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/* C source code for multiprecision arithmetic library routines.
Implemented Nov 86 by Philip Zimmermann
Last revised 27 Nov 91 by PRZ
Boulder Software Engineering
3021 Eleventh Street
Boulder, CO 80304
(303) 541-0140
(c) Copyright 1986-1994 by Philip Zimmermann. All rights reserved.
The author assumes no liability for damages resulting from the use
of this software, even if the damage results from defects in this
software. No warranty is expressed or implied. The use of this
cryptographic software for developing weapon systems is expressly
forbidden.
These routines implement all of the multiprecision arithmetic
necessary for number-theoretic cryptographic algorithms such as
ElGamal, Diffie-Hellman, Rabin, or factoring studies for large
composite numbers, as well as Rivest-Shamir-Adleman (RSA) public
key cryptography.
Although originally developed in Microsoft C for the IBM PC, this code
contains few machine dependencies. It assumes 2's complement
arithmetic. It can be adapted to 8-bit, 16-bit, or 32-bit machines,
lowbyte-highbyte order or highbyte-lowbyte order. This version
has been converted to ANSI C.
The internal representation for these extended precision integer
"registers" is an array of "units". A unit is a machine word, which
is either an 8-bit byte, a 16-bit unsigned integer, or a 32-bit
unsigned integer, depending on the machine's word size. For example,
an IBM PC or AT uses a unit size of 16 bits. To perform arithmetic
on these huge precision integers, we pass pointers to these unit
arrays to various subroutines. A pointer to an array of units is of
type unitptr. This is a pointer to a huge integer "register".
When calling a subroutine, we always pass a pointer to the BEGINNING
of the array of units, regardless of the byte order of the machine.
On a lowbyte-first machine, such as the Intel 80x86, this unitptr
points to the LEAST significant unit, and the array of units increases
significance to the right. On a highbyte-first machine, such as the
Motorola 680x0, this unitptr points to the MOST significant unit, and
the array of units decreases significance to the right.
Modified 8 Apr 92 - HAJK
Implement new VAX/VMS primitive support.
Modified 30 Sep 92 -Castor Fu <castor@drizzle.stanford.edu>
Upgraded PORTABLE support to allow sizeof(unit) == sizeof(long)
Modified 28 Nov 92 - Thad Smith
Added Smith modmult, generalized non-portable support.
*/
/* #define COUNTMULTS *//* count modmults for performance studies */
#ifdef DEBUG
#ifdef MSDOS
#ifdef __GO32__ /* DJGPP */
#include <pc.h>
#else
#include <conio.h>
#endif /* __GO32__ */
#define poll_for_break() {while (kbhit()) getch();}
#endif /* MSDOS */
#endif /* DEBUG */
#ifndef poll_for_break
#define poll_for_break() /* stub */
#endif
#include "mpilib.h"
#ifdef mp_smula
#ifdef mp_smul
Error:Both mp_smula and mp_smul cannot be defined.
#else
#define mp_smul mp_smula
#endif
#endif
/* set macros for MULTUNIT */
#ifdef MUNIT8
#define MULTUNITSIZE 8
typedef unsigned char MULTUNIT;
#ifdef UNIT8
#define MULTUNIT_SIZE_SAME
#endif
#else /* not MUNIT8 */
#ifdef MUNIT32
#define MULTUNITSIZE 32
typedef unsigned long MULTUNIT;
#ifdef UNIT32
#define MULTUNIT_SIZE_SAME
#else
/* #error is not portable, this has the same effect */
#include "UNITSIZE cannot be smaller than MULTUNITSIZE"
#endif
#else /* assume MUNIT16 */
#define MULTUNITSIZE 16
typedef unsigned short MULTUNIT;
#ifdef UNIT16
#define MULTUNIT_SIZE_SAME
#endif /* UNIT16 */
#ifdef UNIT8
#include "UNITSIZE cannot be smaller than MULTUNITSIZE"
#endif /* UNIT8 */
#endif /* MUNIT32 */
#endif /* MUNIT8 */
#define MULTUNIT_msb ((MULTUNIT)1 << (MULTUNITSIZE-1)) /* msb of
MULTUNIT */
#define DMULTUNIT_msb (1L << (2*MULTUNITSIZE-1))
#define MULTUNIT_mask ((MULTUNIT)((1L << MULTUNITSIZE)-1))
#define MULTUNITs_perunit (UNITSIZE/MULTUNITSIZE)
void mp_smul(MULTUNIT * prod, MULTUNIT * multiplicand, MULTUNIT multiplier);
void mp_dmul(unitptr prod, unitptr multiplicand, unitptr multiplier);
short global_precision = 0; /* units of precision for all routines */
/* global_precision is the unit precision last set by set_precision.
Initially, set_precision() should be called to define global_precision
before using any of these other multiprecision library routines.
i.e.: set_precision(MAX_UNIT_PRECISION);
*/
/*************** multiprecision library primitives ****************/
/* The following portable C primitives should be recoded in assembly.
The entry point name should be defined, in "mpilib.h" to the external
entry point name. If undefined, the C version will be used.
*/
typedef unsigned long int ulint;
#ifndef mp_addc
/*
multiprecision add with carry r2 to r1, result in r1
carry is incoming carry flag-- value should be 0 or 1
*/
boolean mp_addc
(register unitptr r1, register unitptr r2, register boolean carry)
{
register unit x;
short precision; /* number of units to add */
precision = global_precision;
make_lsbptr(r1, precision);
make_lsbptr(r2, precision);
while (precision--) {
if (carry) {
x = *r1 + *r2 + 1;
carry = (*r2 >= (unit) (~*r1));
} else {
x = *r1 + *r2;
carry = (x < *r1);
}
post_higherunit(r2);
*post_higherunit(r1) = x;
}
return carry; /* return the final carry flag bit */
} /* mp_addc */
#endif /* mp_addc */
#ifndef mp_subb
/*
multiprecision subtract with borrow, r2 from r1, result in r1
borrow is incoming borrow flag-- value should be 0 or 1
*/
boolean mp_subb
(register unitptr r1, register unitptr r2, register boolean borrow)
{
register unit x;
short precision; /* number of units to subtract */
precision = global_precision;
make_lsbptr(r1, precision);
make_lsbptr(r2, precision);
while (precision--) {
if (borrow) {
x = *r1 - *r2 - 1;
borrow = (*r1 <= *r2);
} else {
x = *r1 - *r2;
borrow = (*r1 < *r2);
}
post_higherunit(r2);
*post_higherunit(r1) = x;
}
return borrow; /* return the final carry/borrow flag bit */
} /* mp_subb */
#endif /* mp_subb */
#ifndef mp_rotate_left
/*
multiprecision rotate left 1 bit with carry, result in r1.
carry is incoming carry flag-- value should be 0 or 1
*/
boolean mp_rotate_left(register unitptr r1, register boolean carry)
{
register int precision; /* number of units to rotate */
unsigned int mcarry = carry, nextcarry; /* int is supposed to be
* the efficient size for ops*/
precision = global_precision;
make_lsbptr(r1, precision);
while (precision--) {
nextcarry = (((signedunit) * r1) < 0);
*r1 = (*r1 << 1) | mcarry;
mcarry = nextcarry;
pre_higherunit(r1);
}
return nextcarry; /* return the final carry flag bit */
} /* mp_rotate_left */
#endif /* mp_rotate_left */
/************** end of primitives that should be in assembly *************/
/* The mp_shift_right_bits function is not called in any time-critical
situations in public-key cryptographic functions, so it doesn't
need to be coded in assembly language.
*/
/*
multiprecision shift right bits, result in r1.
bits is how many bits to shift, must be <= UNITSIZE.
