From b43e639bf65cb0935fa75da81d1da282c0cdc8f2 Mon Sep 17 00:00:00 2001 From: Brian Anderson Date: Tue, 8 Jan 2013 13:53:45 -0800 Subject: [PATCH] Remove unused bigint from runtime --- configure | 2 +- src/rt/bigint/bigint.h | 294 ------- src/rt/bigint/bigint_ext.cpp | 553 ------------- src/rt/bigint/bigint_int.cpp | 1428 ---------------------------------- src/rt/bigint/low_primes.h | 1069 ------------------------- 5 files changed, 1 insertion(+), 3345 deletions(-) delete mode 100644 src/rt/bigint/bigint.h delete mode 100644 src/rt/bigint/bigint_ext.cpp delete mode 100644 src/rt/bigint/bigint_int.cpp delete mode 100644 src/rt/bigint/low_primes.h diff --git a/configure b/configure index f2afa2d26ef..1e1b725ff47 100755 --- a/configure +++ b/configure @@ -578,7 +578,7 @@ for t in $CFG_TARGET_TRIPLES do make_dir rt/$t for i in \ - isaac linenoise bigint sync test arch/i386 arch/x86_64 \ + isaac linenoise sync test arch/i386 arch/x86_64 \ libuv libuv/src/ares libuv/src/eio libuv/src/ev do make_dir rt/$t/$i diff --git a/src/rt/bigint/bigint.h b/src/rt/bigint/bigint.h deleted file mode 100644 index b4c48f0376c..00000000000 --- a/src/rt/bigint/bigint.h +++ /dev/null @@ -1,294 +0,0 @@ -/* bigint.h - include file for bigint package -** -** This library lets you do math on arbitrarily large integers. It's -** pretty fast - compared with the multi-precision routines in the "bc" -** calculator program, these routines are between two and twelve times faster, -** except for division which is maybe half as fast. -** -** The calling convention is a little unusual. There's a basic problem -** with writing a math library in a language that doesn't do automatic -** garbage collection - what do you do about intermediate results? -** You'd like to be able to write code like this: -** -** d = bi_sqrt( bi_add( bi_multiply( x, x ), bi_multiply( y, y ) ) ); -** -** That works fine when the numbers being passed back and forth are -** actual values - ints, floats, or even fixed-size structs. However, -** when the numbers can be any size, as in this package, then you have -** to pass them around as pointers to dynamically-allocated objects. -** Those objects have to get de-allocated after you are done with them. -** But how do you de-allocate the intermediate results in a complicated -** multiple-call expression like the above? -** -** There are two common solutions to this problem. One, switch all your -** code to a language that provides automatic garbage collection, for -** example Java. This is a fine idea and I recommend you do it wherever -** it's feasible. Two, change your routines to use a calling convention -** that prevents people from writing multiple-call expressions like that. -** The resulting code will be somewhat clumsy-looking, but it will work -** just fine. -** -** This package uses a third method, which I haven't seen used anywhere -** before. It's simple: each number can be used precisely once, after -** which it is automatically de-allocated. This handles the anonymous -** intermediate values perfectly. Named values still need to be copied -** and freed explicitly. Here's the above example using this convention: -** -** d = bi_sqrt( bi_add( -** bi_multiply( bi_copy( x ), bi_copy( x ) ), -** bi_multiply( bi_copy( y ), bi_copy( y ) ) ) ); -** bi_free( x ); -** bi_free( y ); -** -** Or, since the package contains a square routine, you could just write: -** -** d = bi_sqrt( bi_add( bi_square( x ), bi_square( y ) ) ); -** -** This time the named values are only being used once, so you don't -** have to copy and free them. -** -** This really works, however you do have to be very careful when writing -** your code. If you leave out a bi_copy() and use a value more than once, -** you'll get a runtime error about "zero refs" and a SIGFPE. Run your -** code in a debugger, get a backtrace to see where the call was, and then -** eyeball the code there to see where you need to add the bi_copy(). -** -** -** Copyright © 2000 by Jef Poskanzer . -** All rights reserved. -** -** Redistribution and use in source and binary forms, with or without -** modification, are permitted provided that the following conditions -** are met: -** 1. Redistributions of source code must retain the above copyright -** notice, this list of conditions and the following disclaimer. -** 2. Redistributions in binary form must reproduce the above copyright -** notice, this list of conditions and the following disclaimer in the -** documentation and/or other materials provided with the distribution. -** -** THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND -** ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE -** IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE -** ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE -** FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL -** DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS -** OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) -** HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT -** LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY -** OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF -** SUCH DAMAGE. -*/ - - -/* Type definition for bigints - it's an opaque type, the real definition -** is in bigint.c. -*/ -typedef void* bigint; - - -/* Some convenient pre-initialized numbers. These are all permanent, -** so you can use them as many times as you want without calling bi_copy(). -*/ -extern bigint bi_0, bi_1, bi_2, bi_10, bi_m1, bi_maxint, bi_minint; - - -/* Initialize the bigint package. You must call this when your program -** starts up. -*/ -void bi_initialize( void ); - -/* Shut down the bigint package. You should call this when your program -** exits. It's not actually required, but it does do some consistency -** checks which help keep your program bug-free, so you really ought -** to call it. -*/ -void bi_terminate( void ); - -/* Run in unsafe mode, skipping most runtime checks. Slightly faster. -** Once your code is debugged you can add this call after bi_initialize(). -*/ -void bi_no_check( void ); - -/* Make a copy of a bigint. You must call this if you want to use a -** bigint more than once. (Or you can make the bigint permanent.) -** Note that this routine is very cheap - all it actually does is -** increment a reference counter. -*/ -bigint bi_copy( bigint bi ); - -/* Make a bigint permanent, so it doesn't get automatically freed when -** used as an operand. -*/ -void bi_permanent( bigint bi ); - -/* Undo bi_permanent(). The next use will free the bigint. */ -void bi_depermanent( bigint bi ); - -/* Explicitly free a bigint. Normally bigints get freed automatically -** when they are used as an operand. This routine lets you free one -** without using it. If the bigint is permanent, this doesn't do -** anything, you have to depermanent it first. -*/ -void bi_free( bigint bi ); - -/* Compare two bigints. Returns -1, 0, or 1. */ -int bi_compare( bigint bia, bigint bib ); - -/* Convert an int to a bigint. */ -bigint int_to_bi( int i ); - -/* Convert a string to a bigint. */ -bigint str_to_bi( char* str ); - -/* Convert a bigint to an int. SIGFPE on overflow. */ -int bi_to_int( bigint bi ); - -/* Write a bigint to a file. */ -void bi_print( FILE* f, bigint bi ); - -/* Read a bigint from a file. */ -bigint bi_scan( FILE* f ); - - -/* Operations on a bigint and a regular int. */ - -/* Add an int to a bigint. */ -bigint bi_int_add( bigint bi, int i ); - -/* Subtract an int from a bigint. */ -bigint bi_int_subtract( bigint bi, int i ); - -/* Multiply a bigint by an int. */ -bigint bi_int_multiply( bigint bi, int i ); - -/* Divide a bigint by an int. SIGFPE on divide-by-zero. */ -bigint bi_int_divide( bigint binumer, int denom ); - -/* Take