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| author | Daniel Baumann <daniel.baumann@progress-linux.org> | 2024-04-07 19:33:14 +0000 |
|---|---|---|
| committer | Daniel Baumann <daniel.baumann@progress-linux.org> | 2024-04-07 19:33:14 +0000 |
| commit | 36d22d82aa202bb199967e9512281e9a53db42c9 (patch) | |
| tree | 105e8c98ddea1c1e4784a60a5a6410fa416be2de /security/nss/lib/freebl/mpi/doc | |
| parent | Initial commit. (diff) | |
| download | firefox-esr-36d22d82aa202bb199967e9512281e9a53db42c9.tar.xz firefox-esr-36d22d82aa202bb199967e9512281e9a53db42c9.zip | |
Adding upstream version 115.7.0esr.upstream/115.7.0esr
Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>
Diffstat (limited to 'security/nss/lib/freebl/mpi/doc')
19 files changed, 7610 insertions, 0 deletions
diff --git a/security/nss/lib/freebl/mpi/doc/LICENSE b/security/nss/lib/freebl/mpi/doc/LICENSE new file mode 100644 index 0000000000..35cca68ce9 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/LICENSE @@ -0,0 +1,11 @@ +Within this directory, each of the file listed below is licensed under +the terms given in the file LICENSE-MPL, also in this directory. + +basecvt.pod +gcd.pod +invmod.pod +isprime.pod +lap.pod +mpi-test.pod +prime.txt +prng.pod diff --git a/security/nss/lib/freebl/mpi/doc/LICENSE-MPL b/security/nss/lib/freebl/mpi/doc/LICENSE-MPL new file mode 100644 index 0000000000..41dc2327f1 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/LICENSE-MPL @@ -0,0 +1,3 @@ +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/. diff --git a/security/nss/lib/freebl/mpi/doc/basecvt.pod b/security/nss/lib/freebl/mpi/doc/basecvt.pod new file mode 100644 index 0000000000..c3d87fbc7e --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/basecvt.pod @@ -0,0 +1,65 @@ +# 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/. + +=head1 NAME + + basecvt - radix conversion for arbitrary precision integers + +=head1 SYNOPSIS + + basecvt <ibase> <obase> [values] + +=head1 DESCRIPTION + +The B<basecvt> program is a command-line tool for converting integers +of arbitrary precision from one radix to another. The current version +supports radix values from 2 (binary) to 64, inclusive. The first two +command line arguments specify the input and output radix, in base 10. +Any further arguments are taken to be integers notated in the input +radix, and these are converted to the output radix. The output is +written, one integer per line, to standard output. + +When reading integers, only digits considered "valid" for the input +radix are considered. Processing of an integer terminates when an +invalid input digit is encountered. So, for example, if you set the +input radix to 10 and enter '10ACF', B<basecvt> would assume that you +had entered '10' and ignore the rest of the string. + +If no values are provided, no output is written, but the program +simply terminates with a zero exit status. Error diagnostics are +written to standard error in the event of out-of-range radix +specifications. Regardless of the actual values of the input and +output radix, the radix arguments are taken to be in base 10 (decimal) +notation. + +=head1 DIGITS + +For radices from 2-10, standard ASCII decimal digits 0-9 are used for +both input and output. For radices from 11-36, the ASCII letters A-Z +are also included, following the convention used in hexadecimal. In +this range, input is accepted in either upper or lower case, although +on output only lower-case letters are used. + +For radices from 37-62, the output includes both upper- and lower-case +ASCII letters, and case matters. In this range, case is distinguished +both for input and for output values. + +For radices 63 and 64, the characters '+' (plus) and '/' (forward +solidus) are also used. These are derived from the MIME base64 +encoding scheme. The overall encoding is not the same as base64, +because the ASCII digits are used for the bottom of the range, and the +letters are shifted upward; however, the output will consist of the +same character set. + +This input and output behaviour is inherited from the MPI library used +by B<basecvt>, and so is not configurable at runtime. + +=head1 SEE ALSO + + dec2hex(1), hex2dec(1) + +=head1 AUTHOR + + Michael J. Fromberger <sting@linguist.dartmouth.edu> + Thayer School of Engineering, Hanover, New Hampshire, USA diff --git a/security/nss/lib/freebl/mpi/doc/build b/security/nss/lib/freebl/mpi/doc/build new file mode 100755 index 0000000000..4d75b1e5a2 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/build @@ -0,0 +1,30 @@ +#!/bin/sh +# 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/. + +VERS="1.7p6" +SECT="1" +NAME="MPI Tools" + +echo "Building manual pages ..." +case $# in + 0) + files=`ls *.pod` + ;; + *) + files=$* + ;; +esac + +for name in $files +do + echo -n "$name ... " +# sname=`noext $name` + sname=`basename $name .pod` + pod2man --section="$SECT" --center="$NAME" --release="$VERS" $name > $sname.