*/
void mp_shift_right_bits(register unitptr r1, register short bits)
{
register short precision; /* number of units to shift */
register unit carry, nextcarry, bitmask;
register short unbits;
if (bits == 0)
return; /* shift zero bits is a no-op */
carry = 0;
bitmask = power_of_2(bits) - 1;
unbits = UNITSIZE - bits; /* shift bits must be <= UNITSIZE */
precision = global_precision;
make_msbptr(r1, precision);
if (bits == UNITSIZE) {
while (precision--) {
nextcarry = *r1;
*r1 = carry;
carry = nextcarry;
pre_lowerunit(r1);
}
} else {
while (precision--) {
nextcarry = *r1 & bitmask;
*r1 >>= bits;
*r1 |= carry << unbits;
carry = nextcarry;
pre_lowerunit(r1);
}
}
} /* mp_shift_right_bits */
#ifndef mp_compare
/*
* Compares multiprecision integers *r1, *r2, and returns:
* -1 iff *r1 < *r2
* 0 iff *r1 == *r2
* +1 iff *r1 > *r2
*/
short mp_compare(register unitptr r1, register unitptr r2)
{
register short precision; /* number of units to compare */
precision = global_precision;
make_msbptr(r1, precision);
make_msbptr(r2, precision);
do {
if (*r1 < *r2)
return -1;
if (*post_lowerunit(r1) > *post_lowerunit(r2))
return 1;
} while (--precision);
return 0; /* *r1 == *r2 */
} /* mp_compare */
#endif /* mp_compare */
/* Increment multiprecision integer r. */
boolean mp_inc(register unitptr r)
{
register short precision;
precision = global_precision;
make_lsbptr(r, precision);
do {
if (++(*r))
return 0; /* no carry */
post_higherunit(r);
} while (--precision);
return 1; /* return carry set */
} /* mp_inc */
/* Decrement multiprecision integer r. */
boolean mp_dec(register unitptr r)
{
register short precision;
precision = global_precision;
make_lsbptr(r, precision);
do {
if ((signedunit) (--(*r)) != (signedunit) - 1)
return 0; /* no borrow */
post_higherunit(r);
} while (--precision);
return 1; /* return borrow set */
} /* mp_dec */
/* Compute 2's complement, the arithmetic negative, of r */
void mp_neg(register unitptr r)
{
register short precision; /* number of units to negate */
precision = global_precision;
mp_dec(r); /* 2's complement is 1's complement plus 1 */
do { /* now do 1's complement */
*r = ~(*r);
r++;
} while (--precision);
} /* mp_neg */
#ifndef mp_move
void mp_move(register unitptr dst, register unitptr src)
{
register short precision; /* number of units to move */
precision = global_precision;
do {
*dst++ = *src++;
} while (--precision);
} /* mp_move */
#endif /* mp_move */
/* Init multiprecision register r with short value. */
void mp_init(register unitptr r, word16 value)
{ /* Note that mp_init doesn't extend sign bit for >32767 */
unitfill0(r, global_precision);
make_lsbptr(r, global_precision);
*post_higherunit(r) = value;
#ifdef UNIT8
*post_higherunit(r) = value >> UNITSIZE;
#endif
} /* mp_init */
/* Returns number of significant units in r */
short significance(register unitptr r)
{
register short precision;
precision = global_precision;
make_msbptr(r, precision);
do {
if (*post_lowerunit(r))
return precision;
} while (--precision);
return precision;
} /* significance */
#ifndef unitfill0
/* Zero-fill the unit buffer r. */
void unitfill0(unitptr r, word16 unitcount)
{
while (unitcount--)
*r++ = 0;
} /* unitfill0 */
#endif /* unitfill0 */
/* Unsigned divide, treats both operands as positive. */
int mp_udiv(register unitptr remainder, register unitptr quotient,
register unitptr dividend, register unitptr divisor)
{
int bits;
short dprec;
register unit bitmask;
if (testeq(divisor, 0))
return -1; /* zero divisor means divide error */
mp_init0(remainder);
mp_init0(quotient);
/* normalize and compute number of bits in dividend first */
init_bitsniffer(dividend, bitmask, dprec, bits);
/* rescale quotient to same precision (dprec) as dividend */
rescale(quotient, global_precision, dprec);
make_msbptr(quotient, dprec);
while (bits--) {
mp_rotate_left(remainder,
(boolean) (sniff_bit(dividend, bitmask) != 0));
if (mp_compare(remainder, divisor) >= 0) {
mp_sub(remainder, divisor);
stuff_bit(quotient, bitmask);
}
bump_2bitsniffers(dividend, quotient, bitmask);
}
return 0;
} /* mp_udiv */
#ifdef UPTON_OR_SMITH
#define RECIPMARGIN 0 /* extra margin bits used by mp_recip() */
/* Compute reciprocal (quotient) as 1/divisor. Used by faster modmult. */
int mp_recip(register unitptr quotient, register unitptr divisor)
{
int bits;
short qprec;
register unit bitmask;
unit remainder[MAX_UNIT_PRECISION];
if (testeq(divisor, 0))
return -1; /* zero divisor means divide error */
mp_init0(remainder);
mp_init0(quotient);
/* normalize and compute number of bits in quotient first */
bits = countbits(divisor) + RECIPMARGIN;
bitmask = bitmsk(bits); /* bitmask within a single unit */
qprec = bits2units(bits + 1);
mp_setbit(remainder, (bits - RECIPMARGIN) - 1);
/* rescale quotient to precision of divisor + RECIPMARGIN bits */
rescale(quotient, global_precision, qprec);
make_msbptr(quotient, qprec);
while (bits--) {
mp_shift_left(remainder);
if (mp_compare(remainder, divisor) >= 0) {
mp_sub(remainder, divisor);
stuff_bit(quotient, bitmask);
}
bump_bitsniffer(quotient, bitmask);
}
mp_init0(remainder); /* burn sensitive data left on stack */
return 0;
} /* mp_recip */
#endif /* UPTON_OR_SMITH */
/* Signed divide, either or both operands may be negative. */
int mp_div(register unitptr remainder, register unitptr quotient,
register unitptr dividend, register unitptr divisor)
{
boolean dvdsign, dsign;
int status;
dvdsign = (boolean) (mp_tstminus(dividend) != 0);
dsign = (boolean) (mp_tstminus(divisor) != 0);
if (dvdsign)
mp_neg(dividend);
if (dsign)
mp_neg(divisor);
status = mp_udiv(remainder, quotient, dividend, divisor);
if (dvdsign)
mp_neg(dividend); /* repair caller's dividend */
if (dsign)
mp_neg(divisor); /* repair caller's divisor */
if (status < 0)
return status; /* divide error? */
if (dvdsign)
mp_neg(remainder);
if (dvdsign ^ dsign)
mp_neg(quotient);
return status;
} /* mp_div */
/*
* This function does a fast divide and mod on a multiprecision dividend
* using a short integer divisor returning a short integer remainder.
* This is an unsigned divide. It treats both operands as positive.
* It is used mainly for faster printing of large numbers in base 10.
*/
word16 mp_shortdiv(register unitptr quotient,
register unitptr dividend, register word16 divisor)
{
int bits;
short dprec;
register unit bitmask;
register word16 remainder;
if (!divisor) /* if divisor == 0 */
return -1; /* zero divisor means divide error */
remainder = 0;
mp_init0(quotient);
/* normalize and compute number of bits in dividend first */
init_bitsniffer(dividend, bitmask, dprec, bits);
/* rescale quotient to same precision (dprec) as dividend */
rescale(quotient, global_precision, dprec);
make_msbptr(quotient, dprec);
while (bits--) {
remainder <<= 1;
if (sniff_bit(dividend, bitmask))
remainder++;
if (remainder >= divisor) {
remainder -= divisor;
stuff_bit(quotient, bitmask);
}
bump_2bitsniffers(dividend, quotient, bitmask);
}
return remainder;
} /* mp_shortdiv */
/* Unsigned divide, treats both operands as positive. */
int mp_mod(register unitptr remainder,
register unitptr dividend, register unitptr divisor)
{
int bits;
short dprec;
register unit bitmask;
if (testeq(divisor, 0))
return -1; /* zero divisor means divide error */
mp_init0(remainder);
/* normalize and compute number of bits in dividend first */
init_bitsniffer(dividend, bitmask, dprec, bits);
while (bits--) {
mp_rotate_left(remainder,
(boolean) (sniff_bit(dividend, bitmask) != 0));
msub(remainder, divisor);
bump_bitsniffer(dividend, bitmask);
}
return 0;
} /* mp_mod */
/*
* This function does a fast mod operation on a multiprecision dividend
* using a short integer modulus returning a short integer remainder.
* This is an unsigned divide. It treats both operands as positive.
* It is used mainly for fast sieve searches for large primes.
*/
word16 mp_shortmod(register unitptr dividend, register word16 divisor)
{
int bits;
short dprec;
register unit bitmask;
register word16 remainder;
if (!divisor) /* if divisor == 0 */
return -1; /* zero divisor means divide error */
remainder = 0;
/* normalize and compute number of bits in dividend first */
init_bitsniffer(dividend, bitmask, dprec, bits);
while (bits--) {
remainder <<= 1;
if (sniff_bit(dividend, bitmask))
remainder++;
if (remainder >= divisor)
remainder -= divisor;
bump_bitsniffer(dividend, bitmask);
}
return remainder;
} /* mp_shortmod */
#ifdef COMB_MULT /* use faster "comb" multiply algorithm */
/* We are skipping this code because it has a bug... */
/*
* Computes multiprecision prod = multiplicand * multiplier
*
* Uses interleaved comb multiply algorithm.
