver program in order to seed the random generator; this function doesn't call it. Primality Testing (mpprime.h) ----------------- mpp_divis_vector(a, v, s, w) - is a divisible by any of the s values in v, and if so, w = which. mpp_divis_primes(a, np) - is a divisible by any of the first np primes? mpp_fermat(a, w) - is a pseudoprime with respect to witness w? mpp_pprime(a, nt) - run nt iterations of Rabin-Miller on a. The mpp_divis_vector() function tests a for divisibility by each member of an array of digits. The array is v, the size of that array is s. Returns MP_YES if a is divisible, and stores the index of the offending digit in w. Returns MP_NO if a is not divisible by any of the digits in the array. A small table of primes is compiled into the library (typically the first 128 primes, although you can change this by editing the file 'primes.c' before you build). The global variable prime_tab_size contains the number of primes in the table, and the values themselves are in the array prime_tab[], which is an array of mp_digit. The mpp_divis_primes() function is basically just a wrapper around mpp_divis_vector() that uses prime_tab[] as the test vector. The np parameter is a pointer to an mp_digit -- on input, it should specify the number of primes to be tested against. If a is divisible by any of the primes, MP_YES is returned and np is given the prime value that divided a (you can use this if you're factoring, for example). Otherwise, MP_NO is returned and np is untouched. The function mpp_fermat() performs Fermat's test, using w as a witness. This test basically relies on the fact that if a is prime, and w is relatively prime to a, then: w^a = w (mod a) That is, w^(a - 1) = 1 (mod a) The function returns MP_YES if the test passes, MP_NO if it fails. If w is relatively prime to a, and the test fails, a is definitely composite. If w is relatively prime to a and the test passes, then a is either prime, or w is a false witness (the probability of this happening depends on the choice of w and of a ... consult a number theory textbook for more information about this). Note: If (w, a) != 1, the output of this test is meaningless. ---- The function mpp_pprime() performs the Rabin-Miller probabilistic primality test for nt rounds. If all the tests pass, MP_YES is returned, and a is probably prime. The probability that an answer of MP_YES is incorrect is no greater than 1 in 4^nt, and in fact is usually much less than that (this is a pessimistic estimate). If any test fails, MP_NO is returned, and a is definitely composite. Bruce Schneier recommends at least 5 iterations of this test for most cryptographic applications; Knuth suggests that 25 are reasonable. Run it as many times as you feel are necessary. See the programs 'makeprime.c' and 'isprime.c' for reasonable examples of how to use these functions for primality testing. Bitwise Logic (mplogic.c) ------------- The four commonest logical operations are implemented as: mpl_not(a, b) - Compute bitwise (one's) complement, b = ~a mpl_and(a, b, c) - Compute bitwise AND, c = a & b mpl_or(a, b, c) - Compute bitwise OR, c = a | b mpl_xor(a, b, c) - Compute bitwise XOR, c = a ^ b Left and right shifts are available as well. These take a number to shift, a destination, and a shift amount. The shift amount must be a digit value between 0 and DIGIT_BIT inclusive; if it is not, MP_RANGE will be returned and the shift will not happen. mpl_rsh(a, b, d) - Compute logical right shift, b = a >> d mpl_lsh(a, b, d) - Compute logical left shift, b = a << d Since these are logical shifts, they fill with zeroes (the library uses a signed magnitude representation, so there are no sign bits to extend anyway). Command-line Utilities ---------------------- A handful of interesting command-line utilities are provided. These are: lap.c - Find the order of a mod m. Usage is 'lap <a> <m>'. This uses a dumb algorithm, so don't use it for a really big modulus. invmod.c - Find the inverse of a mod m, if it exists. Usage is 'invmod <a> <m>' sieve.c - A simple bitmap-based implementation of the Sieve of Eratosthenes. Used to generate the table of primes in primes.c. Usage is 'sieve <nbits>' prng.c - Uses the routines in bbs_rand.{h,c} to generate one or more 32-bit pseudo-random integers. This is mainly an example, not intended for use in a cryptographic application (the system time is the only source of entropy used) dec2hex.c - Convert decimal to hexadecimal hex2dec.c - Convert hexadecimal to decimal basecvt.c - General radix conversion tool (supports 2-64) isprime.c - Probabilistically test an integer for primality using the Rabin-Miller pseudoprime test combined with division by small primes. primegen.c - Generate primes at random. exptmod.c - Perform modular exponentiation ptab.pl - A Perl script to munge the output of the sieve program into a compilable C structure. Other Files ----------- PRIMES - Some randomly generated numbers which are prime with extremely high probability. README - You're reading me already. About the Author ---------------- This software was written by Michael J. Fromberger. You can contact the author as follows: E-mail: <sting@linguist.dartmouth.edu> Postal: 8000 Cummings Hall, Thayer School of Engineering Dartmouth College, Hanover, New Hampshire, USA PGP key: http://linguist.dartmouth.edu/~sting/keys/mjf.html 9736 188B 5AFA 23D6 D6AA BE0D 5856 4525 289D 9907