the remainder of a bigint by an int, with an int result. -** SIGFPE if m is zero. -*/ -int bi_int_rem( bigint bi, int m ); - -/* Take the modulus of a bigint by an int, with an int result. -** Note that mod is not rem: mod is always within [0..m), while -** rem can be negative. SIGFPE if m is zero or negative. -*/ -int bi_int_mod( bigint bi, int m ); - - -/* Basic operations on two bigints. */ - -/* Add two bigints. */ -bigint bi_add( bigint bia, bigint bib ); - -/* Subtract bib from bia. */ -bigint bi_subtract( bigint bia, bigint bib ); - -/* Multiply two bigints. */ -bigint bi_multiply( bigint bia, bigint bib ); - -/* Divide one bigint by another. SIGFPE on divide-by-zero. */ -bigint bi_divide( bigint binumer, bigint bidenom ); - -/* Binary division of one bigint by another. SIGFPE on divide-by-zero. -** This is here just for testing. It's about five times slower than -** regular division. -*/ -bigint bi_binary_divide( bigint binumer, bigint bidenom ); - -/* Take the remainder of one bigint by another. SIGFPE if bim is zero. */ -bigint bi_rem( bigint bia, bigint bim ); - -/* Take the modulus of one bigint by another. Note that mod is not rem: -** mod is always within [0..bim), while rem can be negative. SIGFPE if -** bim is zero or negative. -*/ -bigint bi_mod( bigint bia, bigint bim ); - - -/* Some less common operations. */ - -/* Negate a bigint. */ -bigint bi_negate( bigint bi ); - -/* Absolute value of a bigint. */ -bigint bi_abs( bigint bi ); - -/* Divide a bigint in half. */ -bigint bi_half( bigint bi ); - -/* Multiply a bigint by two. */ -bigint bi_double( bigint bi ); - -/* Square a bigint. */ -bigint bi_square( bigint bi ); - -/* Raise bi to the power of biexp. SIGFPE if biexp is negative. */ -bigint bi_power( bigint bi, bigint biexp ); - -/* Integer square root. */ -bigint bi_sqrt( bigint bi ); - -/* Factorial. */ -bigint bi_factorial( bigint bi ); - - -/* Some predicates. */ - -/* 1 if the bigint is odd, 0 if it's even. */ -int bi_is_odd( bigint bi ); - -/* 1 if the bigint is even, 0 if it's odd. */ -int bi_is_even( bigint bi ); - -/* 1 if the bigint equals zero, 0 if it's nonzero. */ -int bi_is_zero( bigint bi ); - -/* 1 if the bigint equals one, 0 otherwise. */ -int bi_is_one( bigint bi ); - -/* 1 if the bigint is less than zero, 0 if it's zero or greater. */ -int bi_is_negative( bigint bi ); - - -/* Now we get into the esoteric number-theory stuff used for cryptography. */ - -/* Modular exponentiation. Much faster than bi_mod(bi_power(bi,biexp),bim). -** Also, biexp can be negative. -*/ -bigint bi_mod_power( bigint bi, bigint biexp, bigint bim ); - -/* Modular inverse. mod( bi * modinv(bi), bim ) == 1. SIGFPE if bi is not -** relatively prime to bim. -*/ -bigint bi_mod_inverse( bigint bi, bigint bim ); - -/* Produce a random number in the half-open interval [0..bi). You need -** to have called srandom() before using this. -*/ -bigint bi_random( bigint bi ); - -/* Greatest common divisor of two bigints. Euclid's algorithm. */ -bigint bi_gcd( bigint bim, bigint bin ); - -/* Greatest common divisor of two bigints, plus the corresponding multipliers. -** Extended Euclid's algorithm. -*/ -bigint bi_egcd( bigint bim, bigint bin, bigint* bim_mul, bigint* bin_mul ); - -/* Least common multiple of two bigints. */ -bigint bi_lcm( bigint bia, bigint bib ); - -/* The Jacobi symbol. SIGFPE if bib is even. */ -bigint bi_jacobi( bigint bia, bigint bib ); - -/* Probabalistic prime checking. A non-zero return means the probability -** that bi is prime is at least 1 - 1/2 ^ certainty. -*/ -int bi_is_probable_prime( bigint bi, int certainty ); - -/* Random probabilistic prime with the specified number of bits. */ -bigint bi_generate_prime( int bits, int certainty ); - -/* Number of bits in the number. The log base 2, approximately. */ -int bi_bits( bigint bi ); diff --git a/src/rt/bigint/bigint_ext.cpp b/src/rt/bigint/bigint_ext.cpp deleted file mode 100644 index 66d79106f48..00000000000 --- a/src/rt/bigint/bigint_ext.cpp +++ /dev/null @@ -1,553 +0,0 @@ -/* bigint_ext - external portion of large integer package -** -** Copyright © 2000 by Jef Poskanzer . -** All rights reserved. -** -** Redistribution and use in source and binary forms, with or without -** modification, are permitted provided that the following conditions -** are met: -** 1. Redistributions of source code must retain the above copyright -** notice, this list of conditions and the following disclaimer. -** 2. Redistributions in binary form must reproduce the above copyright -** notice, this list of conditions and the following disclaimer in the -** documentation and/or other materials provided with the distribution. -** -** THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND -** ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE -** IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE -** ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE -** FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL -** DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS -** OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) -** HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT -** LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY -** OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF -** SUCH DAMAGE. -*/ - -#include -#include -#include -#include -#include -#include - -#include "bigint.h" -#include "low_primes.h" - - -bigint bi_0, bi_1, bi_2, bi_10, bi_m1, bi_maxint, bi_minint; - - -/* Forwards. */ -static void print_pos( FILE* f, bigint bi ); - - -bigint -str_to_bi( char* str ) - { - int sign; - bigint biR; - - sign = 1; - if ( *str == '-' ) - { - sign = -1; - ++str; - } - for ( biR = bi_0; *str >= '0' && *str <= '9'; ++str ) - biR = bi_int_add( bi_int_multiply( biR, 10 ), *str - '0' ); - if ( sign == -1 ) - biR = bi_negate( biR ); - return biR; - } - - -void -bi_print( FILE* f, bigint bi ) - { - if ( bi_is_negative( bi_copy( bi ) ) ) - { - putc( '-', f ); - bi = bi_negate( bi ); - } - print_pos( f, bi ); - } - - -bigint -bi_scan( FILE* f ) - { - int sign; - int c; - bigint biR; - - sign = 1; - c = getc( f ); - if ( c == '-' ) - sign = -1; - else - ungetc( c, f ); - - biR = bi_0; - for (;;) - { - c = getc( f ); - if ( c < '0' || c > '9' ) - break; - biR = bi_int_add( bi_int_multiply( biR, 10 ), c - '0' ); - } - - if ( sign == -1 ) - biR = bi_negate( biR ); - return biR; - } - - -static void -print_pos( FILE* f, bigint bi ) - { - if ( bi_compare( bi_copy( bi ), bi_10 ) >= 0 ) - print_pos( f, bi_int_divide( bi_copy( bi ), 10 ) ); - putc( bi_int_mod( bi, 10 ) + '0', f ); - } - - -int -bi_int_mod( bigint bi, int m ) - { - int r; - - if ( m <= 0 ) - { - (void) fprintf( stderr, "bi_int_mod: zero or negative modulus\n" ); - (void) kill( getpid(), SIGFPE ); - } - r = bi_int_rem( bi, m ); - if ( r < 0 ) - r += m; - return r; - } - - -bigint -bi_rem( bigint bia, bigint bim ) - { - return bi_subtract( - bia, bi_multiply( bi_divide( bi_copy( bia ), bi_copy( bim ) ), bim ) ); - } - - -bigint -bi_mod( bigint bia, bigint bim ) - { - bigint biR; - - if ( bi_compare( bi_copy( bim ), bi_0 ) <= 0 ) - { - (void) fprintf( stderr, "bi_mod: zero or negative modulus\n" ); - (void) kill( getpid(), SIGFPE ); - } - biR = bi_rem( bia, bi_copy( bim ) ); - if ( bi_is_negative( bi_copy( biR ) ) ) - biR = bi_add( biR, bim ); - else - bi_free( bim ); - return biR; - } - - -bigint -bi_square( bigint bi ) - { - bigint biR; - - biR = bi_multiply( bi_copy( bi ), bi_copy( bi ) ); - bi_free( bi ); - return biR; - } - - -bigint -bi_power( bigint bi, bigint biexp ) - { - bigint biR; - - if ( bi_is_negative( bi_copy( biexp ) ) ) - { - (void) fprintf( stderr, "bi_power: negative exponent\n" ); - (void) kill( getpid(), SIGFPE ); - } - biR = bi_1; - for (;;) - { - if ( bi_is_odd( bi_copy( biexp ) ) ) - biR = bi_multiply( biR, bi_copy( bi ) ); - biexp = bi_half( biexp ); - if ( bi_compare( bi_copy( biexp ), bi_0 ) <= 0 ) - break; - bi = bi_multiply( bi_copy( bi ), bi ); - } - bi_free( bi ); - bi_free( biexp ); - return biR; - } - - -bigint -bi_factorial( bigint bi ) - { - bigint biR; - - biR = bi_1; - while ( bi_compare( bi_copy( bi ), bi_1 ) > 0 ) - { - biR = bi_multiply( biR, bi_copy( bi ) ); - bi = bi_int_subtract( bi, 1 ); - } - bi_free( bi ); - return biR; - } - - -int -bi_is_even( bigint bi ) - { - return ! bi_is_odd( bi ); - } - - -bigint -bi_mod_power( bigint bi, bigint biexp, bigint bim ) - { - int invert; - bigint biR; - - invert = 0; - if ( bi_is_negative( bi_copy( biexp ) ) ) - { - biexp = bi_negate( biexp ); - invert = 1; - } - - biR = bi_1; - for (;;) - { - if ( bi_is_odd( bi_copy( biexp ) ) ) - biR = bi_mod( bi_multiply( biR, bi_copy( bi ) ), bi_copy( bim ) ); - biexp = bi_half( biexp ); - if ( bi_compare( bi_copy( biexp ), bi_0 ) <= 0 ) - break; - bi = bi_mod( bi_multiply( bi_copy( bi ), bi ), bi_copy( bim ) ); - } - bi_free( bi ); - bi_free( biexp ); - - if ( invert ) - biR = bi_mod_inverse( biR, bim ); - else - bi_free( bim ); - return biR; - } - - -bigint -bi_mod_inverse( bigint bi, bigint bim ) - { - bigint gcd, mul0, mul1; - - gcd = bi_egcd( bi_copy( bim ), bi, &mul0, &mul1 ); - - /* Did we get gcd == 1? */ - if ( ! bi_is_one( gcd ) ) - { - (void) fprintf( stderr, "bi_mod_inverse: not relatively prime\n" ); - (void) kill( getpid(), SIGFPE ); - } - - bi_free( mul0 ); - return bi_mod( mul1, bim ); - } - - -/* Euclid's algorithm. */ -bigint -bi_gcd( bigint bim, bigint bin ) - { - bigint bit; - - bim = bi_abs( bim ); - bin = bi_abs( bin ); - while ( ! bi_is_zero( bi_copy( bin ) ) ) - { - bit = bi_mod( bim, bi_copy( bin ) ); - bim = bin; - bin = bit; - } - bi_free( bin ); - return bim; - } - - -/* Extended Euclidean algorithm. */ -bigint -bi_egcd( bigint bim, bigint bin, bigint* bim_mul, bigint* bin_mul ) - { - bigint a0, b0, c0, a1, b1, c1, q, t; - - if ( bi_is_negative( bi_copy( bim ) ) ) - { - bigint biR; - - biR = bi_egcd( bi_negate( bim ), bin, &t, bin_mul ); - *bim_mul = bi_negate( t ); - return biR; - } - if ( bi_is_negative( bi_copy( bin ) ) ) - { - bigint biR; - - biR = bi_egcd( bim, bi_negate( bin ), bim_mul, &t ); - *bin_mul = bi_negate( t ); - return biR; - } - - a0 = bi_1; b0 = bi_0; c0 = bim; - a1 = bi_0; b1 = bi_1; c1 = bin; - - while ( ! bi_is_zero( bi_copy( c1 ) ) ) - { - q = bi_divide( bi_copy( c0 ), bi_copy( c1 ) ); - t = a0; - a0 = bi_copy( a1 ); - a1 = bi_subtract( t, bi_multiply( bi_copy( q ), a1 ) ); - t = b0; - b0 = bi_copy( b1 ); - b1 = bi_subtract( t, bi_multiply( bi_copy( q ), b1 ) ); - t = c0; - c0 = bi_copy( c1 ); - c1 = bi_subtract( t, bi_multiply( bi_copy( q ), c1 ) ); - bi_free( q ); - } - - bi_free( a1 ); - bi_free( b1 ); - bi_free( c1 ); - *bim_mul = a0; - *bin_mul = b0; - return c0; - } - - -bigint -bi_lcm( bigint bia, bigint bib ) - { - bigint biR; - - biR = bi_divide( - bi_multiply( bi_copy( bia ), bi_copy( bib ) ), - bi_gcd( bi_copy( bia ), bi_copy( bib ) ) ); - bi_free( bia ); - bi_free( bib ); - return biR; - } - - -/* The Jacobi symbol. */ -bigint -bi_jacobi( bigint bia, bigint bib ) - { - bigint biR; - - if ( bi_is_even( bi_copy( bib ) ) ) - { - (void) fprintf( stderr, "bi_jacobi: don't know how to compute Jacobi(n, even)\n" ); - (void) kill( getpid(), SIGFPE ); - } - - if ( bi_compare( bi_copy( bia ), bi_copy( bib ) ) >= 0 ) - return bi_jacobi( bi_mod( bia, bi_copy( bib ) ), bib ); - - if ( bi_is_zero( bi_copy( bia ) ) || bi_is_one( bi_copy( bia ) ) ) - { - bi_free( bib ); - return bia; - } - - if ( bi_compare( bi_copy( bia ), bi_2 ) == 0 ) - { - bi_free( bia ); - switch ( bi_int_mod( bib, 8 ) ) - { - case 1: case 7: - return bi_1; - case 3: case 5: - return bi_m1; - } - } - - if ( bi_is_even( bi_copy( bia ) ) ) - { - biR = bi_multiply( - bi_jacobi( bi_2, bi_copy( bib ) ), - bi_jacobi( bi_half( bia ), bi_copy( bib ) ) ); - bi_free( bib ); - return biR; - } - - if ( bi_int_mod( bi_copy( bia ), 4 ) == 3 && - bi_int_mod( bi_copy( bib ), 4 ) == 3 ) - return bi_negate( bi_jacobi( bib, bia ) ); - else - return bi_jacobi( bib, bia ); - } - - -/* Probabalistic prime checking. */ -int -bi_is_probable_prime( bigint bi, int certainty ) - { - int i, p; - bigint bim1; - - /* First do trial division by a list of small primes. This eliminates - ** many candidates. - */ - for ( i = 0; i < sizeof(low_primes)/sizeof(*low_primes); ++i ) - { - p = low_primes[i]; - switch ( bi_compare( int_to_bi( p ), bi_copy( bi ) ) ) - { - case 0: - bi_free( bi ); - return 1; - case 1: - bi_free( bi ); - return 0; - } - if ( bi_int_mod( bi_copy( bi ), p ) == 0 ) - { - bi_free( bi ); - return 0; - } - } - - /* Now do the probabilistic tests. */ - bim1 = bi_int_subtract( bi_copy( bi ), 1 ); - for ( i = 0; i < certainty; ++i ) - { - bigint a, j, jac; - - /* Pick random test number. */ - a = bi_random( bi_copy( bi ) ); - - /* Decide whether to run the Fermat test or the Solovay-Strassen - ** test. The Fermat test is fast but lets some composite numbers - ** through. Solovay-Strassen runs slower but is more certain. - ** So the compromise here is we run the Fermat test a couple of - ** times to quickly reject most composite numbers, and then do - ** the rest of the iterations with Solovay-Strassen so nothing - ** slips through. - */ - if ( i < 2 && certainty >= 5 ) - { - /* Fermat test. Note that this is not state of the art. There's a - ** class of numbers called Carmichael numbers which are composite - ** but look prime to this test - it lets them slip through no - ** matter how many reps you run. However, it's nice and fast so - ** we run it anyway to help quickly reject most of the composites. - */ - if ( ! bi_is_one( bi_mod_power( bi_copy( a ), bi_copy( bim1 ), bi_copy( bi ) ) ) ) - { - bi_free( bi ); - bi_free( bim1 ); - bi_free( a ); - return 0; - } - } - else - { - /* GCD test. This rarely hits, but we need it for Solovay-Strassen. */ - if ( ! bi_is_one( bi_gcd( bi_copy( bi ), bi_copy( a ) ) ) ) - { - bi_free( bi ); - bi_free( bim1 ); - bi_free( a ); - return 0; - } - - /* Solovay-Strassen test. First compute pseudo Jacobi. */ - j = bi_mod_power( - bi_copy( a ), bi_half( bi_copy( bim1 ) ), bi_copy( bi ) ); - if ( bi_compare( bi_copy( j ), bi_copy( bim1 ) ) == 0 ) - { - bi_free( j ); - j = bi_m1; - } - - /* Now compute real Jacobi. */ - jac = bi_jacobi( bi_copy( a ), bi_copy( bi ) ); - - /* If they're not equal, the number is definitely composite. */ - if ( bi_compare( j, jac ) != 0 ) - { - bi_free( bi ); - bi_free( bim1 ); - bi_free( a ); - return 0; - } - } - - bi_free( a ); - } - - bi_free( bim1 ); - - bi_free( bi ); - return 1; - } - - -bigint -bi_generate_prime( int bits, int certainty ) - { - bigint bimo2, bip; - int i, inc = 0; - - bimo2 = bi_power( bi_2, int_to_bi( bits - 1 ) ); - for (;;) - { - bip = bi_add( bi_random( bi_copy( bimo2 ) ), bi_copy( bimo2 ) ); - /* By shoving the candidate numbers up to the next highest multiple - ** of six plus or minus one, we pre-eliminate all multiples of - ** two and/or three. - */ - switch ( bi_int_mod( bi_copy( bip ), 6 ) ) - { - case 0: inc = 4; bip = bi_int_add( bip, 1 ); break; - case 1: inc = 4; break; - case 2: inc = 2; bip = bi_int_add( bip, 3 ); break; - case 3: inc = 2; bip = bi_int_add( bip, 2 ); break; - case 4: inc = 2; bip = bi_int_add( bip, 1 ); break; - case 5: inc = 2; break; - } - /* Starting from the generated random number, check a bunch of - ** numbers in sequence. This is just to avoid calls to bi_random(), - ** which is more expensive than a simple add. - */ - for ( i = 0; i < 1000; ++i ) /* arbitrary */ - { - if ( bi_is_probable_prime( bi_copy( bip ), certainty ) ) - { - bi_free( bimo2 ); - return bip; - } - bip = bi_int_add( bip, inc ); - inc = 6 - inc; - } - /* We ran through the whole sequence and didn't find a prime. - ** Shrug, just try a different random starting point. - */ - bi_free( bip ); - } - } diff --git a/src/rt/bigint/bigint_int.cpp b/src/rt/bigint/bigint_int.cpp deleted file mode 100644 index 194ddcb5600..00000000000 --- a/src/rt/bigint/bigint_int.cpp +++ /dev/null @@ -1,1428 +0,0 @@ -/* bigint - internal portion of large integer package -** -** Copyright © 2000 by Jef Poskanzer . -** All rights reserved. -** -** Redistribution and use in source and binary forms, with or without -** modification, are permitted provided that the following conditions -** are met: -** 1. Redistributions of source code must retain the above copyright -** notice, this list of conditions and the following disclaimer. -** 2. Redistributions in binary form must reproduce the above copyright -** notice, this list of conditions and the following disclaimer in the -** documentation and/or other materials provided with the distribution. -** -** THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND -** ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE -** IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE -** ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE -** FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL -** DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS -** OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) -** HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT -** LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY -** OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF -** SUCH DAMAGE. -*/ - -#include -#include -#include -#include -#include -#include - -#include "bigint.h" - -#define max(a,b) ((a)>(b)?(a):(b)) -#define min(a,b) ((a)<(b)?(a):(b)) - -/* MAXINT and MININT extracted from , which gives a warning -** message if included. -*/ -#define BITSPERBYTE 8 -#define BITS(type) (BITSPERBYTE * (int)sizeof(type)) -#define INTBITS BITS(int) -#define MININT (1 << (INTBITS - 1)) -#define MAXINT (~MININT) - - -/* The package represents arbitrary-precision integers as a sign and a sum -** of components multiplied by successive powers of the basic radix, i.e.: -** -** sign * ( comp0 + comp1 * radix + comp2 * radix^2 + comp3 * radix^3 ) -** -** To make good use of the computer's word size, the radix is chosen -** to be a power of two. It could be chosen to be the full word size, -** however this would require a lot of finagling in the middle of the -** algorithms to get the inter-word overflows right. That would slow things -** down. Instead, the radix is chosen to be *half* the actual word size. -** With just a little care, this means the words can hold all intermediate -** values, and the overflows can be handled all at once at the end, in a -** normalization step. This simplifies the coding enormously, and is probably -** somewhat faster to run. The cost is that numbers use twice as much -** storage as they would with the most efficient representation, but storage -** is cheap. -** -** A few more notes on the representation: -** -** - The sign is always 1 or -1, never 0. The number 0 is represented -** with a sign of 1. -** - The components are signed numbers, to allow for negative intermediate -** values. After normalization, all components are >= 0 and the sign is -** updated. -*/ - -/* Type definition for bigints. */ -typedef int64_t comp; /* should be the largest signed int type you have */ -struct _real_bigint { - int refs; - struct _real_bigint* next; - int num_comps, max_comps; - int sign; - comp* comps; - }; -typedef struct _real_bigint* real_bigint; - - -#undef DUMP - - -#define PERMANENT 123456789 - -static comp bi_radix, bi_radix_o2; -static int bi_radix_sqrt, bi_comp_bits; - -static real_bigint active_list, free_list; -static int active_count, free_count; -static int check_level; - - -/* Forwards. */ -static bigint regular_multiply( real_bigint bia, real_bigint bib ); -static bigint multi_divide( bigint binumer, real_bigint bidenom ); -static bigint multi_divide2( bigint binumer, real_bigint bidenom ); -static void more_comps( real_bigint bi, int n ); -static real_bigint alloc( int num_comps ); -static real_bigint clone( real_bigint bi ); -static void normalize( real_bigint bi ); -static void check( real_bigint bi ); -static void double_check( void ); -static void triple_check( void ); -#ifdef DUMP -static void dump( char* str, bigint bi ); -#endif /* DUMP */ -static int csqrt( comp c ); -static int cbits( comp c ); - - -void -bi_initialize( void ) - { - /* Set the radix. This does not actually have to be a power of - ** two, that's just the most efficient value. It does have to - ** be even for bi_half() to work. - */ - bi_radix = 1; - bi_radix <<= BITS(comp) / 2 - 1; - - /* Halve the radix. Only used by bi_half(). */ - bi_radix_o2 = bi_radix >> 1; - - /* Take the square root of the radix. Only used by bi_divide(). */ - bi_radix_sqrt = csqrt( bi_radix ); - - /* Figure out how many bits in a component. Only used by bi_bits(). */ - bi_comp_bits = cbits( bi_radix - 1 ); - - /* Init various globals. */ - active_list = (real_bigint) 0; - active_count = 0; - free_list = (real_bigint) 0; - free_count = 0; - - /* This can be 0 through 3. */ - check_level = 3; - - /* Set up some convenient bigints. */ - bi_0 = int_to_bi( 0 ); bi_permanent( bi_0 ); - bi_1 = int_to_bi( 1 ); bi_permanent( bi_1 ); - bi_2 = int_to_bi( 2 ); bi_permanent( bi_2 ); - bi_10 = int_to_bi( 10 ); bi_permanent( bi_10 ); - bi_m1 = int_to_bi( -1 ); bi_permanent( bi_m1 ); - bi_maxint = int_to_bi( MAXINT ); bi_permanent( bi_maxint ); - bi_minint = int_to_bi( MININT ); bi_permanent( bi_minint ); - } - - -void -bi_terminate( void ) - { - real_bigint p, pn; - - bi_depermanent( bi_0 ); bi_free( bi_0 ); - bi_depermanent( bi_1 ); bi_free( bi_1 ); - bi_depermanent( bi_2 ); bi_free( bi_2 ); - bi_depermanent( bi_10 ); bi_free( bi_10 ); - bi_depermanent( bi_m1 ); bi_free( bi_m1 ); - bi_depermanent( bi_maxint ); bi_free( bi_maxint ); - bi_depermanent( bi_minint ); bi_free( bi_minint ); - - if ( active_count != 0 ) - (void) fprintf( - stderr, "bi_terminate: there were %d un-freed bigints\n", - active_count ); - if ( check_level >= 2 ) - double_check(); - if ( check_level >= 3 ) - { - triple_check(); - for ( p = active_list; p != (bigint) 0; p = pn ) - { - pn = p->next; - free( p->comps ); - free( p ); - } - } - for ( p = free_list; p != (bigint) 0; p = pn ) - { - pn = p->next; - free( p->comps ); - free( p ); - } - } - - -void -bi_no_check( void ) - { - check_level = 0; - } - - -bigint -bi_copy( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - - check( bi ); - if ( bi->refs != PERMANENT ) - ++bi->refs; - return bi; - } - - -void -bi_permanent( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - - check( bi ); - if ( check_level >= 1 && bi->refs != 1 ) - { - (void) fprintf( stderr, "bi_permanent: refs was not 1\n" ); - (void) kill( getpid(), SIGFPE ); - } - bi->refs = PERMANENT; - } - - -void -bi_depermanent( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - - check( bi ); - if ( check_level >= 1 && bi->refs != PERMANENT ) - { - (void) fprintf( stderr, "bi_depermanent: bigint was not permanent\n" ); - (void) kill( getpid(), SIGFPE ); - } - bi->refs = 1; - } - - -void -bi_free( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - - check( bi ); - if ( bi->refs == PERMANENT ) - return; - --bi->refs; - if ( bi->refs > 0 ) - return; - if ( check_level >= 3 ) - { - /* The active list only gets maintained at check levels 3 or higher. */ - real_bigint* nextP; - for ( nextP = &active_list; *nextP != (real_bigint) 0; nextP = &((*nextP)->next) ) - if ( *nextP == bi ) - { - *nextP = bi->next; - break; - } - } - --active_count; - bi->next = free_list; - free_list = bi; - ++free_count; - if ( check_level >= 1 && active_count < 0 ) - { - (void) fprintf( stderr, - "bi_free: active_count went negative - double-freed bigint?