* This improved multiply more than twice as fast as a Russian
* peasant multiply, because it does a lot fewer shifts.
* Must have global_precision set to the size of the multiplicand
* plus UNITSIZE-1 SLOP_BITS. Produces a product that is the sum
* of the lengths of the multiplier and multiplicand.
*
* BUG ALERT: Unfortunately, this code has a bug. It fails for
* some numbers. One such example:
* x= 59DE 60CE 2345 8091 A02B 2A1C DBC3 8BE5
* x*x= 59DE 60CE 2345 26B3 993B 67A5 2499 0B7D
* 52C8 CDC7 AFB3 61C8 243C 741B
* --which is obviously wrong. The answer should be:
* x*x= 1F8C 607B 5EA6 C061 2714 04A9 A0C6 A17A
* C9AB 6095 C62F 3756 3843 E4D0 3950 7AD9
* We'll have to fix this some day. In the meantime, we'll
* just have the compiler skip over this code.
*
* BUG NOTE: Possibly fixed. Needs testing.
*/
int mp_mult(register unitptr prod,
register unitptr multiplicand, register unitptr multiplier)
{
int bits;
register unit bitmask;
unitptr product, mplier, temp;
short mprec, mprec2;
unit mplicand[MAX_UNIT_PRECISION];
/* better clear full width--double precision */
mp_init(prod + tohigher(global_precision), 0);
if (testeq(multiplicand, 0))
return 0; /* zero multiplicand means zero product */
mp_move(mplicand, multiplicand); /* save it from damage */
normalize(multiplier, mprec);
if (!mprec)
return 0;
make_lsbptr(multiplier, mprec);
bitmask = 1; /* start scan at LSB of multiplier */
do { /* UNITSIZE times */
/* do for bits 0-15 */
product = prod;
mplier = multiplier;
mprec2 = mprec;
while (mprec2--) { /* do for each word in multiplier */
if (sniff_bit(mplier, bitmask)) {
if (mp_addc(product, multiplicand, 0)) { /* ripple carry */
/* After 1st time thru, this is rarely encountered. */
temp = msbptr(product, global_precision);
pre_higherunit(temp);
/* temp now points to LSU of carry region. */
unmake_lsbptr(temp, global_precision);
mp_inc(temp);
} /* ripple carry */
}
pre_higherunit(mplier);
pre_higherunit(product);
}
if (!(bitmask <<= 1))
break;
mp_shift_left(multiplicand);
} while (TRUE);
mp_move(multiplicand, mplicand); /* recover */
return 0; /* normal return */
} /* mp_mult */
#endif /* COMB_MULT */
/* Because the "comb" multiply has a bug, we will use the slower
Russian peasant multiply instead. Fortunately, the mp_mult
function is not called from any time-critical code.
*/
/*
* Computes multiprecision prod = multiplicand * multiplier
* Uses "Russian peasant" multiply algorithm.
*/
int mp_mult(register unitptr prod,
register unitptr multiplicand, register unitptr multiplier)
{
int bits;
register unit bitmask;
short mprec;
mp_init(prod, 0);
if (testeq(multiplicand, 0))
return 0; /* zero multiplicand means zero product */
/* normalize and compute number of bits in multiplier first */
init_bitsniffer(multiplier, bitmask, mprec, bits);
while (bits--) {
mp_shift_left(prod);
if (sniff_bit(multiplier, bitmask))
mp_add(prod, multiplicand);
bump_bitsniffer(multiplier, bitmask);
}
return 0;
} /* mp_mult */
/*
* mp_modmult computes a multiprecision multiply combined with a
* modulo operation. This is the most time-critical function in
* this multiprecision arithmetic library for performing modulo
* exponentiation. We experimented with different versions of modmult,
* depending on the machine architecture and performance requirements.
* We will either use a Russian Peasant modmult (peasant_modmult),
* Charlie Merritt's modmult (merritt_modmult), Jimmy Upton's
* modmult (upton_modmult), or Thad Smith's modmult (smith_modmult).
* On machines with a hardware atomic multiply instruction,
* Smith's modmult is fastest. It can utilize assembly subroutines to
* speed up the hardware multiply logic and trial quotient calculation.
* If the machine lacks a fast hardware multiply, Merritt's modmult
* is preferred, which doesn't call any assembly multiply routine.
* We use the alias names mp_modmult, stage_modulus, and modmult_burn
* for the corresponding true names, which depend on what flavor of
* modmult we are using.
*
* Before making the first call to mp_modmult, you must set up the
* modulus-dependant precomputated tables by calling stage_modulus.
* After making all the calls to mp_modmult, you call modmult_burn to
* erase the tables created by stage_modulus that were left in memory.
*/
#ifdef COUNTMULTS
/* "number of modmults" counters, used for performance studies. */
static unsigned int tally_modmults = 0;
static unsigned int tally_modsquares = 0;
#endif /* COUNTMULTS */
#ifdef PEASANT
/* Conventional Russian peasant multiply with modulo algorithm. */
static unitptr pmodulus = 0; /* used only by mp_modmult */
/*
* Must pass modulus to stage_modulus before calling modmult.
* Assumes that global_precision has already been adjusted to the
* size of the modulus, plus SLOP_BITS.
*/
int stage_peasant_modulus(unitptr n)
{ /* For this simple version of modmult, just copy unit pointer. */
pmodulus = n;
return 0; /* normal return */
} /* stage_peasant_modulus */
/* "Russian peasant" multiply algorithm, combined with a modulo
* operation. This is a simple naive replacement for Merritt's
* faster modmult algorithm. References global unitptr "modulus".
* Computes: prod = (multiplicand*multiplier) mod modulus
* WARNING: All the arguments must be less than the modulus!
*/
int peasant_modmult(register unitptr prod,
unitptr multiplicand, register unitptr multiplier)
{
int bits;
register unit bitmask;
short mprec;
mp_init(prod, 0);
/* if (testeq(multiplicand,0))
return 0; *//* zero multiplicand means zero product */
/* normalize and compute number of bits in multiplier first */
init_bitsniffer(multiplier, bitmask, mprec, bits);
while (bits--) {
mp_shift_left(prod);
msub(prod, pmodulus); /* turns mult into modmult */
if (sniff_bit(multiplier, bitmask)) {
mp_add(prod, multiplicand);
msub(prod, pmodulus); /* turns mult into modmult */
}
bump_bitsniffer(multiplier, bitmask);
}
return 0;
} /* peasant_modmult */
/*
* If we are using a version of mp_modmult that uses internal tables
* in memory, we have to call modmult_burn() at the end of mp_modexp.
* This is so that no sensitive data is left in memory after the program
* exits. The Russian peasant method doesn't use any such tables.
*/
/*
* Alias for modmult_burn, called only from mp_modexp(). Destroys
* internal modmult tables. This version does nothing because no
* tables are used by the Russian peasant modmult.
*/
void peasant_burn(void)
{
} /* peasant_burn */
#endif /* PEASANT */
#ifdef MERRITT
/*=========================================================================*/
/*
This is Charlie Merritt's MODMULT algorithm, implemented in C by PRZ.
Also refined by Zhahai Stewart to reduce the number of subtracts.
Modified by Raymond Brand to reduce the number of SLOP_BITS by 1.
It performs a multiply combined with a modulo operation, without
going into "double precision". It is faster than the Russian peasant
method, and still works well on machines that lack a fast hardware
multiply instruction.
*/
/* The following support functions, macros, and data structures
are used only by Merritt's modmult algorithm... */
/* Shift r1 1 whole unit to the left. Used only by modmult function. */
static void mp_lshift_unit(register unitptr r1)
{
register short precision;
register unitptr r2;
precision = global_precision;
make_msbptr(r1, precision);
r2 = r1;
while (--precision)
*post_lowerunit(r1) = *pre_lowerunit(r2);
*r1 = 0;
} /* mp_lshift_unit */
/* moduli_buf contains shifted images of the modulus, set by stage_modulus */
static unit moduli_buf[UNITSIZE - 1][MAX_UNIT_PRECISION] =
{0};
static unitptr moduli[UNITSIZE] = /* contains pointers into moduli_buf */
{0, moduli_buf[0], moduli_buf[1], moduli_buf[2],
moduli_buf[3], moduli_buf[4], moduli_buf[5], moduli_buf[6]
#ifndef UNIT8
,moduli_buf[7], moduli_buf[8], moduli_buf[9], moduli_buf[10],
moduli_buf[11], moduli_buf[12], moduli_buf[13], moduli_buf[14]
#ifndef UNIT16 /* and not UNIT8 */
,moduli_buf[15], moduli_buf[16], moduli_buf[17], moduli_buf[18],
moduli_buf[19], moduli_buf[20], moduli_buf[21], moduli_buf[22],
moduli_buf[23], moduli_buf[24], moduli_buf[25], moduli_buf[26],
moduli_buf[27], moduli_buf[28], moduli_buf[29], moduli_buf[30]
#endif /* UNIT16 and UNIT8 not defined */
#endif /* UNIT8 not defined */
};
/*
* To optimize msubs, need following 2 unit arrays, each filled
* with the most significant unit and next-to-most significant unit
* of the preshifted images of the modulus.