$SECT + echo "(done)" +done + +echo "Finished building." + diff --git a/security/nss/lib/freebl/mpi/doc/div.txt b/security/nss/lib/freebl/mpi/doc/div.txt new file mode 100644 index 0000000000..c13fb6ef18 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/div.txt @@ -0,0 +1,64 @@ +Division + +This describes the division algorithm used by the MPI library. + +Input: a, b; a > b +Compute: Q, R; a = Qb + R + +The input numbers are normalized so that the high-order digit of b is +at least half the radix. This guarantees that we have a reasonable +way to guess at the digits of the quotient (this method was taken from +Knuth, vol. 2, with adaptations). + +To normalize, test the high-order digit of b. If it is less than half +the radix, multiply both a and b by d, where: + + radix - 1 + d = ----------- + bmax + 1 + +...where bmax is the high-order digit of b. Otherwise, set d = 1. + +Given normalize values for a and b, let the notation a[n] denote the +nth digit of a. Let #a be the number of significant figures of a (not +including any leading zeroes). + + Let R = 0 + Let p = #a - 1 + + while(p >= 0) + do + R = (R * radix) + a[p] + p = p - 1 + while(R < b and p >= 0) + + if(R < b) + break + + q = (R[#R - 1] * radix) + R[#R - 2] + q = q / b[#b - 1] + + T = b * q + + while(T > L) + q = q - 1 + T = T - b + endwhile + + L = L - T + + Q = (Q * radix) + q + + endwhile + +At this point, Q is the quotient, and R is the normalized remainder. +To denormalize R, compute: + + R = (R / d) + +At this point, you are finished. + +------------------------------------------------------------------ + 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/. diff --git a/security/nss/lib/freebl/mpi/doc/expt.txt b/security/nss/lib/freebl/mpi/doc/expt.txt new file mode 100644 index 0000000000..bd9d6f1960 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/expt.txt @@ -0,0 +1,94 @@ +Exponentiation + +For exponentiation, the MPI library uses a simple and fairly standard +square-and-multiply method. The algorithm is this: + +Input: a, b +Output: a ** b + + s = 1 + + while(b != 0) + if(b is odd) + s = s * a + endif + + b = b / 2 + + x = x * x + endwhile + + return s + +The modular exponentiation is done the same way, except replacing: + + s = s * a + +with + s = (s * a) mod m + +and replacing + + x = x * x + +with + + x = (x * x) mod m + +Here is a sample exponentiation using the MPI library, as compared to +the same problem solved by the Unix 'bc' program on my system: + +Computation of 2,381,283 ** 235 + +'bc' says: + +4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\ +4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\ +6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\ +4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\ +6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\ +FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\ +CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\ +5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\ +CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\ +49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\ +5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\ +A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\ +D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\ +92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\ +A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\ +AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\ +E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\ +1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\ +CFFF2E1AC93F3CA264A1B + +MPI says: + +4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\ +4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\ +6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\ +4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\ +6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\ +FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\ +CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\ +5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\ +CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\ +49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\ +5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\ +A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\ +D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\ +92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\ +A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\ +AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\ +E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\ +1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\ +CFFF2E1AC93F3CA264A1B + +Diff says: +% diff bc.txt mp.txt +% + +------------------------------------------------------------------ + 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/. diff --git a/security/nss/lib/freebl/mpi/doc/gcd.pod b/security/nss/lib/freebl/mpi/doc/gcd.pod new file mode 100644 index 0000000000..b5b8fa34fd --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/gcd.pod @@ -0,0 +1,28 @@ +# 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/. + +=head1 NAME + + gcd - compute greatest common divisor of two integers + +=head1 SYNOPSIS + + gcd <a> <b> + +=head1 DESCRIPTION + +The B<gcd> program