\n" ); - (void) kill( getpid(), SIGFPE ); - } - } - - -int -bi_compare( bigint obia, bigint obib ) - { - real_bigint bia = (real_bigint) obia; - real_bigint bib = (real_bigint) obib; - int r, c; - - check( bia ); - check( bib ); - - /* First check for pointer equality. */ - if ( bia == bib ) - r = 0; - else - { - /* Compare signs. */ - if ( bia->sign > bib->sign ) - r = 1; - else if ( bia->sign < bib->sign ) - r = -1; - /* Signs are the same. Check the number of components. */ - else if ( bia->num_comps > bib->num_comps ) - r = bia->sign; - else if ( bia->num_comps < bib->num_comps ) - r = -bia->sign; - else - { - /* Same number of components. Compare starting from the high end - ** and working down. - */ - r = 0; /* if we complete the loop, the numbers are equal */ - for ( c = bia->num_comps - 1; c >= 0; --c ) - { - if ( bia->comps[c] > bib->comps[c] ) - { r = bia->sign; break; } - else if ( bia->comps[c] < bib->comps[c] ) - { r = -bia->sign; break; } - } - } - } - - bi_free( bia ); - bi_free( bib ); - return r; - } - - -bigint -int_to_bi( int i ) - { - real_bigint biR; - - biR = alloc( 1 ); - biR->sign = 1; - biR->comps[0] = i; - normalize( biR ); - check( biR ); - return biR; - } - - -int -bi_to_int( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - comp v, m; - int c, r; - - check( bi ); - if ( bi_compare( bi_copy( bi ), bi_maxint ) > 0 || - bi_compare( bi_copy( bi ), bi_minint ) < 0 ) - { - (void) fprintf( stderr, "bi_to_int: overflow\n" ); - (void) kill( getpid(), SIGFPE ); - } - v = 0; - m = 1; - for ( c = 0; c < bi->num_comps; ++c ) - { - v += bi->comps[c] * m; - m *= bi_radix; - } - r = (int) ( bi->sign * v ); - bi_free( bi ); - return r; - } - - -bigint -bi_int_add( bigint obi, int i ) - { - real_bigint bi = (real_bigint) obi; - real_bigint biR; - - check( bi ); - biR = clone( bi ); - if ( biR->sign == 1 ) - biR->comps[0] += i; - else - biR->comps[0] -= i; - normalize( biR ); - check( biR ); - return biR; - } - - -bigint -bi_int_subtract( bigint obi, int i ) - { - real_bigint bi = (real_bigint) obi; - real_bigint biR; - - check( bi ); - biR = clone( bi ); - if ( biR->sign == 1 ) - biR->comps[0] -= i; - else - biR->comps[0] += i; - normalize( biR ); - check( biR ); - return biR; - } - - -bigint -bi_int_multiply( bigint obi, int i ) - { - real_bigint bi = (real_bigint) obi; - real_bigint biR; - int c; - - check( bi ); - biR = clone( bi ); - if ( i < 0 ) - { - i = -i; - biR->sign = -biR->sign; - } - for ( c = 0; c < biR->num_comps; ++c ) - biR->comps[c] *= i; - normalize( biR ); - check( biR ); - return biR; - } - - -bigint -bi_int_divide( bigint obinumer, int denom ) - { - real_bigint binumer = (real_bigint) obinumer; - real_bigint biR; - int c; - comp r; - - check( binumer ); - if ( denom == 0 ) - { - (void) fprintf( stderr, "bi_int_divide: divide by zero\n" ); - (void) kill( getpid(), SIGFPE ); - } - biR = clone( binumer ); - if ( denom < 0 ) - { - denom = -denom; - biR->sign = -biR->sign; - } - r = 0; - for ( c = biR->num_comps - 1; c >= 0; --c ) - { - r = r * bi_radix + biR->comps[c]; - biR->comps[c] = r / denom; - r = r % denom; - } - normalize( biR ); - check( biR ); - return biR; - } - - -int -bi_int_rem( bigint obi, int m ) - { - real_bigint bi = (real_bigint) obi; - comp rad_r, r; - int c; - - check( bi ); - if ( m == 0 ) - { - (void) fprintf( stderr, "bi_int_rem: divide by zero\n" ); - (void) kill( getpid(), SIGFPE ); - } - if ( m < 0 ) - m = -m; - rad_r = 1; - r = 0; - for ( c = 0; c < bi->num_comps; ++c ) - { - r = ( r + bi->comps[c] * rad_r ) % m; - rad_r = ( rad_r * bi_radix ) % m; - } - if ( bi->sign < 1 ) - r = -r; - bi_free( bi ); - return (int) r; - } - - -bigint -bi_add( bigint obia, bigint obib ) - { - real_bigint bia = (real_bigint) obia; - real_bigint bib = (real_bigint) obib; - real_bigint biR; - int c; - - check( bia ); - check( bib ); - biR = clone( bia ); - more_comps( biR, max( biR->num_comps, bib->num_comps ) ); - for ( c = 0; c < bib->num_comps; ++c ) - if ( biR->sign == bib->sign ) - biR->comps[c] += bib->comps[c]; - else - biR->comps[c] -= bib->comps[c]; - bi_free( bib ); - normalize( biR ); - check( biR ); - return biR; - } - - -bigint -bi_subtract( bigint obia, bigint obib ) - { - real_bigint bia = (real_bigint) obia; - real_bigint bib = (real_bigint) obib; - real_bigint biR; - int c; - - check( bia ); - check( bib ); - biR = clone( bia ); - more_comps( biR, max( biR->num_comps, bib->num_comps ) ); - for ( c = 0; c < bib->num_comps; ++c ) - if ( biR->sign == bib->sign ) - biR->comps[c] -= bib->comps[c]; - else - biR->comps[c] += bib->comps[c]; - bi_free( bib ); - normalize( biR ); - check( biR ); - return biR; - } - - -/* Karatsuba multiplication. This is supposedly O(n^1.59), better than -** regular multiplication for large n. The define below sets the crossover -** point - below that we use regular multiplication, above it we -** use Karatsuba. Note that Karatsuba is a recursive algorithm, so -** all Karatsuba calls involve regular multiplications as the base -** steps. -*/ -#define KARATSUBA_THRESH 12 -bigint -bi_multiply( bigint obia, bigint obib ) - { - real_bigint bia = (real_bigint) obia; - real_bigint bib = (real_bigint) obib; - - check( bia ); - check( bib ); - if ( min( bia->num_comps, bib->num_comps ) < KARATSUBA_THRESH ) - return regular_multiply( bia, bib ); - else - { - /* The factors are large enough that Karatsuba multiplication - ** is a win. The basic idea here is you break each factor up - ** into two parts, like so: - ** i * r^n + j k * r^n + l - ** r is the radix we're representing numbers with, so this - ** breaking up just means shuffling components around, no - ** math required. With regular multiplication the product - ** would be: - ** ik * r^(n*2) + ( il + jk ) * r^n + jl - ** That's four sub-multiplies and one addition, not counting the - ** radix-shifting. With Karatsuba, you instead do: - ** ik * r^(n*2) + ( (i+j)(k+l) - ik - jl ) * r^n + jl - ** This is only three sub-multiplies. The number of adds - ** (and subtracts) increases to four, but those run in linear time - ** so they are cheap. The sub-multiplies are accomplished by - ** recursive calls, eventually reducing to regular multiplication. - */ - int n, c; - real_bigint bi_i, bi_j, bi_k, bi_l; - real_bigint bi_ik, bi_mid, bi_jl; - - n = ( max( bia->num_comps, bib->num_comps ) + 1 ) / 2; - bi_i = alloc( n ); - bi_j = alloc( n ); - bi_k = alloc( n ); - bi_l = alloc( n ); - for ( c = 0; c < n; ++c ) - { - if ( c + n < bia->num_comps ) - bi_i->comps[c] = bia->comps[c + n]; - else - bi_i->comps[c] = 0; - if ( c < bia->num_comps ) - bi_j->comps[c] = bia->comps[c]; - else - bi_j->comps[c] = 0; - if ( c + n < bib->num_comps ) - bi_k->comps[c] = bib->comps[c + n]; - else - bi_k->comps[c] = 0; - if ( c < bib->num_comps ) - bi_l->comps[c] = bib->comps[c]; - else - bi_l->comps[c] = 0; - } - bi_i->sign = bi_j->sign = bi_k->sign = bi_l->sign = 1; - normalize( bi_i ); - normalize( bi_j ); - normalize( bi_k ); - normalize( bi_l ); - bi_ik = bi_multiply( bi_copy( bi_i ), bi_copy( bi_k ) ); - bi_jl = bi_multiply( bi_copy( bi_j ), bi_copy( bi_l ) ); - bi_mid = bi_subtract( - bi_subtract( - bi_multiply( bi_add( bi_i, bi_j ), bi_add( bi_k, bi_l ) ), - bi_copy( bi_ik ) ), - bi_copy( bi_jl ) ); - more_comps( - bi_jl, max( bi_mid->num_comps + n, bi_ik->num_comps + n * 2 ) ); - for ( c = 0; c < bi_mid->num_comps; ++c ) - bi_jl->comps[c + n] += bi_mid->comps[c]; - for ( c = 0; c < bi_ik->num_comps; ++c ) - bi_jl->comps[c + n * 2] += bi_ik->comps[c]; - bi_free( bi_ik ); - bi_free( bi_mid ); - bi_jl->sign = bia->sign * bib->sign; - bi_free( bia ); - bi_free( bib ); - normalize( bi_jl ); - check( bi_jl ); - return bi_jl; - } - } - - -/* Regular O(n^2) multiplication. */ -static bigint -regular_multiply( real_bigint bia, real_bigint bib ) - { - real_bigint biR; - int new_comps, c1, c2; - - check( bia ); - check( bib ); - biR = clone( bi_0 ); - new_comps = bia->num_comps + bib->num_comps; - more_comps( biR, new_comps ); - for ( c1 = 0; c1 < bia->num_comps; ++c1 ) - { - for ( c2 = 0; c2 < bib->num_comps; ++c2 ) - biR->comps[c1 + c2] += bia->comps[c1] * bib->comps[c2]; - /* Normalize after each inner loop to avoid overflowing any - ** components. But be sure to reset biR's components count, - ** in case a previous normalization lowered it. - */ - biR->num_comps = new_comps; - normalize( biR ); - } - check( biR ); - if ( ! bi_is_zero( bi_copy( biR ) ) ) - biR->sign = bia->sign * bib->sign; - bi_free( bia ); - bi_free( bib ); - return biR; - } - - -/* The following three routines implement a multi-precision divide method -** that I haven't seen used anywhere else. It is not quite as fast as -** the standard divide method, but it is a lot simpler. In fact it's -** about as simple as the binary shift-and-subtract method, which goes -** about five times slower than this. -** -** The method assumes you already have multi-precision multiply and subtract -** routines, and also a multi-by-single precision divide routine. The latter -** is used to generate approximations, which are then checked and corrected -** using the former. The result converges to the correct value by about -** 16 bits per loop. -*/ - -/* Public routine to divide two arbitrary numbers. */ -bigint -bi_divide( bigint binumer, bigint obidenom ) - { - real_bigint bidenom = (real_bigint) obidenom; - int sign; - bigint biquotient; - - /* Check signs and trivial cases. */ - sign = 1; - switch ( bi_compare( bi_copy( bidenom ), bi_0 ) ) - { - case 0: - (void) fprintf( stderr, "bi_divide: divide by zero\n" ); - (void) kill( getpid(), SIGFPE ); - case -1: - sign *= -1; - bidenom = bi_negate( bidenom ); - break; - } - switch ( bi_compare( bi_copy( binumer ), bi_0 ) ) - { - case 0: - bi_free( binumer ); - bi_free( bidenom ); - return bi_0; - case -1: - sign *= -1; - binumer = bi_negate( binumer ); - break; - } - switch ( bi_compare( bi_copy( binumer ), bi_copy( bidenom ) ) ) - { - case -1: - bi_free( binumer ); - bi_free( bidenom ); - return bi_0; - case 0: - bi_free( binumer ); - bi_free( bidenom ); - if ( sign == 1 ) - return bi_1; - else - return bi_m1; - } - - /* Is the denominator small enough to do an int divide? */ - if ( bidenom->num_comps == 1 ) - { - /* Win! */ - biquotient = bi_int_divide( binumer, bidenom->comps[0] ); - bi_free( bidenom ); - } - else - { - /* No, we have to do a full multi-by-multi divide. */ - biquotient = multi_divide( binumer, bidenom ); - } - - if ( sign == -1 ) - biquotient = bi_negate( biquotient ); - return biquotient; - } - - -/* Divide two multi-precision positive numbers. */ -static bigint -multi_divide( bigint binumer, real_bigint bidenom ) - { - /* We use a successive approximation method that is kind of like a - ** continued fraction. The basic approximation is to do an int divide - ** by the high-order component of the denominator. Then we correct - ** based on the remainder from that. - ** - ** However, if the high-order component is too small, this doesn't - ** work well. In particular, if the high-order component is 1 it - ** doesn't work at all. Easily fixed, though - if the component - ** is too small, increase it! - */ - if ( bidenom->comps[bidenom->num_comps-1] < bi_radix_sqrt ) - { - /* We use the square root of the radix as the threshhold here - ** because that's the largest value guaranteed to not make the - ** high-order component overflow and become too small again. - ** - ** We increase binumer along with bidenom to keep the end result - ** the same. - */ - binumer = bi_int_multiply( binumer, bi_radix_sqrt ); - bidenom = bi_int_multiply( bidenom, bi_radix_sqrt ); - } - - /* Now start the recursion. */ - return multi_divide2( binumer, bidenom ); - } - - -/* Divide two multi-precision positive conditioned numbers. */ -static bigint -multi_divide2( bigint binumer, real_bigint bidenom ) - { - real_bigint biapprox; - bigint birem, biquotient; - int c, o; - - /* Figure out the approximate quotient. Since we're dividing by only - ** the top component of the denominator, which is less than or equal to - ** the full denominator, the result is guaranteed to be greater than or - ** equal to the correct quotient. - */ - o = bidenom->num_comps - 1; - biapprox = bi_int_divide( bi_copy( binumer ), bidenom->comps[o] ); - /* And downshift the result to get the approximate quotient. */ - for ( c = o; c < biapprox->num_comps; ++c ) - biapprox->comps[c - o] = biapprox->comps[c]; - biapprox->num_comps -= o; - - /* Find the remainder from the approximate quotient. */ - birem = bi_subtract( - bi_multiply( bi_copy( biapprox ), bi_copy( bidenom ) ), binumer ); - - /* If the remainder is negative, zero, or in fact any value less - ** than bidenom, then we have the correct quotient and we're done. - */ - if ( bi_compare( bi_copy( birem ), bi_copy( bidenom ) ) < 0 ) - { - biquotient = biapprox; - bi_free( birem ); - bi_free( bidenom ); - } - else - { - /* The real quotient is now biapprox - birem / bidenom. We still - ** have to do a divide. However, birem is smaller than binumer, - ** so the next divide will go faster. We do the divide by - ** recursion. Since this is tail-recursion or close to it, we - ** could probably re-arrange things and make it a non-recursive - ** loop, but the overhead of recursion is small and the bookkeeping - ** is simpler this way. - ** - ** Note that since the sub-divide uses the same denominator, it - ** doesn't have to adjust the values again - the high-order component - ** will still be good. - */ - biquotient = bi_subtract( biapprox, multi_divide2( birem, bidenom ) ); - } - - return biquotient; - } - - -/* Binary division - about five times slower than the above. */ -bigint -bi_binary_divide( bigint binumer, bigint obidenom ) - { - real_bigint bidenom = (real_bigint) obidenom; - int sign; - bigint biquotient; - - /* Check signs and trivial cases. */ - sign = 1; - switch ( bi_compare( bi_copy( bidenom ), bi_0 ) ) - { - case 0: - (void) fprintf( stderr, "bi_divide: divide by zero\n" ); - (void) kill( getpid(), SIGFPE ); - case -1: - sign *= -1; - bidenom = bi_negate( bidenom ); - break; - } - switch ( bi_compare( bi_copy( binumer ), bi_0 ) ) - { - case 0: - bi_free( binumer ); - bi_free( bidenom ); - return bi_0; - case -1: - sign *= -1; - binumer = bi_negate( binumer ); - break; - } - switch ( bi_compare( bi_copy( binumer ), bi_copy( bidenom ) ) ) - { - case -1: - bi_free( binumer ); - bi_free( bidenom ); - return bi_0; - case 0: - bi_free( binumer ); - bi_free( bidenom ); - if ( sign == 1 ) - return bi_1; - else - return bi_m1; - } - - /* Is the denominator small enough to do an int divide? */ - if ( bidenom->num_comps == 1 ) - { - /* Win! */ - biquotient = bi_int_divide( binumer, bidenom->comps[0] ); - bi_free( bidenom ); - } - else - { - /* No, we have to do a full multi-by-multi divide. */ - int num_bits, den_bits, i; - - num_bits = bi_bits( bi_copy( binumer ) ); - den_bits = bi_bits( bi_copy( bidenom ) ); - bidenom = bi_multiply( bidenom, bi_power( bi_2, int_to_bi( num_bits - den_bits ) ) ); - biquotient = bi_0; - for ( i = den_bits; i <= num_bits; ++i ) - { - biquotient = bi_double( biquotient ); - if ( bi_compare( bi_copy( binumer ), bi_copy( bidenom ) ) >= 0 ) - { - biquotient = bi_int_add( biquotient, 