*/
static unit msu_moduli[UNITSIZE] =
{0}; /* most signif. unit */
static unit nmsu_moduli[UNITSIZE] =
{0}; /* next-most signif. unit */
/*
* mpdbuf contains preshifted images of the multiplicand, mod n.
* It is used only by mp_modmult. It could be staticly declared
* inside of mp_modmult, but we put it outside mp_modmult so that
* it can be wiped clean by modmult_burn(), which is called at the
* end of mp_modexp. This is so that no sensitive data is left in
* memory after the program exits.
*/
static unit mpdbuf[UNITSIZE - 1][MAX_UNIT_PRECISION] =
{0};
/*
* Computes UNITSIZE images of r, each shifted left 1 more bit.
* Used only by modmult function.
*/
static void stage_mp_images(unitptr images[UNITSIZE], unitptr r)
{
short int i;
images[0] = r; /* no need to move the first image, just copy ptr */
for (i = 1; i < UNITSIZE; i++) {
mp_move(images[i], images[i - 1]);
mp_shift_left(images[i]);
msub(images[i], moduli[0]);
}
} /* stage_mp_images */
/*
* Computes UNITSIZE images of modulus, each shifted left 1 more bit.
* Before calling mp_modmult, you must first stage the moduli images by
* calling stage_modulus. n is the pointer to the modulus.
* Assumes that global_precision has already been adjusted to the
* size of the modulus, plus SLOP_BITS.
*/
int stage_merritt_modulus(unitptr n)
{
short int i;
unitptr msu; /* ptr to most significant unit, for faster msubs */
moduli[0] = n; /* no need to move the first image, just copy ptr */
/* used by optimized msubs macro... */
msu = msbptr(n, global_precision); /* needed by msubs */
msu_moduli[0] = *post_lowerunit(msu);
nmsu_moduli[0] = *msu;
for (i = 1; i < UNITSIZE; i++) {
mp_move(moduli[i], moduli[i - 1]);
mp_shift_left(moduli[i]);
/* used by optimized msubs macro... */
msu = msbptr(moduli[i], global_precision); /* needed by msubs */
msu_moduli[i] = *post_lowerunit(msu); /* for faster msubs */
nmsu_moduli[i] = *msu;
}
return 0; /* normal return */
} /* stage_merritt_modulus */
/* The following macros, sniffadd and msubs, are used by modmult... */
#define sniffadd(i) if (*multiplier & power_of_2(i)) mp_add(prod,mpd[i])
/* Unoptimized msubs macro (msubs0) follows... */
/* #define msubs0(i) msub(prod,moduli[i])
*/
/*
* To optimize msubs, compare the most significant units of the
* product and the shifted modulus before deciding to call the full
* mp_compare routine. Better still, compare the upper two units
* before deciding to call mp_compare.
* Optimization of msubs relies heavily on standard C left-to-right
* parsimonious evaluation of logical expressions.
*/
/* Partially-optimized msubs macro (msubs1) follows... */
/* #define msubs1(i) if ( \
((p_m = (*msu_prod-msu_moduli[i])) >= 0) && \
(p_m || (mp_compare(prod,moduli[i]) >= 0) ) \
) mp_sub(prod,moduli[i])
*/
/* Fully-optimized msubs macro (msubs2) follows... */
#define msubs(i) if (((p_m = *msu_prod-msu_moduli[i])>0) || ( \
(p_m==0) && ( (*nmsu_prod>nmsu_moduli[i]) || ( \
(*nmsu_prod==nmsu_moduli[i]) && ((mp_compare(prod,moduli[i]) >= 0)) ))) ) \
mp_sub(prod,moduli[i])
/*
* Performs combined multiply/modulo operation.
* Computes: prod = (multiplicand*multiplier) mod modulus
* WARNING: All the arguments must be less than the modulus!
* Assumes the modulus has been predefined by first calling
* stage_modulus.
*/
int merritt_modmult(register unitptr prod,
unitptr multiplicand, register unitptr multiplier)
{
/* p_m, msu_prod, and nmsu_prod are used by the optimized msubs macro... */
register signedunit p_m;
register unitptr msu_prod; /* ptr to most significant unit of product */
register unitptr nmsu_prod; /* next-most signif. unit of product */
short mprec; /* precision of multiplier, in units */
/* Array mpd contains a list of pointers to preshifted images of
the multiplicand: */
static unitptr mpd[UNITSIZE] =
{
0, mpdbuf[0], mpdbuf[1], mpdbuf[2],
mpdbuf[3], mpdbuf[4], mpdbuf[5], mpdbuf[6]
#ifndef UNIT8
,mpdbuf[7], mpdbuf[8], mpdbuf[9], mpdbuf[10],
mpdbuf[11], mpdbuf[12], mpdbuf[13], mpdbuf[14]
#ifndef UNIT16 /* and not UNIT8 */
,mpdbuf[15], mpdbuf[16], mpdbuf[17], mpdbuf[18],
mpdbuf[19], mpdbuf[20], mpdbuf[21], mpdbuf[22],
mpdbuf[23], mpdbuf[24], mpdbuf[25], mpdbuf[26],
mpdbuf[27], mpdbuf[28], mpdbuf[29], mpdbuf[30]
#endif /* UNIT16 and UNIT8 not defined */
#endif /* UNIT8 not defined */
};
/* Compute preshifted images of multiplicand, mod n: */
stage_mp_images(mpd, multiplicand);
/* To optimize msubs, set up msu_prod and nmsu_prod: */
msu_prod = msbptr(prod, global_precision); /* Get ptr to MSU of prod */
nmsu_prod = msu_prod;
post_lowerunit(nmsu_prod); /* Get next-MSU of prod */
/*
* To understand this algorithm, it would be helpful to first
* study the conventional Russian peasant modmult algorithm.
* This one does about the same thing as Russian peasant, but
* is organized differently to save some steps. It loops
* through the multiplier a word (unit) at a time, instead of
* a bit at a time. It word-shifts the product instead of
* bit-shifting it, so it should be faster. It also does about
* half as many subtracts as Russian peasant.
*/
mp_init(prod, 0); /* Initialize product to 0. */
/*
* The way mp_modmult is actually used in cryptographic
* applications, there will NEVER be a zero multiplier or
* multiplicand. So there is no need to optimize for that
* condition.
*/
/* if (testeq(multiplicand,0))
return 0; *//* zero multiplicand means zero product */
/* Normalize and compute number of units in multiplier first: */
normalize(multiplier, mprec);
if (mprec == 0) /* if precision of multiplier is 0 */
return 0; /* zero multiplier means zero product */
make_msbptr(multiplier, mprec); /* start at MSU of multiplier */
while (mprec--) { /* Loop for the number of units in the multiplier */
/*
* Shift the product one whole unit to the left.
* This is part of the multiply phase of modmult.
*/
mp_lshift_unit(prod);
/*
* The product may have grown by as many as UNITSIZE
* bits. That's why we have global_precision set to the
* size of the modulus plus UNITSIZE slop bits.
* Now reduce the product back down by conditionally
* subtracting the preshifted images of the modulus.
* This is part of the modulo reduction phase of modmult.
* The following loop is unrolled for speed, using macros...
for (i=UNITSIZE-1; i>=LOG_UNITSIZE; i--)
if (mp_compare(prod,moduli[i]) >= 0)
mp_sub(prod,moduli[i]);
*/
#ifndef UNIT8
#ifndef UNIT16 /* and not UNIT8 */
msubs(31);
msubs(30);
msubs(29);
msubs(28);
msubs(27);
msubs(26);
msubs(25);
msubs(24);
msubs(23);
msubs(22);
msubs(21);
msubs(20);
msubs(19);
msubs(18);
msubs(17);
msubs(16);
#endif /* not UNIT16 and not UNIT8 */
msubs(15);
msubs(14);
msubs(13);
msubs(12);
msubs(11);
msubs(10);
msubs(9);
msubs(8);
#endif /* not UNIT8 */
msubs(7);
msubs(6);
msubs(5);
#ifndef UNIT32
msubs(4);
#ifndef UNIT16
msubs(3);
#endif
#endif
/*
* Sniff each bit in the current unit of the multiplier,
* and conditionally add the corresponding preshifted
* image of the multiplicand to the product.