computes the greatest common divisor of two +arbitrary-precision integers I<a> and I<b>. The result is written in +standard decimal notation to the standard output. + +If I<b> is zero, B<gcd> will print an error message and exit. + +=head1 SEE ALSO + +invmod(1), isprime(1), lap(1) + +=head1 AUTHOR + + Michael J. Fromberger <sting@linguist.dartmouth.edu> + Thayer School of Engineering, Hanover, New Hampshire, USA diff --git a/security/nss/lib/freebl/mpi/doc/invmod.pod b/security/nss/lib/freebl/mpi/doc/invmod.pod new file mode 100644 index 0000000000..0194f44884 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/invmod.pod @@ -0,0 +1,34 @@ +# 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/. + +=head1 NAME + + invmod - compute modular inverse of an integer + +=head1 SYNOPSIS + + invmod <a> <m> + +=head1 DESCRIPTION + +The B<invmod> program computes the inverse of I<a>, modulo I<m>, if +that inverse exists. Both I<a> and I<m> are arbitrary-precision +integers in decimal notation. The result is written in standard +decimal notation to the standard output. + +If there is no inverse, the message: + + No inverse + +...will be printed to the standard output (an inverse exists if and +only if the greatest common divisor of I<a> and I<m> is 1). + +=head1 SEE ALSO + +gcd(1), isprime(1), lap(1) + +=head1 AUTHOR + + Michael J. Fromberger <sting@linguist.dartmouth.edu> + Thayer School of Engineering, Hanover, New Hampshire, USA diff --git a/security/nss/lib/freebl/mpi/doc/isprime.pod b/security/nss/lib/freebl/mpi/doc/isprime.pod new file mode 100644 index 0000000000..a8ec1f7ee3 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/isprime.pod @@ -0,0 +1,63 @@ +# 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/. + +=head1 NAME + + isprime - probabilistic primality testing + +=head1 SYNOPSIS + + isprime <a> + +=head1 DESCRIPTION + +The B<isprime> program attempts to determine whether the arbitrary +precision integer I<a> is prime. It first tests I<a> for divisibility +by the first 170 or so small primes, and assuming I<a> is not +divisible by any of these, applies 15 iterations of the Rabin-Miller +probabilistic primality test. + +If the program discovers that the number is composite, it will print: + + Not prime (reason) + +Where I<reason> is either: + + divisible by small prime x + +Or: + + failed nth pseudoprime test + +In the first case, I<x> indicates the first small prime factor that +was found. In the second case, I<n> indicates which of the +pseudoprime tests failed (numbered from 1) + +If this happens, the number is definitely not prime. However, if the +number succeeds, this message results: + + Probably prime, 1 in 4^15 chance of false positive + +If this happens, the number is prime with very high probability, but +its primality has not been absolutely proven, only demonstrated to a +very convincing degree. + +The value I<a> can be input in standard decimal notation, or, if it is +prefixed with I<Ox>, it will be read as hexadecimal. + +=head1 ENVIRONMENT + +You can control how many iterations of Rabin-Miller are performed on +the candidate number by setting the I<RM_TESTS> environment variable +to an integer value before starting up B<isprime>. This will change +the output slightly if the number passes all the tests. + +=head1 SEE ALSO + +gcd(1), invmod(1), lap(1) + +=head1 AUTHOR + + Michael J. Fromberger <sting@linguist.dartmouth.edu> + Thayer School of Engineering, Hanover, New Hampshire, USA diff --git a/security/nss/lib/freebl/mpi/doc/lap.pod b/security/nss/lib/freebl/mpi/doc/lap.pod new file mode 100644 index 0000000000..47539fbbf9 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/lap.pod @@ -0,0 +1,36 @@ +# 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/. + +=head1 NAME + + lap - compute least annihilating power of a number + +=head1 SYNOPSIS + + lap <a> <m> + +=head1 DESCRIPTION + +The B<lap> program computes the order of I<a> modulo I<m>, for +arbitrary precision integers I<a> and I<m>. The B<order> of I<a> +modulo I<m> is defined as the smallest positive value I<n> for which +I<a> raised to the I<n>th power, modulo I<m>, is equal to 1. The +order may not exist, if I<m> is composite. + +=head1 RESTRICTIONS + +This program is very slow, especially for large moduli. It is +intended as a way to help find primitive elements in a modular field, +but it does not do so in a particularly inefficient manner. It was +written simply to help verify that a particular candidate does not +have an obviously short cycle mod I<m>. + +=head1 SEE ALSO + +gcd(1), invmod(1), isprime(1) + +=head1 AUTHOR + + Michael J. Fromberger <sting@linguist.dartmouth.edu> + Thayer School of Engineering, Hanover, New Hampshire, USA diff --git a/security/nss/lib/freebl/mpi/doc/mpi-test.pod b/security/nss/lib/freebl/mpi/doc/mpi-test.pod new file mode 100644 index 0000000000..b05f866e5e --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/mpi-test.pod @@ -0,0 +1,51 @@ +# 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/. + +=head1 NAME + + mpi-test - automated test program for MPI library + +=head1 SYNOPSIS + + mpi-test <suite-name> [quiet] + mpi-test list + mpi-test help + +=head1 DESCRIPTION + +The B<mpi-test> program is a general unit test driver for the MPI +library. It is used to verify that the library works as it is +supposed to on your architecture. As with most such things, passing +all the tests in B<mpi-test> does not guarantee the code is correct, +but if any of them fail, there are certainly problems. + +Each major function of the library can be tested individually. For a +list of the test suites understood by B<mpi-test>, run it with the +I<list> command line option: + + mpi-test list + +This will display a list of the available test suites and a brief +synopsis of what each one does. For a brief overview of this +document, run B<mpi-test> I<help>. + +B<mpi-test> exits with a zero status if the selected test succeeds, or +a nonzero status if it fails. If a I<suite-name> which is not +understood by B<mpi-test> is given, a diagnostic is printed to the +standard error, and the program exits with a result code of 2. If a +test fails, the result code will be 1, and a diagnostic is ordinarily +printed to the standard error. However, if the I<quiet> option is +provided, these diagnostics will be suppressed. + +=head1 RESTRICTIONS + +Only a few canned test cases are provided. The solutions have been +verified using the GNU bc(1) program, so bugs there may cause problems +here; however, this is very unlikely, so if a test fails, it is almost +certainly my fault, not bc(1)'s. + +=head1 AUTHOR + + Michael J. Fromberger <sting@linguist.dartmouth.edu> + Thayer School of Engineering, Hanover, New Hampshire, USA diff --git a/security/nss/lib/freebl/mpi/doc/mul.txt b/security/nss/lib/freebl/mpi/doc/mul.txt new file mode 100644 index 0000000000..975f56ddbe --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/mul.txt @@ -0,0 +1,77 @@ +Multiplication + +This describes the multiplication algorithm used by the MPI library. + +This is basically a standard "schoolbook" algorithm. It is slow -- +O(mn) for m = #a, n = #b -- but easy to implement and verify. +Basically, we run two nested loops, as illustrated here (R is the +radix): + +k = 0 +for j <- 0 to (#b - 1) + for i <- 0 to (#a - 1) + w = (a[j] * b[i]) + k + c[i+j] + c[i+j] = w mod R + k = w div R + endfor + c[i+j] = k; + k = 0; +endfor + +It is necessary that 'w' have room for at least two radix R digits. +The product of any two digits in radix R is at most: + + (R - 1)(R - 1) = R^2 - 2R + 1 + +Since a two-digit radix-R number can hold R^2 - 1 distinct values, +this insures that the product will fit into the two-digit register. + +To insure that two digits is enough for w, we must also show that +there is room for the carry-in from the previous multiplication, and +the current value of the product digit that is being recomputed. +Assuming each of these may be as big as R - 1 (and no larger, +certainly), two digits will be enough if and only if: + + (R^2 - 2R + 1) + 2(R - 1) <= R^2 - 1 + +Solving this equation shows that, indeed, this is the case: + + R^2 - 2R + 1 + 2R - 2 <= R^2 - 1 + + R^2 - 1 <= R^2 - 1 + +This suggests that a good radix would be one more than the largest +value that can be held in half a machine word -- so, for example, as +in this implementation, where we used a radix of 65536 on a machine +with 4-byte words. Another advantage of a radix of this sort is that +binary-level operations are easy on numbers in this representation. + +Here's an example multiplication worked out longhand in radix-10, +using the above algorithm: + + a = 999 + b = x 999 + ------------- + p = 98001 + +w = (a[jx] * b[ix]) + kin + c[ix + jx] +c[ix+jx] = w % RADIX +k = w / RADIX + product +ix jx a[jx] b[ix] kin w c[i+j] kout 000000 +0 0 9 9 0 81+0+0 1 8 000001 +0 1 9 9 8 81+8+0 9 8 000091 +0 2 9 9 8 81+8+0 9 8 000991 + 8 0 008991 +1 0 9 9 0 81+0+9 0 9 008901 +1 1 9 9 9 81+9+9 9 9 008901 +1 2 9 9 9 81+9+8 8 9 008901 + 9 0 098901 +2 0 9 9 0 81+0+9 0 9 098001 +2 1 9 9 9 81+9+8 8 9 098001 +2 2 9 9 9 81+9+9 9 9 098001 + +------------------------------------------------------------------ + 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/. diff --git a/security/nss/lib/freebl/mpi/doc/pi.txt b/security/nss/lib/freebl/mpi/doc/pi.txt new file mode 100644 index 0000000000..a6ef91137f --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/pi.txt @@ -0,0 +1,53 @@ +This file describes how pi is computed by the program in 'pi.c' (see +the utils subdirectory). + +Basically, we use Machin's formula, which is what everyone in the +world uses as a simple method for computing approximations to pi. +This works for up to a few thousand digits without too much effort. +Beyond that, though, it gets too slow. + +Machin's formula states: + + pi := 16 * arctan(1/5) - 4 * arctan(1/239) + +We compute this in integer arithmetic by first multiplying everything +through by 10^d, where 'd' is the number of digits of pi we wanted to +compute. It turns out, the last few digits will be wrong, but the +number that are wrong is usually very small (ordinarly only 2-3). +Having done this, we compute the