1 ); - binumer = bi_subtract( binumer, bi_copy( bidenom ) ); - } - bidenom = bi_half( bidenom ); - } - bi_free( binumer ); - bi_free( bidenom ); - } - - if ( sign == -1 ) - biquotient = bi_negate( biquotient ); - return biquotient; - } - - -bigint -bi_negate( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - real_bigint biR; - - check( bi ); - biR = clone( bi ); - biR->sign = -biR->sign; - check( biR ); - return biR; - } - - -bigint -bi_abs( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - real_bigint biR; - - check( bi ); - biR = clone( bi ); - biR->sign = 1; - check( biR ); - return biR; - } - - -bigint -bi_half( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - real_bigint biR; - int c; - - check( bi ); - /* This depends on the radix being even. */ - biR = clone( bi ); - for ( c = 0; c < biR->num_comps; ++c ) - { - if ( biR->comps[c] & 1 ) - if ( c > 0 ) - biR->comps[c - 1] += bi_radix_o2; - biR->comps[c] = biR->comps[c] >> 1; - } - /* Avoid normalization. */ - if ( biR->num_comps > 1 && biR->comps[biR->num_comps-1] == 0 ) - --biR->num_comps; - check( biR ); - return biR; - } - - -bigint -bi_double( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - real_bigint biR; - int c; - - check( bi ); - biR = clone( bi ); - for ( c = biR->num_comps - 1; c >= 0; --c ) - { - biR->comps[c] = biR->comps[c] << 1; - if ( biR->comps[c] >= bi_radix ) - { - if ( c + 1 >= biR->num_comps ) - more_comps( biR, biR->num_comps + 1 ); - biR->comps[c] -= bi_radix; - biR->comps[c + 1] += 1; - } - } - check( biR ); - return biR; - } - - -/* Find integer square root by Newton's method. */ -bigint -bi_sqrt( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - bigint biR, biR2, bidiff; - - switch ( bi_compare( bi_copy( bi ), bi_0 ) ) - { - case -1: - (void) fprintf( stderr, "bi_sqrt: imaginary result\n" ); - (void) kill( getpid(), SIGFPE ); - case 0: - return bi; - } - if ( bi_is_one( bi_copy( bi ) ) ) - return bi; - - /* Newton's method converges reasonably fast, but it helps to have - ** a good initial guess. We can make a *very* good initial guess - ** by taking the square root of the top component times the square - ** root of the radix part. Both of those are easy to compute. - */ - biR = bi_int_multiply( - bi_power( int_to_bi( bi_radix_sqrt ), int_to_bi( bi->num_comps - 1 ) ), - csqrt( bi->comps[bi->num_comps - 1] ) ); - - /* Now do the Newton loop until we have the answer. */ - for (;;) - { - biR2 = bi_divide( bi_copy( bi ), bi_copy( biR ) ); - bidiff = bi_subtract( bi_copy( biR ), bi_copy( biR2 ) ); - if ( bi_is_zero( bi_copy( bidiff ) ) || - bi_compare( bi_copy( bidiff ), bi_m1 ) == 0 ) - { - bi_free( bi ); - bi_free( bidiff ); - bi_free( biR2 ); - return biR; - } - if ( bi_is_one( bi_copy( bidiff ) ) ) - { - bi_free( bi ); - bi_free( bidiff ); - bi_free( biR ); - return biR2; - } - bi_free( bidiff ); - biR = bi_half( bi_add( biR, biR2 ) ); - } - } - - -int -bi_is_odd( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - int r; - - check( bi ); - r = bi->comps[0] & 1; - bi_free( bi ); - return r; - } - - -int -bi_is_zero( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - int r; - - check( bi ); - r = ( bi->sign == 1 && bi->num_comps == 1 && bi->comps[0] == 0 ); - bi_free( bi ); - return r; - } - - -int -bi_is_one( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - int r; - - check( bi ); - r = ( bi->sign == 1 && bi->num_comps == 1 && bi->comps[0] == 1 ); - bi_free( bi ); - return r; - } - - -int -bi_is_negative( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - int r; - - check( bi ); - r = ( bi->sign == -1 ); - bi_free( bi ); - return r; - } - - -bigint -bi_random( bigint bi ) - { - real_bigint biR; - int c; - - biR = bi_multiply( bi_copy( bi ), bi_copy( bi ) ); - for ( c = 0; c < biR->num_comps; ++c ) - biR->comps[c] = random(); - normalize( biR ); - biR = bi_mod( biR, bi ); - return biR; - } - - -int -bi_bits( bigint obi ) - { - real_bigint bi = (real_bigint) obi; - int bits; - - bits = - bi_comp_bits * ( bi->num_comps - 1 ) + - cbits( bi->comps[bi->num_comps - 1] ); - bi_free( bi ); - return bits; - } - - -/* Allocate and zero more components. Does not consume bi, of course. */ -static void -more_comps( real_bigint bi, int n ) - { - if ( n > bi->max_comps ) - { - bi->max_comps = max( bi->max_comps * 2, n ); - bi->comps = (comp*) realloc( - (void*) bi->comps, bi->max_comps * sizeof(comp) ); - if ( bi->comps == (comp*) 0 ) - { - (void) fprintf( stderr, "out of memory\n" ); - exit( 1 ); - } - } - for ( ; bi->num_comps < n; ++bi->num_comps ) - bi->comps[bi->num_comps] = 0; - } - - -/* Make a new empty bigint. Fills in everything except sign and the -** components. -*/ -static real_bigint -alloc( int num_comps ) - { - real_bigint biR; - - /* Can we recycle an old bigint? */ - if ( free_list != (real_bigint) 0 ) - { - biR = free_list; - free_list = biR->next; - --free_count; - if ( check_level >= 1 && biR->refs != 0 ) - { - (void) fprintf( stderr, "alloc: refs was not 0\n" ); - (void) kill( getpid(), SIGFPE ); - } - more_comps( biR, num_comps ); - } - else - { - /* No free bigints available - create a new one. */ - biR = (real_bigint) malloc( sizeof(struct _real_bigint) ); - if ( biR == (real_bigint) 0 ) - { - (void) fprintf( stderr, "out of memory\n" ); - exit( 1 ); - } - biR->comps = (comp*) malloc( num_comps * sizeof(comp) ); - if ( biR->comps == (comp*) 0 ) - { - (void) fprintf( stderr, "out of memory\n" ); - exit( 1 ); - } - biR->max_comps = num_comps; - } - biR->num_comps = num_comps; - biR->refs = 1; - if ( check_level >= 3 ) - { - /* The active list only gets maintained at check levels 3 or higher. */ - biR->next = active_list; - active_list = biR; - } - else - biR->next = (real_bigint) 0; - ++active_count; - return biR; - } - - -/* Make a modifiable copy of bi. DOES consume bi. */ -static real_bigint -clone( real_bigint bi ) - { - real_bigint biR; - int c; - - /* Very clever optimization. */ - if ( bi->refs != PERMANENT && bi->refs == 1 ) - return bi; - - biR = alloc( bi->num_comps ); - biR->sign = bi->sign; - for ( c = 0; c < bi->num_comps; ++c ) - biR->comps[c] = bi->comps[c]; - bi_free( bi ); - return biR; - } - - -/* Put bi into normal form. Does not consume bi, of course. -** -** Normal form is: -** - All components >= 0 and < bi_radix. -** - Leading 0 components removed. -** - Sign either 1 or -1. -** - The number zero represented by a single 0 component and a sign of 1. -*/ -static void -normalize( real_bigint bi ) - { - int c; - - /* Borrow for negative components. Got to be careful with the math here: - ** -9 / 10 == 0 -9 % 10 == -9 - ** -10 / 10 == -1 -10 % 10 == 0 - ** -11 / 10 == -1 -11 % 10 == -1 - */ - for ( c = 0; c < bi->num_comps - 1; ++c ) - if ( bi->comps[c] < 0 ) - { - bi->comps[c+1] += bi->comps[c] / bi_radix - 1; - bi->comps[c] = bi->comps[c] % bi_radix; - if ( bi->comps[c] != 0 ) - bi->comps[c] += bi_radix; - else - bi->comps[c+1] += 1; - } - /* Is the top component negative? */ - if ( bi->comps[bi->num_comps - 1] < 0 ) - { - /* Switch the sign of the number, and fix up the components. */ - bi->sign = -bi->sign; - for ( c = 0; c < bi->num_comps - 1; ++c ) - { - bi->comps[c] = bi_radix - bi->comps[c]; - bi->comps[c + 1] += 1; - } - bi->comps[bi->num_comps - 1] = -bi->comps[bi->num_comps - 1]; - } - - /* Carry for components larger than the radix. */ - for ( c = 0; c < bi->num_comps; ++c ) - if ( bi->comps[c] >= bi_radix ) - { - if ( c + 1 >= bi->num_comps ) - more_comps( bi, bi->num_comps + 1 ); - bi->comps[c+1] += bi->comps[c] / bi_radix; - bi->comps[c] = bi->comps[c] % bi_radix; - } - - /* Trim off any leading zero components. */ - for ( ; bi->num_comps > 1 && bi->comps[bi->num_comps-1] == 0; --bi->num_comps ) - ; - - /* Check for -0. */ - if ( bi->num_comps == 1 && bi->comps[0] == 0 && bi->sign == -1 ) - bi->sign = 1; - } - - -static void -check( real_bigint bi ) - { - if ( check_level == 0 ) - return; - if ( bi->refs == 0 ) - { - (void) fprintf( stderr, "check: zero refs in bigint\n" ); - (void) kill( getpid(), SIGFPE ); - } - if ( bi->refs < 0 ) - { - (void) fprintf( stderr, "check: negative refs in bigint\n" ); - (void) kill( getpid(), SIGFPE ); - } - if ( check_level < 3 ) - { - /* At check levels less than 3, active bigints have a zero next. */ - if ( bi->next != (real_bigint) 0 ) - { - (void) fprintf( - stderr, "check: attempt to use a bigint from the free list\n" ); - (void) kill( getpid(), SIGFPE ); - } - } - else - { - /* At check levels 3 or higher, active bigints must be on the active - ** list. - */ - real_bigint p; - - for ( p = active_list; p != (real_bigint) 0; p = p->next ) - if ( p == bi ) - break; - if ( p == (real_bigint) 0 ) - { - (void) fprintf( stderr, - "check: attempt to use a bigint not on the active list\n" ); - (void) kill( getpid(), SIGFPE ); - } - } - if ( check_level >= 2 ) - double_check(); - if ( check_level >= 3 ) - triple_check(); - } - - -static void -double_check( void ) - { - real_bigint p; - int c; - - for ( p = free_list, c = 0; p != (real_bigint) 0; p = p->next, ++c ) - if ( p->refs != 0 ) - { - (void) fprintf( stderr, - "double_check: found a non-zero ref on the free list\n" ); - (void) kill( getpid(), SIGFPE ); - } - if ( c != free_count ) - { - (void) fprintf( stderr, - "double_check: free_count is %d but the free list has %d items\n", - free_count, c ); - (void) kill( getpid(), SIGFPE ); - } - } - - -static void -triple_check( void ) - { - real_bigint p; - int c; - - for ( p = active_list, c = 0; p != (real_bigint) 0; p = p->next, ++c ) - if ( p->refs == 0 ) - { - (void) fprintf( stderr, - "triple_check: found a zero ref on the active list\n" ); - (void) kill( getpid(), SIGFPE ); - } - if ( c != active_count ) - { - (void) fprintf( stderr, - "triple_check: active_count is %d but active_list has %d items\n", - free_count, c ); - (void) kill( getpid(), SIGFPE ); - } - } - - -#ifdef DUMP -/* Debug routine to dump out a complete bigint. Does not consume bi. */ -static void -dump( char* str, bigint obi ) - { - int c; - real_bigint bi = (real_bigint) obi; - - (void) fprintf( stdout, "dump %s at 0x%08x:\n", str, (unsigned int) bi ); - (void) fprintf( stdout, " refs: %d\n", bi->refs ); - (void) fprintf( stdout, " next: 0x%08x\n", (unsigned int) bi->next ); - (void) fprintf( stdout, " num_comps: %d\n", bi->num_comps ); - (void) fprintf( stdout, " max_comps: %d\n", bi->max_comps ); - (void) fprintf( stdout, " sign: %d\n", bi->sign ); - for ( c = bi->num_comps - 1; c >= 0; --c ) - (void) fprintf( stdout, " comps[%d]: %11lld (0x%016llx)\n", c, (long long) bi->comps[c], (long long) bi->comps[c] ); - (void) fprintf( stdout, " print: " ); - bi_print( stdout, bi_copy( bi ) ); - (void) fprintf( stdout, "\n" ); - } -#endif /* DUMP */ - - -/* Trivial square-root routine so that we don't have to link in the math lib. */ -static int -csqrt( comp c ) - { - comp r, r2, diff; - - if ( c < 0 ) - { - (void) fprintf( stderr, "csqrt: imaginary result\n" ); - (void) kill( getpid(), SIGFPE ); - } - - r = c / 2; - for (;;) - { - r2 = c / r; - diff = r - r2; - if ( diff == 0 || diff == -1 ) - return (int) r; - if ( diff == 1 ) - return (int) r2; - r = ( r + r2 ) / 2; - } - } - - -/* Figure out how many bits are in a number. */ -static int -cbits( comp c ) - { - int b; - - for ( b = 0; c != 0; ++b ) - c >>= 1; - return b; - } diff --git a/src/rt/bigint/low_primes.h b/src/rt/bigint/low_primes.h deleted file mode 100644 index c9d3df0b3f4..00000000000 --- a/src/rt/bigint/low_primes.h +++ /dev/null @@ -1,1069 +0,0 @@ -/* Primes up to 100000. */ -static long low_primes[] = { - 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, 787, 797, 809, 811, 821, 823, 827, - 829, 839, 853, 857, 859, 863, 877, 881, 883, - 887, 907, 911, 919, 929, 937, 941, 947, 953, - 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, - 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, - 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, - 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, - 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, - 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, - 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, - 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, - 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, - 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, - 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, - 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, - 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, - 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, - 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, - 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, - 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, - 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, - 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, - 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, - 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, - 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, - 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, - 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, - 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, - 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, - 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, - 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, - 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, - 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, - 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, - 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, - 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, - 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, - 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, - 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, - 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, - 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, - 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, - 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, - 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, - 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, - 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, - 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, - 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, - 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, - 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, - 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, - 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, - 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, - 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, - 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, - 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, - 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, - 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, - 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, - 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, - 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, - 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, - 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, - 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, - 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, - 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, - 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, - 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, - 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