* This is also part of the multiply phase of modmult.
*
* The following loop is unrolled for speed, using macros...
for (i=UNITSIZE-1; i>=0; i--)
if (*multiplier & power_of_2(i))
mp_add(prod,mpd[i]);
*/
#ifndef UNIT8
#ifndef UNIT16 /* and not UNIT8 */
sniffadd(31);
sniffadd(30);
sniffadd(29);
sniffadd(28);
sniffadd(27);
sniffadd(26);
sniffadd(25);
sniffadd(24);
sniffadd(23);
sniffadd(22);
sniffadd(21);
sniffadd(20);
sniffadd(19);
sniffadd(18);
sniffadd(17);
sniffadd(16);
#endif /* not UNIT16 and not UNIT8 */
sniffadd(15);
sniffadd(14);
sniffadd(13);
sniffadd(12);
sniffadd(11);
sniffadd(10);
sniffadd(9);
sniffadd(8);
#endif /* not UNIT8 */
sniffadd(7);
sniffadd(6);
sniffadd(5);
sniffadd(4);
sniffadd(3);
sniffadd(2);
sniffadd(1);
sniffadd(0);
/*
* The product may have grown by as many as LOG_UNITSIZE+1
* bits.
* Now reduce the product back down by conditionally
* subtracting LOG_UNITSIZE+1 preshifted images of the
* modulus. This is the modulo reduction phase of modmult.
*
* The following loop is unrolled for speed, using macros...
for (i=LOG_UNITSIZE; i>=0; i--)
if (mp_compare(prod,moduli[i]) >= 0)
mp_sub(prod,moduli[i]);
*/
#ifndef UNIT8
#ifndef UNIT16
msubs(5);
#endif
msubs(4);
#endif
msubs(3);
msubs(2);
msubs(1);
msubs(0);
/* Bump pointer to next lower unit of multiplier: */
post_lowerunit(multiplier);
} /* Loop for the number of units in the
multiplier */
return 0; /* normal return */
} /* merritt_modmult */
#undef msubs
#undef sniffadd
/*
* Merritt's mp_modmult function leaves some internal tables in memory,
* so we have to call modmult_burn() at the end of mp_modexp.
* This is so that no cryptographically sensitive data is left in memory
* after the program exits.
*/
/* Alias for modmult_burn, merritt_burn() is called only by mp_modexp. */
void merritt_burn(void)
{
unitfill0(&(mpdbuf[0][0]), (UNITSIZE - 1) * MAX_UNIT_PRECISION);
unitfill0(&(moduli_buf[0][0]), (UNITSIZE - 1) * MAX_UNIT_PRECISION);
unitfill0(msu_moduli, UNITSIZE);
unitfill0(nmsu_moduli, UNITSIZE);
} /* merritt_burn() */
/******* end of Merritt's MODMULT stuff. *******/
/*=========================================================================*/
#endif /* MERRITT */
#ifdef UPTON_OR_SMITH /* Used by Upton's and Smith's
modmult algorithms */
/*
* Jimmy Upton's multiprecision modmult algorithm in C.
* Performs a multiply combined with a modulo operation.
*
* The following support functions and data structures
* are used only by Upton's modmult algorithm...
*/
short munit_prec; /* global_precision expressed in MULTUNITs */
/*
* Note that since the SPARC CPU has no hardware integer multiply
* instruction, there is not that much advantage in having an
* assembly version of mp_smul on that machine. It might be faster
* to use Merritt's modmult instead of Upton's modmult on the SPARC.
*/
/*
* Multiply the single-word multiplier times the multiprecision integer
* in multiplicand, accumulating result in prod. The resulting
* multiprecision prod will be 1 word longer than the multiplicand.
* multiplicand is munit_prec words long. We add into prod, so caller
* should zero it out first. For best results, this time-critical
* function should be implemented in assembly.
* NOTE: Unlike other functions in the multiprecision arithmetic
* library, both multiplicand and prod are pointing at the LSB,
* regardless of byte order of the machine. On an 80x86, this makes
* no difference. But if this assembly function is implemented
* on a 680x0, it becomes important.
*/
/*
* Note that this has been modified from the previous version to allow
* better support for Smith's modmult:
* The final carry bit is added to the existing product
* array, rather than simply stored.
*/
#ifndef mp_smul
void mp_smul(MULTUNIT * prod, MULTUNIT * multiplicand, MULTUNIT multiplier)
{
short i;
unsigned long p, carry;
carry = 0;
for (i = 0; i < munit_prec; ++i) {
p = (unsigned long) multiplier **post_higherunit(multiplicand);
p += *prod + carry;
*post_higherunit(prod) = (MULTUNIT) p;
carry = p >> MULTUNITSIZE;
}
/* Add carry to the next higher word of product / dividend */
*prod += (MULTUNIT) carry;
} /* mp_smul */
#endif
/*
* mp_dmul is a double-precision multiply multiplicand times multiplier,
* result into prod. prod must be pointing at a "double precision"
* buffer. E.g. If global_precision is 10 words, prod must be
* pointing at a 20-word buffer.
*/
#ifndef mp_dmul
void mp_dmul(unitptr prod, unitptr multiplicand, unitptr multiplier)
{
register int i;
register MULTUNIT *p_multiplicand, *p_multiplier;
register MULTUNIT *prodp;
unitfill0(prod, global_precision * 2); /* Pre-zero prod */
/* Calculate precision in units of MULTUNIT */
munit_prec = global_precision * UNITSIZE / MULTUNITSIZE;
p_multiplicand = (MULTUNIT *) multiplicand;
p_multiplier = (MULTUNIT *) multiplier;
prodp = (MULTUNIT *) prod;
make_lsbptr(p_multiplicand, munit_prec);
make_lsbptr(p_multiplier, munit_prec);
make_lsbptr(prodp, munit_prec * 2);
/* Multiply multiplicand by each word in multiplier, accumulating prod: */
for (i = 0; i < munit_prec; ++i)
mp_smul(post_higherunit(prodp),
p_multiplicand, *post_higherunit(p_multiplier));
} /* mp_dmul */
#endif /* mp_dmul */
static unit ALIGN modulus[MAX_UNIT_PRECISION];
static short nbits; /* number of modulus significant bits */
#endif /* UPTON_OR_SMITH */
#ifdef UPTON
/*
* These scratchpad arrays are used only by upton_modmult (mp_modmult).
* Some of them could be staticly declared inside of mp_modmult, but we
* put them outside mp_modmult so that they can be wiped clean by
* modmult_burn(), which is called at the end of mp_modexp. This is
* so that no sensitive data is left in memory after the program exits.
*/
static unit ALIGN reciprocal[MAX_UNIT_PRECISION];
static unit ALIGN dhi[MAX_UNIT_PRECISION];
static unit ALIGN d_data[MAX_UNIT_PRECISION * 2]; /* double-wide
register */
static unit ALIGN e_data[MAX_UNIT_PRECISION * 2]; /* double-wide
register */
static unit ALIGN f_data[MAX_UNIT_PRECISION * 2]; /* double-wide
register */
static short nbitsDivUNITSIZE;
static short nbitsModUNITSIZE;
/*
* stage_upton_modulus() is aliased to stage_modulus().
* Prepare for an Upton modmult. Calculate the reciprocal of modulus,
* and save both. Note that reciprocal will have one more bit than
* modulus.
* Assumes that global_precision has already been adjusted to the
* size of the modulus, plus SLOP_BITS.
*/
int stage_upton_modulus(unitptr n)
{
mp_move(modulus, n);
mp_recip(reciprocal, modulus);
nbits = countbits(modulus);
nbitsDivUNITSIZE = nbits / UNITSIZE;
nbitsModUNITSIZE = nbits % UNITSIZE;
return 0; /* normal return */
} /* stage_upton_modulus */
/*
* Upton's algorithm performs a multiply combined with a modulo operation.
* Computes: prod = (multiplicand*multiplier) mod modulus
* WARNING: All the arguments must be less than the modulus!
* References global unitptr modulus and reciprocal.