arctan() function using the formula: + + 1 1 1 1 1 + arctan(1/x) := --- - ----- + ----- - ----- + ----- - ... + x 3 x^3 5 x^5 7 x^7 9 x^9 + +This is done iteratively by computing the first term manually, and +then iteratively dividing x^2 and k, where k = 3, 5, 7, ... out of the +current figure. This is then added to (or subtracted from) a running +sum, as appropriate. The iteration continues until we overflow our +available precision and the current figure goes to zero under integer +division. At that point, we're finished. + +Actually, we get a couple extra bits of precision out of the fact that +we know we're computing y * arctan(1/x), by setting up the multiplier +as: + + y * 10^d + +... instead of just 10^d. There is also a bit of cleverness in how +the loop is constructed, to avoid special-casing the first term. +Check out the code for arctan() in 'pi.c', if you are interested in +seeing how it is set up. + +Thanks to Jason P. for this algorithm, which I assembled from notes +and programs found on his cool "Pile of Pi Programs" page, at: + + http://www.isr.umd.edu/~jasonp/pipage.html + +Thanks also to Henrik Johansson <Henrik.Johansson@Nexus.Comm.SE>, from +whose pi program I borrowed the clever idea of pre-multiplying by x in +order to avoid a special case on the loop iteration. + +------------------------------------------------------------------ + 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/. diff --git a/security/nss/lib/freebl/mpi/doc/prime.txt b/security/nss/lib/freebl/mpi/doc/prime.txt new file mode 100644 index 0000000000..694797d5f3 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/prime.txt @@ -0,0 +1,6542 @@ +2 +3 +5 +7 +11 +13 +17 +19 +23 +29 +31 +37 +41 +43 +47 +53 +59 +61 +67 +71 +73 +79 +83 +89 +97 +101 +103 +107 +109 +113 +127 +131 +137 +139 +149 +151 +157 +163 +167 +173 +179 +181 +191 +193 +197 +199 +211 +223 +227 +229 +233 +239 +241 +251 +257 +263 +269 +271 +277 +281 +283 +293 +307 +311 +313 +317 +331 +337 +347 +349 +353 +359 +367 +373 +379 +383 +389 +397 +401 +409 +419 +421 +431 +433 +439 +443 +449 +457 +461 +463 +467 +479 +487 +491 +499 +503 +509 +521 +523 +541 +547 +557 +563 +569 +571 +577 +587 +593 +599 +601 +607 +613 +617 +619 +631 +641 +643 +647 +653 +659 +661 +673 +677 +683 +691 +701 +709 +719 +727 +733 +739 +743 +751 +757 +761 +769 +773 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+64373 +64381 +64399 +64403 +64433 +64439 +64451 +64453 +64483 +64489 +64499 +64513 +64553 +64567 +64577 +64579 +64591 +64601 +64609 +64613 +64621 +64627 +64633 +64661 +64663 +64667 +64679 +64693 +64709 +64717 +64747 +64763 +64781 +64783 +64793 +64811 +64817 +64849 +64853 +64871 +64877 +64879 +64891 +64901 +64919 +64921 +64927 +64937 +64951 +64969 +64997 +65003 +65011 +65027 +65029 +65033 +65053 +65063 +65071 +65089 +65099 +65101 +65111 +65119 +65123 +65129 +65141 +65147 +65167 +65171 +65173 +65179 +65183 +65203 +65213 +65239 +65257 +65267 +65269 +65287 +65293 +65309 +65323 +65327 +65353 +65357 +65371 +65381 +65393 +65407 +65413 +65419 +65423 +65437 +65447 +65449 +65479 +65497 +65519 +65521 diff --git a/security/nss/lib/freebl/mpi/doc/prng.pod b/security/nss/lib/freebl/mpi/doc/prng.pod new file mode 100644 index 0000000000..6da4d4a9c4 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/prng.pod @@ -0,0 +1,38 @@ +=head1 NAME + + prng - pseudo-random number generator + +=head1 SYNOPSIS + + prng [count] + +=head1 DESCRIPTION + +B<Prng> generates 32-bit pseudo-random integers using the +Blum-Blum-Shub (BBS) quadratic residue generator. It is seeded using +the standard C library's rand() function, which itself seeded from the +system clock and the process ID number. Thus, the values generated +are not particularly useful for cryptographic applications, but they +are in general much better than the typical output of the usual +multiplicative congruency generator used by most runtime libraries. + +You may optionally specify how many random values should be generated +by giving a I<count> argument on the command line. If you do not +specify a count, only one random value will be generated. The results +are output to the standard output in decimal notation, one value per +line. + +=head1 RESTRICTIONS + +As stated above, B<prng> uses the C library's rand() function to seed +the generator, so it is not terribly suitable for cryptographic +applications. Also note that each time you run the program, a new +seed is generated, so it is better to run it once with a I<count> +parameter than it is to run it multiple times to generate several +values. + +=head1 AUTHOR + + Michael J. Fromberger <sting@linguist.dartmouth.edu> + Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved + Thayer School of Engineering, Dartmouth College, Hanover, NH USA diff --git a/security/nss/lib/freebl/mpi/doc/redux.txt b/security/nss/lib/freebl/mpi/doc/redux.txt new file mode 100644 index 0000000000..0df0f0390a --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/redux.txt @@ -0,0 +1,86 @@ +Modular Reduction + +Usually, modular reduction is accomplished by long division, using the +mp_div() or mp_mod() functions. However, when