* The reciprocal of modulus is 1 bit longer than the modulus.
* upton_modmult() is aliased to mp_modmult().
*/
int upton_modmult(unitptr prod, unitptr multiplicand, unitptr multiplier)
{
unitptr d = d_data;
unitptr d1 = d_data;
unitptr e = e_data;
unitptr f = f_data;
short orig_precision;
orig_precision = global_precision;
mp_dmul(d, multiplicand, multiplier);
/* Throw off low nbits of d */
#ifndef HIGHFIRST
d1 = d + nbitsDivUNITSIZE;
#else
d1 = d + orig_precision - nbitsDivUNITSIZE;
#endif
mp_move(dhi, d1); /* Don't screw up d, we need it later */
mp_shift_right_bits(dhi, nbitsModUNITSIZE);
mp_dmul(e, dhi, reciprocal); /* Note - reciprocal has nbits+1 bits */
#ifndef HIGHFIRST
e += nbitsDivUNITSIZE;
#else
e += orig_precision - nbitsDivUNITSIZE;
#endif
mp_shift_right_bits(e, nbitsModUNITSIZE);
mp_dmul(f, e, modulus);
/* Now for the only double-precision call to mpilib: */
set_precision(orig_precision * 2);
mp_sub(d, f);
/* d's precision should be <= orig_precision */
rescale(d, orig_precision * 2, orig_precision);
set_precision(orig_precision);
/* Should never have to do this final subtract more than twice: */
while (mp_compare(d, modulus) > 0)
mp_sub(d, modulus);
mp_move(prod, d);
return 0; /* normal return */
} /* upton_modmult */
/* Upton's mp_modmult function leaves some internal arrays in memory,
so we have to call modmult_burn() at the end of mp_modexp.
This is so that no cryptographically sensitive data is left in memory
after the program exits.
upton_burn() is aliased to modmult_burn().
*/
void upton_burn(void)
{
unitfill0(modulus, MAX_UNIT_PRECISION);
unitfill0(reciprocal, MAX_UNIT_PRECISION);
unitfill0(dhi, MAX_UNIT_PRECISION);
unitfill0(d_data, MAX_UNIT_PRECISION * 2);
unitfill0(e_data, MAX_UNIT_PRECISION * 2);
unitfill0(f_data, MAX_UNIT_PRECISION * 2);
nbits = nbitsDivUNITSIZE = nbitsModUNITSIZE = 0;
} /* upton_burn */
/******* end of Upton's MODMULT stuff. *******/
/*=========================================================================*/
#endif /* UPTON */
#ifdef SMITH /* using Thad Smith's modmult algorithm */
/*
* Thad Smith's implementation of multiprecision modmult algorithm in C.
* Performs a multiply combined with a modulo operation.
* The multiplication is done with mp_dmul, the same as for Upton's
* modmult. The modulus reduction is done by long division, in
* which a trial quotient "digit" is determined, then the product of
* that digit and the divisor are subtracted from the dividend.
*
* In this case, the digit is MULTUNIT in size and the subtraction
* is done by adding the product to the one's complement of the
* dividend, which allows use of the existing mp_smul routine.
*
* The following support functions and data structures
* are used only by Smith's modmult algorithm...
*/
/*
* These scratchpad arrays are used only by smith_modmult (mp_modmult).
* Some of them could be statically declared inside of mp_modmult, but we
* put them outside mp_modmult so that they can be wiped clean by
* modmult_burn(), which is called at the end of mp_modexp. This is
* so that no sensitive data is left in memory after the program exits.
*/
static unit ALIGN ds_data[MAX_UNIT_PRECISION * 2 + 2];
static unit mod_quotient[4];
static unit mod_divisor[4]; /* 2 most signif. MULTUNITs of modulus */
static MULTUNIT *modmpl; /* ptr to modulus least significant
** MULTUNIT */
static int mshift; /* number of bits for
** recip scaling */
static MULTUNIT reciph; /* MSunit of scaled recip */
static MULTUNIT recipl; /* LSunit of scaled recip */
static short modlenMULTUNITS; /* length of modulus in MULTUNITs */
static MULTUNIT mutemp; /* temporary */
/*
* The routines mp_smul and mp_dmul are the same as for UPTON and
* should be coded in assembly. Note, however, that the previous
* Upton's mp_smul version has been modified to compatible with
* Smith's modmult. The new version also still works for Upton's
* modmult.
*/
#ifndef mp_set_recip
#define mp_set_recip(rh,rl,m) /* null */
#else
/* setup routine for external mp_quo_digit */
void mp_set_recip(MULTUNIT rh, MULTUNIT rl, int m);
#endif
MULTUNIT mp_quo_digit(MULTUNIT * dividend);
#ifdef MULTUNIT_SIZE_SAME
#define mp_musubb mp_subb /* use existing routine */
#else /* ! MULTUNIT_SIZE_SAME */
/*
* This performs the same function as mp_subb, but with MULTUNITs.
* Note: Processors without alignment requirements may be able to use
* mp_subb, even though MULTUNITs are smaller than units. In that case,
* use mp_subb, since it would be faster if coded in assembly. Note that
* this implementation won't work for MULTUNITs which are long -- use
* mp_subb in that case.
*/
#ifndef mp_musubb
/*
* multiprecision subtract of MULTUNITs with borrow, r2 from r1, result in r1
* borrow is incoming borrow flag-- value should be 0 or 1
*/
boolean mp_musubb
(register MULTUNIT * r1, register MULTUNIT * r2, register boolean borrow)
{
register ulint x; /* won't work if sizeof(MULTUNIT)==
sizeof(long) */
short precision; /* number of MULTUNITs to subtract */
precision = global_precision * MULTUNITs_perunit;
make_lsbptr(r1, precision);
make_lsbptr(r2, precision);
while (precision--) {
x = (ulint) * r1 - (ulint) * post_higherunit(r2) - (ulint) borrow;
*post_higherunit(r1) = x;
borrow = (((1L << MULTUNITSIZE) & x) != 0L);
}
return (borrow);
} /* mp_musubb */
#endif /* mp_musubb */
#endif /* MULTUNIT_SIZE_SAME */
/*
* The function mp_quo_digit is the heart of Smith's modulo reduction,
* which uses a form of long division. It computes a trial quotient
* "digit" (MULTUNIT-sized digit) by multiplying the three most
* significant MULTUNITs of the dividend by the two most significant
* MULTUNITs of the reciprocal of the modulus. Note that this function
* requires that MULTUNITSIZE * 2 <= sizeof(unsigned long).
*
* An important part of this technique is that the quotient never be
* too small, although it may occasionally be too large. This was
* done to eliminate the need to check and correct for a remainder
* exceeding the divisor. It is easier to check for a negative
* remainder. The following technique rarely needs correction for
* MULTUNITs of at least 16 bits.
*
* The following routine has two implementations:
*
* ASM_PROTOTYPE defined: written to be an executable prototype for
* an efficient assembly language implementation. Note that several
* of the following masks and shifts can be done by directly
* manipulating single precision registers on some architectures.
*
* ASM_PROTOTYPE undefined: a slightly more efficient implementation
* in C. Although this version returns a result larger than the
* optimum (which is corrected in smith_modmult) more often than the
* prototype version, the faster execution in C more than makes up
* for the difference.
*
* Parameter: dividend - points to the most significant MULTUNIT
* of the dividend. Note that dividend actually contains the
* one's complement of the actual dividend value (see comments for
* smith_modmult).
*
* Return: the trial quotient digit resulting from dividing the first
* three MULTUNITs at dividend by the upper two MULTUNITs of the
* modulus.
*/
/* #define ASM_PROTOTYPE *//* undefined: use C-optimized version */
#ifndef mp_quo_digit
MULTUNIT mp_quo_digit(MULTUNIT * dividend)
{
unsigned long q, q0, q1, q2;
unsigned short lsb_factor;
/*
* Compute the least significant product group.
* The last terms of q1 and q2 perform upward rounding, which is
* needed to guarantee that the result not be too small.
*/
q1 = (dividend[tohigher(-2)] ^ MULTUNIT_mask) * (unsigned long) reciph
+ reciph;
q2 = (dividend[tohigher(-1)] ^ MULTUNIT_mask) * (unsigned long) recipl
+ (1L << MULTUNITSIZE);
#ifdef ASM_PROTOTYPE
lsb_factor = 1 & (q1 >> MULTUNITSIZE) & (q2 >> MULTUNITSIZE);
q = q1 + q2;
/* The following statement is equivalent to shifting the sum right
one bit while shifting in the carry bit.
*/
q0 = (q1 > ~q2 ? DMULTUNIT_msb : 0) | (q >> 1);
#else /* optimized C version */
q0 = (q1 >> 1) + (q2 >> 1) + 1;
#endif
/* Compute the middle significant product group. */
q1 = (dividend[tohigher(-1)] ^ MULTUNIT_mask) * (unsigned long) reciph;
q2 = (dividend[0] ^ MULTUNIT_mask) * (unsigned long) recipl;
#ifdef ASM_PROTOTYPE
q = q1 + q2;
q = (q1 > ~q2 ? DMULTUNIT_msb : 0) | (q >> 1);
/* Add in the most significant word of the first group.
The last term takes care of the carry from adding the lsb's
that were shifted out prior to addition.
*/
q = (q0 >> MULTUNITSIZE) + q + (lsb_factor & (q1 ^ q2));
#else /* optimized C version */
q = (q0 >> MULTUNITSIZE) + (q1 >> 1) + (q2 >> 1) + 1;
#endif
/* Compute the most significant term and add in the others */
q = (q >> (MULTUNITSIZE - 2)) +
(((dividend[0] ^ MULTUNIT_mask) * (unsigned long) reciph) << 1);
q >>= mshift;
/* Prevent overflow and then wipe out the intermediate results. */
mutemp = (MULTUNIT) min(q, (1L << MULTUNITSIZE) - 1);
q = q0 = q1 = q2 = 0;
lsb_factor = 0;
(void) lsb_factor;
return mutemp;
}
#endif /* mp_quo_digit */
/*
* stage_smith_modulus() - Prepare for a Smith modmult.
*
* Calculate the reciprocal of modulus with a precision of two MULTUNITs.
* Assumes that global_precision has already been adjusted to the
* size of the modulus, plus SLOP_BITS.
*
* Note: This routine was designed to work with large values and
* doesn't have the necessary testing or handling to work with a
* modulus having less than three significant units. For such cases,
* the separate multiply and modulus routines can be used.
*
* stage_smith_modulus() is aliased to stage_modulus().
*/
int stage_smith_modulus(unitptr n_modulus)
{
int original_precision;
int sigmod; /* significant units in modulus */
unitptr mp; /* modulus most significant pointer */
MULTUNIT *mpm; /* reciprocal pointer */
int prec; /* precision of reciprocal calc in units */
mp_move(modulus, n_modulus);
modmpl = (MULTUNIT *) modulus;
modmpl = lsbptr(modmpl, global_precision * MULTUNITs_perunit);
nbits = countbits(modulus);
modlenMULTUNITS = (nbits + MULTUNITSIZE - 1) / MULTUNITSIZE;
original_precision = global_precision;
/* The following code copies the three most significant units of
* modulus to mod_divisor.
*/
mp = modulus;
sigmod = significance(modulus);
rescale(mp, original_precision, sigmod);
/* prec is the unit precision required for 3 MULTUNITs */
prec = (3 + (MULTUNITs_perunit - 1)) / MULTUNITs_perunit;
set_precision(prec);
/* set mp = ptr to most significant units of modulus, then move
* the most significant units to mp_divisor
*/
mp = msbptr(mp, sigmod) - tohigher(prec - 1);
unmake_lsbptr(mp, prec);
mp_move(mod_divisor, mp);
/* Keep 2*MULTUNITSIZE bits in mod_divisor.
* This will (normally) result in a reciprocal of 2*MULTUNITSIZE+1 bits.
*/
mshift = countbits(mod_divisor) - 2 * MULTUNITSIZE;
mp_shift_right_bits(mod_divisor, mshift);
mp_recip(mod_quotient, mod_divisor);
mp_shift_right_bits(mod_quotient, 1);
/* Reduce to: 0 < mshift <= MULTUNITSIZE */
mshift = ((mshift + (MULTUNITSIZE - 1)) % MULTUNITSIZE) + 1;
/* round up, rescaling if necessary */
mp_inc(mod_quotient);
if (countbits(mod_quotient) > 2 * MULTUNITSIZE) {
mp_shift_right_bits(mod_quotient, 1);
mshift--; /* now 0 <= mshift <= MULTUNITSIZE */
}
mpm = lsbptr((MULTUNIT *) mod_quotient, prec * MULTUNITs_perunit);
recipl = *post_higherunit(mpm);
reciph = *mpm;
mp_set_recip(reciph, recipl, mshift);
set_precision(original_precision);
return 0; /* normal return */
} /* stage_smith_modulus */
/* Smith's algorithm performs a multiply combined with a modulo operation.
Computes: prod = (multiplicand*multiplier) mod modulus
The modulus must first be set by stage_smith_modulus().
smith_modmult() is aliased to mp_modmult().
*/
int smith_modmult(unitptr prod, unitptr multiplicand, unitptr multiplier)
{
unitptr d; /* ptr to product */
MULTUNIT *dmph, *dmpl, *dmp; /* ptrs to dividend (high, low, first)
* aligned for subtraction */
/* Note that dmph points one MULTUNIT higher than indicated by
global precision. This allows us to zero out a word one higher than
the normal precision.
*/
short orig_precision;
short nqd; /* number of quotient digits remaining to
* be generated */
short dmi; /* number of significant MULTUNITs in
product */
d = ds_data + 1; /* room for leading MSB if HIGHFIRST */
orig_precision = global_precision;
mp_dmul(d, multiplicand, multiplier);
rescale(d, orig_precision * 2, orig_precision * 2 + 1);
set_precision(orig_precision * 2 + 1);
*msbptr(d, global_precision) = 0; /* leading 0 unit */
/* We now start working with MULTUNITs.
Determine the most significant MULTUNIT of the product so we don't
have to process leading zeros in our divide loop.
*/
dmi = significance(d) * MULTUNITs_perunit;
if (dmi >= modlenMULTUNITS) {
/* Make dividend negative. This allows the use of mp_smul to
* "subtract" the product of the modulus and the trial divisor
* by actually adding to a negative dividend.
* The one's complement of the dividend is used, since it causes
* a zero value to be represented as all ones. This facilitates
* testing the result for possible overflow, since a sign bit
* indicates that no adjustment is necessary, and we should not
* attempt to adjust if the result of the addition is zero.
*/
mp_inc(d);
mp_neg(d);
set_precision(orig_precision);
munit_prec = global_precision * UNITSIZE / MULTUNITSIZE;
/* Set msb, lsb, and normal ptrs of dividend */
dmph = lsbptr((MULTUNIT *) d, (orig_precision * 2 + 1) *
MULTUNITs_perunit) + tohigher(dmi);
nqd = dmi + 1 - modlenMULTUNITS;
dmpl = dmph - tohigher(modlenMULTUNITS);
/*
* Divide loop.
* Each iteration computes the next quotient MULTUNIT digit, then
* multiplies the divisor (modulus) by the quotient digit and adds
* it to the one's complement of the dividend (equivalent to
* subtracting). If the product was greater than the remaining dividend,
* we get a non-negative result, in which case we subtract off the
* modulus to get the proper negative remainder.
*/
for (; nqd; nqd--) {
MULTUNIT q; /* quotient trial digit */
q = mp_quo_digit(dmph);
if (q > 0) {
mp_smul(dmpl, modmpl, q);
/* Perform correction if q too large.
* This rarely occurs.
*/
if (!(*dmph & MULTUNIT_msb)) {
dmp = dmpl;
unmake_lsbptr(dmp, orig_precision *
MULTUNITs_perunit);
if (mp_musubb(dmp,
(MULTUNIT *) modulus, 0))
(*dmph)--;
}
}
pre_lowerunit(dmph);
pre_lowerunit(dmpl);
}
/* d contains the one's complement of the remainder. */
rescale(d, orig_precision * 2 + 1, orig_precision);
set_precision(orig_precision);
mp_neg(d);
mp_dec(d);
} else {
/* Product was less than modulus. Return it. */
rescale(d, orig_precision * 2 + 1, orig_precision);
set_precision(orig_precision);
}
mp_move(prod, d);
return (0); /* normal return */
} /* smith_modmult */
/* Smith's mp_modmult function leaves some internal arrays in memory,
so we have to call modmult_burn() at the end of mp_modexp.
This is so that no cryptographically sensitive data is left in memory
after the program exits.
smith_burn() is aliased to modmult_burn().