performing modular +exponentiation, you spend a lot of time reducing by the same modulus +again and again. For this purpose, doing a full division for each +multiplication is quite inefficient. + +For this reason, the mp_exptmod() function does not perform modular +reductions in the usual way, but instead takes advantage of an +algorithm due to Barrett, as described by Menezes, Oorschot and +VanStone in their book _Handbook of Applied Cryptography_, published +by the CRC Press (see Chapter 14 for details). This method reduces +most of the computation of reduction to efficient shifting and masking +operations, and avoids the multiple-precision division entirely. + +Here is a brief synopsis of Barrett reduction, as it is implemented in +this library. + +Let b denote the radix of the computation (one more than the maximum +value that can be denoted by an mp_digit). Let m be the modulus, and +let k be the number of significant digits of m. Let x be the value to +be reduced modulo m. By the Division Theorem, there exist unique +integers Q and R such that: + + x = Qm + R, 0 <= R < m + +Barrett reduction takes advantage of the fact that you can easily +approximate Q to within two, given a value M such that: + + 2k + b + M = floor( ----- ) + m + +Computation of M requires a full-precision division step, so if you +are only doing a single reduction by m, you gain no advantage. +However, when multiple reductions by the same m are required, this +division need only be done once, beforehand. Using this, we can use +the following equation to compute Q', an approximation of Q: + + x + floor( ------ ) M + k-1 + b +Q' = floor( ----------------- ) + k+1 + b + +The divisions by b^(k-1) and b^(k+1) and the floor() functions can be +efficiently implemented with shifts and masks, leaving only a single +multiplication to be performed to get this approximation. It can be +shown that Q - 2 <= Q' <= Q, so in the worst case, we can get out with +two additional subtractions to bring the value into line with the +actual value of Q. + +Once we've got Q', we basically multiply that by m and subtract from +x, yielding: + + x - Q'm = Qm + R - Q'm + +Since we know the constraint on Q', this is one of: + + R + m + R + 2m + R + +Since R < m by the Division Theorem, we can simply subtract off m +until we get a value in the correct range, which will happen with no +more than 2 subtractions: + + v = x - Q'm + + while(v >= m) + v = v - m + endwhile + + +In random performance trials, modular exponentiation using this method +of reduction gave around a 40% speedup over using the division for +reduction. + +------------------------------------------------------------------ + 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/. diff --git a/security/nss/lib/freebl/mpi/doc/sqrt.txt b/security/nss/lib/freebl/mpi/doc/sqrt.txt new file mode 100644 index 0000000000..4529cbfc46 --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/sqrt.txt @@ -0,0 +1,50 @@ +Square Root + +A simple iterative algorithm is used to compute the greatest integer +less than or equal to the square root. Essentially, this is Newton's +linear approximation, computed by finding successive values of the +equation: + + x[k]^2 - V +x[k+1] = x[k] - ------------ + 2 x[k] + +...where V is the value for which the square root is being sought. In +essence, what is happening here is that we guess a value for the +square root, then figure out how far off we were by squaring our guess +and subtracting the target. Using this value, we compute a linear +approximation for the error, and adjust the "guess". We keep doing +this until the precision gets low enough that the above equation +yields a quotient of zero. At this point, our last guess is one +greater than the square root we're seeking. + +The initial guess is computed by dividing V by 4, which is a heuristic +I have found to be fairly good on average. This also has the +advantage of being very easy to compute efficiently, even for large +values. + +So, the resulting algorithm works as follows: + + x = V / 4 /* compute initial guess */ + + loop + t = (x * x) - V /* Compute absolute error */ + u = 2 * x /* Adjust by tangent slope */ + t = t / u + + /* Loop is done if error is zero */ + if(t == 0) + break + + /* Adjust guess by error term */ + x = x - t + end + + x = x - 1 + +The result of the computation is the value of x. + +------------------------------------------------------------------ + 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/. diff --git a/security/nss/lib/freebl/mpi/doc/square.txt b/security/nss/lib/freebl/mpi/doc/square.txt new file mode 100644 index 0000000000..edbb97882c --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/square.txt @@ -0,0 +1,72 @@ +Squaring Algorithm + +When you are squaring a value, you can take advantage of the fact that +half the multiplications performed by the more general multiplication +algorithm (see 'mul.txt' for a description) are redundant when the +multiplicand equals the multiplier. + +In particular, the modified algorithm is: + +k = 0 +for j <- 0 to (#a - 1) + w = c[2*j] + (a[j] ^ 2); + k = w div R + + for i <- j+1 to (#a - 1) + w = (2 * a[j] * a[i]) + k + c[i+j] + c[i+j] = w mod R + k = w div R + endfor + c[i+j] = k; + k = 0; +endfor + +On the surface, this looks identical to the multiplication algorithm; +however, note the following differences: + + - precomputation of the leading term in the outer loop + + - i runs from j+1 instead of from zero + + - doubling of a[i] * a[j] in the inner product + +Unfortunately, the construction of the inner product is such that we +need more than two digits to represent the inner product, in some +cases. In a C implementation, this means that some gymnastics must be +performed in order to handle overflow, for which C has no direct +abstraction. We do this by observing the following: + +If we have multiplied a[i] and a[j], and the product is more than half +the maximum value expressible in two digits, then doubling this result +will overflow into a third digit. If this occurs, we take note of the +overflow, and double it anyway -- C integer arithmetic ignores +overflow, so the two digits we get back should still be valid, modulo +the overflow. + +Having doubled this value, we now have to add in the remainders and +the digits already computed by earlier steps. If we did not overflow +in the previous step, we might still cause an overflow here. That +will happen whenever the maximum value expressible in two digits, less +the amount we have to add, is greater than the result of the previous +step. Thus, the overflow computation is: + + + u = 0 + w = a[i] * a[j] + + if(w > (R - 1)/ 2) + u = 1; + + w = w * 2 + v = c[i + j] + k + + if(u == 0 && (R - 1 - v) < w) + u = 1 + +If there is an overflow, u will be 1, otherwise u will be 0. The rest +of the parameters are the same as they are in the above description. + +------------------------------------------------------------------ + 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/. diff --git a/security/nss/lib/freebl/mpi/doc/timing.txt b/security/nss/lib/freebl/mpi/doc/timing.txt new file mode 100644 index 0000000000..58f37c9dff --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/timing.txt @@ -0,0 +1,213 @@ +MPI Library Timing Tests + +Hardware/OS +(A) SGI O2 1 x MIPS R10000 250MHz IRIX 6.5.3 +(B) IBM RS/6000 43P-240 1 x PowerPC 603e 223MHz AIX 4.3 +(C) Dell GX1/L+ 1 x Pentium III 550MHz Linux 2.2.12-20 +(D) PowerBook G3 1 x PowerPC 750 266MHz LinuxPPC 2.2.6-15apmac +(E) PowerBook G3 1 x PowerPC 750 266MHz MacOS 8.5.1 +(F) PowerBook G3 1 x PowerPC 750 400MHz MacOS 9.0.2 + +Compiler +(1) MIPSpro C 7.2.1 -O3 optimizations +(2) GCC 2.95.1 -O3 optimizations +(3) IBM AIX xlc -O3 optimizations (version unknown) +(4) EGCS 2.91.66 -O3 optimizations +(5) Metrowerks CodeWarrior 5.0 C, all optimizations +(6) MIPSpro C 7.30 -O3 optimizations +(7) same as (6), with optimized libmalloc.so + +Timings are given in seconds, computed using the C library's clock() +function. The first column gives the hardware and compiler +configuration used for the test. The second column indicates the +number of tests that were aggregated to get the statistics for that +size. These were compiled using 16 bit digits. + +Source data were generated randomly using a fixed seed, so they should +be internally consistent, but may vary on different systems depending +on the C library. Also, since the resolution of the timer accessed by +clock() varies, there may be some variance in the precision of these +measurements. + +Prime Generation (primegen) + +128 bits: +A1 200 min=0.03, avg=0.19, max=0.72, sum=38.46 +A2 200 min=0.02, avg=0.16, max=0.62, sum=32.55 +B3 200 min=0.01, avg=0.07, max=0.22, sum=13.29 +C4 200 min=0.00, avg=0.03, max=0.20, sum=6.14 +D4 200 min=0.00, avg=0.05, max=0.33, sum=9.70 +A6 200 min=0.01, avg=0.09, max=0.36, sum=17.48 +A7 200 min=0.00, avg=0.05, max=0.24, sum=10.07 + +192 bits: +A1 200 min=0.05, avg=0.45, max=3.13, sum=89.96 +A2 200 min=0.04, avg=0.39, max=2.61, sum=77.55 +B3 200 min=0.02, avg=0.18, max=1.25, sum=36.97 +C4 200 min=0.01, avg=0.09, max=0.33, sum=18.24 +D4 200 min=0.02, avg=0.15, max=0.54, sum=29.63 +A6 200 min=0.02, avg=0.24, max=1.70, sum=47.84 +A7 200 min=0.01, avg=0.15, max=1.05, sum=30.88 + +256 bits: +A1 200 min=0.08, avg=0.92, max=6.13, sum=184.79 +A2 200 min=0.06, avg=0.76, max=5.03, sum=151.11 +B3 200 min=0.04, avg=0.41, max=2.68, sum=82.35 +C4 200 min=0.02, avg=0.19, max=0.69, sum=37.91 +D4 200 min=0.03, avg=0.31, max=1.15, sum=63.00 +A6 200 min=0.04, avg=0.48, max=3.13, sum=95.46 +A7 200 min=0.03, avg=0.37, max=2.36, sum=73.60 + +320 bits: +A1 200 min=0.11, avg=1.59, max=6.14, sum=318.81 +A2 200 min=0.09, avg=1.27, max=4.93, sum=254.03 +B3 200 min=0.07, avg=0.82, max=3.13, sum=163.80 +C4 200 min=0.04, avg=0.44, max=1.91, sum=87.59 +D4 200 min=0.06, avg=0.73, max=3.22, sum=146.73 +A6 200 min=0.07, avg=0.93, max=3.50, sum=185.01 +A7 200 min=0.05, avg=0.76, max=2.94, sum=151.78 + +384 bits: +A1 200 min=0.16, avg=2.69, max=11.41, sum=537.89 +A2 200 min=0.13, avg=2.15, max=9.03, sum=429.14 +B3 200 min=0.11, avg=1.54, max=6.49, sum=307.78 +C4 200 min=0.06, avg=0.81, max=4.84, sum=161.13 +D4 200 min=0.10, avg=1.38, max=8.31, sum=276.81 +A6 200 min=0.11, avg=1.73, max=7.36, sum=345.55 +A7 200 min=0.09, avg=1.46, max=6.12, sum=292.02 + +448 bits: +A1 200 min=0.23, avg=3.36, max=15.92, sum=672.63 +A2 200 min=0.17, avg=2.61, max=12.25, sum=522.86 +B3 200 min=0.16, avg=2.10, max=9.83, sum=420.86 +C4 200 min=0.09, avg=1.44, max=7.64, sum=288.36 +D4 200 min=0.16, avg=2.50, max=13.29, sum=500.17 +A6 200 min=0.15, avg=2.31, max=10.81, sum=461.58 +A7 200 min=0.14, avg=2.03, max=9.53, sum=405.16 + +512 bits: +A1 200 min=0.30, avg=6.12, max=22.18, sum=1223.35 +A2 200 min=0.25, avg=4.67, max=16.90, sum=933.18 +B3 200 min=0.23, avg=4.13, max=14.94, sum=825.45 +C4 200 min=0.13, avg=2.08, max=9.75, sum=415.22 +D4 200 min=0.24, avg=4.04, max=20.18, sum=808.11 +A6 200 min=0.22, avg=4.47, max=16.19, sum=893.83 +A7 200 min=0.20, avg=4.03, max=14.65, sum=806.02 + +Modular Exponentation (metime) + +The following results are aggregated from 200 pseudo-randomly +generated tests, based on a fixed seed. + + base, exponent, and modulus size (bits) +P/C 128 192 256 320 384 448 512 640 768 896 1024 +------- ----------------------------------------------------------------- +A1 0.015 0.027 0.047 0.069 0.098 0.133 0.176 0.294 0.458 0.680 1.040 +A2 0.013 0.024 0.037 0.053 0.077 0.102 0.133 0.214 0.326 0.476 0.668 +B3 0.005 0.011 0.021 0.036 0.056 0.084 0.121 0.222 0.370 0.573 0.840 +C4 0.002 0.006 0.011 0.020 0.032 0.048 0.069 0.129 0.223 0.344 0.507 +D4 0.004 0.010 0.019 0.034 0.056 0.085 0.123 0.232 0.390 0.609 0.899 +E5 0.007 0.015 0.031 0.055 0.088 0.133 0.183 0.342 0.574 0.893 1.317 +A6 0.008 0.016 0.038 0.042 0.064 0.093 0.133 0.239 0.393 0.604 0.880 +A7 0.005 0.011 0.020 0.036 0.056 0.083 0.121 0.223 0.374 0.583 0.855 + +Multiplication and Squaring tests, (mulsqr) + +The following results are aggregated from 500000 pseudo-randomly +generated tests, based on a per-run wall-clock seed. Times are given +in seconds, except where indicated in microseconds (us). + +(A1) + +bits multiply square ad percent time/mult time/square +64 9.33 9.15 > 1.9 18.7us 18.3us +128 10.88 10.44 > 4.0 21.8us 20.9us +192 13.30 11.89 > 10.6 26.7us 23.8us +256 14.88 12.64 > 15.1 29.8us 25.3us +320 18.64 15.01 > 19.5 37.3us 30.0us +384 23.11 17.70 > 23.4 46.2us 35.4us +448 28.28 20.88 > 26.2 56.6us 41.8us +512 34.09 24.51 > 28.1 68.2us 49.0us +640 47.86 33.25 > 30.5 95.7us 66.5us +768 64.91 43.54 > 32.9 129.8us 87.1us +896 84.49 55.48 > 34.3 169.0us 111.0us +1024 107.25 69.21 > 35.5 214.5us 138.4us +1536 227.97 141.91 > 37.8 456.0us 283.8us +2048 394.05 242.15 > 38.5 788.1us 484.3us + +(A2) + +bits multiply square ad percent time/mult time/square +64 7.87 7.95 < 1.0 15.7us 15.9us +128 9.40 9.19 > 2.2 18.8us 18.4us +192 11.15 10.59 > 5.0 22.3us 21.2us +256 12.02 11.16 > 7.2 24.0us 22.3us +320 14.62 13.43 > 8.1 29.2us 26.9us +384 17.72 15.80 > 10.8 35.4us 31.6us +448 21.24 18.51 > 12.9 42.5us 37.0us +512 25.36 21.78 > 14.1 50.7us 43.6us +640 34.57 29.00 > 16.1 69.1us 58.0us +768 46.10 37.60 > 18.4 92.2us 75.2us +896 58.94 47.72 > 19.0 117.9us 95.4us +1024 73.76 59.12 > 19.8 147.5us 118.2us +1536 152.00 118.80 > 21.8 304.0us 237.6us +2048 259.41 199.57 > 23.1 518.8us 399.1us + +(B3) + +bits multiply square ad percent time/mult time/square +64 2.60 2.47 > 5.0 5.20us 4.94us +128 4.43 4.06 > 8.4 8.86us 8.12us +192 7.03 6.10 > 13.2 14.1us 12.2us +256 10.44 8.59 > 17.7 20.9us 17.2us +320 14.44 11.64 > 19.4 28.9us 23.3us +384 19.12 15.08 > 21.1 38.2us 30.2us +448 24.55 19.09 > 22.2 49.1us 38.2us +512 31.03 23.53 > 24.2 62.1us 47.1us +640 45.05 33.80 > 25.0 90.1us 67.6us +768 63.02 46.05 > 26.9 126.0us 92.1us +896 83.74 60.29 > 28.0 167.5us 120.6us +1024 106.73 76.65 > 28.2 213.5us 153.3us +1536 228.94 160.98 > 29.7 457.9us 322.0us +2048 398.08 275.93 > 30.7 796.2us 551.9us + +(C4) + +bits multiply square ad percent time/mult time/square +64 1.34 1.28 > 4.5 2.68us 2.56us +128 2.76 2.59 > 6.2 5.52us 5.18us +192 4.52 4.16 > 8.0 9.04us 8.32us +256 6.64 5.99 > 9.8 13.3us 12.0us +320 9.20 8.13 > 11.6 18.4us 16.3us +384 12.01 10.58 > 11.9 24.0us 21.2us +448 15.24 13.33 > 12.5 30.5us 26.7us +512 19.02 16.46 > 13.5 38.0us 32.9us +640 27.56 23.54 > 14.6 55.1us 47.1us +768 37.89 31.78 > 16.1 75.8us 63.6us +896 49.24 41.42 > 15.9 98.5us 82.8us +1024 62.59 52.18 > 16.6 125.2us 104.3us +1536 131.66 107.72 > 18.2 263.3us 215.4us +2048 226.45 182.95 > 19.2 453.0us 365.9us + +(A7) + +bits multiply square ad percent time/mult time/square +64 1.74 1.71 > 1.7 3.48us 3.42us +128 3.48 2.96 > 14.9 6.96us 5.92us +192 5.74 4.60 > 19.9 11.5us 9.20us +256 8.75 6.61 > 24.5 17.5us 13.2us +320 12.5 8.99 > 28.1 25.0us 18.0us +384 16.9 11.9 > 29.6 33.8us 23.8us +448 22.2 15.2 > 31.7 44.4us 30.4us +512 28.3 19.0 > 32.7 56.6us 38.0us +640 42.4 28.0 > 34.0 84.8us 56.0us +768 59.4 38.5 > 35.2 118.8us 77.0us +896 79.5 51.2 > 35.6 159.0us 102.4us +1024 102.6 65.5 > 36.2 205.2us 131.0us +1536 224.3 140.6 > 37.3 448.6us 281.2us +2048 393.4 244.3 > 37.9 786.8us 488.6us + +------------------------------------------------------------------ + 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/. |