*/
void smith_burn(void)
{
empty_array(modulus);
empty_array(ds_data);
empty_array(mod_quotient);
empty_array(mod_divisor);
modmpl = 0;
mshift = nbits = 0;
reciph = recipl = 0;
modlenMULTUNITS = mutemp = 0;
mp_set_recip(0, 0, 0);
} /* smith_burn */
/* End of Thad Smith's implementation of modmult. */
#endif /* SMITH */
/* Returns number of significant bits in r */
int countbits(unitptr r)
{
int bits;
short prec;
register unit bitmask;
init_bitsniffer(r, bitmask, prec, bits);
return bits;
} /* countbits */
char *copyright_notice(void)
/* force linker to include copyright notice in the executable object image. */
{
return ("(c)1986 Philip Zimmermann");
} /* copyright_notice */
/*
* Russian peasant combined exponentiation/modulo algorithm.
* Calls modmult instead of mult.
* Computes: expout = (expin**exponent) mod modulus
* WARNING: All the arguments must be less than the modulus!
*/
int mp_modexp(register unitptr expout, register unitptr expin,
register unitptr exponent, register unitptr modulus)
{
int bits;
short oldprecision;
register unit bitmask;
unit product[MAX_UNIT_PRECISION];
short eprec;
#ifdef COUNTMULTS
tally_modmults = 0; /* clear "number of modmults" counter */
tally_modsquares = 0; /* clear "number of modsquares" counter */
#endif /* COUNTMULTS */
mp_init(expout, 1);
if (testeq(exponent, 0)) {
if (testeq(expin, 0))
return -1; /* 0 to the 0th power means return error */
return 0; /* otherwise, zero exponent means expout
is 1 */
}
if (testeq(modulus, 0))
return -2; /* zero modulus means error */
#if SLOP_BITS > 0 /* if there's room for sign bits */
if (mp_tstminus(modulus))
return -2; /* negative modulus means error */
#endif /* SLOP_BITS > 0 */
if (mp_compare(expin, modulus) >= 0)
return -3; /* if expin >= modulus, return error */
if (mp_compare(exponent, modulus) >= 0)
return -4; /* if exponent >= modulus, return error */
oldprecision = global_precision; /* save global_precision */
/* set smallest optimum precision for this modulus */
set_precision(bits2units(countbits(modulus) + SLOP_BITS));
/* rescale all these registers to global_precision we just defined */
rescale(modulus, oldprecision, global_precision);
rescale(expin, oldprecision, global_precision);
rescale(exponent, oldprecision, global_precision);
rescale(expout, oldprecision, global_precision);
if (stage_modulus(modulus)) {
set_precision(oldprecision); /* restore original precision */
return -5; /* unstageable modulus (STEWART algorithm) */
}
/* normalize and compute number of bits in exponent first */
init_bitsniffer(exponent, bitmask, eprec, bits);
/* We can "optimize out" the first modsquare and modmult: */
bits--; /* We know for sure at this point that
bits>0 */
mp_move(expout, expin); /* expout = (1*1)*expin; */
bump_bitsniffer(exponent, bitmask);
while (bits--) {
poll_for_break(); /* polls keyboard, allows ctrl-C to
abort program */
#ifdef COUNTMULTS
tally_modsquares++; /* bump "number of modsquares" counter */
#endif /* COUNTMULTS */
mp_modsquare(product, expout);
if (sniff_bit(exponent, bitmask)) {
mp_modmult(expout, product, expin);
#ifdef COUNTMULTS
tally_modmults++; /* bump "number of modmults" counter */
#endif /* COUNTMULTS */
} else {
mp_move(expout, product);
}
bump_bitsniffer(exponent, bitmask);
} /* while bits-- */
mp_burn(product); /* burn the evidence on the stack */
modmult_burn(); /* ask mp_modmult to also burn its
own evidence */
#ifdef COUNTMULTS /* diagnostic analysis */
{
long atomic_mults;
unsigned int unitcount, totalmults;
unitcount = bits2units(countbits(modulus));
/* calculation assumes modsquare takes as long as a modmult: */
atomic_mults = (long) tally_modmults *(unitcount * unitcount);
atomic_mults += (long) tally_modsquares *(unitcount * unitcount);
printf("%ld atomic mults for ", atomic_mults);
printf("%d+%d = %d modsqr+modmlt, at %d bits, %d words.\n",
tally_modsquares, tally_modmults,
tally_modsquares + tally_modmults,
countbits(modulus), unitcount);
}
#endif /* COUNTMULTS */
set_precision(oldprecision); /* restore original precision */
/* Do an explicit reference to the copyright notice so that the linker
will be forced to include it in the executable object image... */
copyright_notice(); /* has no real effect at run time */
return 0; /* normal return */
} /* mp_modexp */
/*
* This is a faster modexp for moduli with a known factorisation into two
* relatively prime factors p and q, and an input relatively prime to the
* modulus, the Chinese Remainder Theorem to do the computation mod p and
* mod q, and then combine the results. This relies on a number of
* precomputed values, but does not actually require the modulus n or the
* exponent e.
*
* expout = expin ^ e mod (p*q).
* We form this by evaluating
* p2 = (expin ^ e) mod p and
* q2 = (expin ^ e) mod q
* and then combining the two by the CRT.
*
* Two optimisations of this are possible. First, we can reduce expin
* modulo p and q before starting.
*
* Second, since we know the factorisation of p and q (trivially derived
* from the factorisation of n = p*q), and expin is relatively prime to
* both p and q, we can use Euler's theorem, expin^phi(m) = 1 (mod m),
* to throw away multiples of phi(p) or phi(q) in e.
* Letting ep = e mod phi(p) and
* eq = e mod phi(q)
* then combining these two speedups, we only need to evaluate
* p2 = ((expin mod p) ^ ep) mod p and
* q2 = ((expin mod q) ^ eq) mod q.
*
* Now we need to apply the CRT. Starting with
* expout = p2 (mod p) and
* expout = q2 (mod q)
* we can say that expout = p2 + p * k, and if we assume that 0 <= p2 < p,
* then 0 <= expout < p*q for some 0 <= k < q. Since we want expout = q2
* (mod q), then p*k = q2-p2 (mod q). Since p and q are relatively prime,
* p has a multiplicative inverse u mod q. In other words, u = 1/p (mod q).
*
* Multiplying by u on both sides gives k = u*(q2-p2) (mod q).
* Since we want 0 <= k < q, we can thus find k as
* k = (u * (q2-p2)) mod q.
*
* Once we have k, evaluating p2 + p * k is easy, and
* that gives us the result.
*
* In the detailed implementation, there is a temporary, temp, used to
* hold intermediate results, p2 is held in expout, and q2 is used as a
* temporary in the final step when it is no longer needed. With that,
* you should be able to understand the code below.
*/
int mp_modexp_crt(unitptr expout, unitptr expin,
unitptr p, unitptr q, unitptr ep, unitptr eq, unitptr u)
{
unit q2[MAX_UNIT_PRECISION];
unit temp[MAX_UNIT_PRECISION];
int status;
/* First, compiute p2 (physically held in M) */
/* p2 = [ (expin mod p) ^ ep ] mod p */
mp_mod(temp, expin, p); /* temp = expin mod p */
status = mp_modexp(expout, temp, ep, p);
if (status < 0) { /* mp_modexp returned an error. */
mp_init(expout, 1);
return status; /* error return */
}
/* And the same thing for q2 */
/* q2 = [ (expin mod q) ^ eq ] mod q */
mp_mod(temp, expin, q); /* temp = expin mod q */
status = mp_modexp(q2, temp, eq, q);
if (status < 0) { /* mp_modexp returned an error. */
mp_init(expout, 1);
return status; /* error return */
}
/* Now use the multiplicative inverse u to glue together the
two halves.
*/
#if 0
/* This optimisation is useful if you commonly get small results,
but for random results and large q, the odds of (1/q) of it
being useful do not warrant the test.
*/
if (mp_compare(expout, q2) != 0) {
#endif
/* Find q2-p2 mod q */
if (mp_sub(q2, expout)) /* if the result went negative */
mp_add(q2, q); /* add q to q2 */
/* expout = p2 + ( p * [(q2*u) mod q] ) */
mp_mult(temp, q2, u); /* q2*u */
mp_mod(q2, temp, q); /* (q2*u) mod q */
mp_mult(temp, p, q2); /* p * [(q2*u) mod q] */
mp_add(expout, temp); /* expout = p2 + p * [...] */
#if 0
}
#endif
mp_burn(q2); /* burn the evidence on the stack... */
mp_burn(temp);
return 0; /* normal return */
} /* mp_modexp_crt */
/****************** end of MPI library ****************************/
These are the contents of the former NiCE NeXT User Group NeXTSTEP/OpenStep software archive, currently hosted by Netfuture.ch.