/*
* Copyright (c) 2007, Cameron Rich
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* * Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
* * 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.
* * Neither the name of the axTLS project nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS 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 COPYRIGHT OWNER 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.
*/
/**
* @defgroup bigint_api Big Integer API
* @brief The bigint implementation as used by the axTLS project.
*
* The bigint library is for RSA encryption/decryption as well as signing.
* This code tries to minimise use of malloc/free by maintaining a small
* cache. A bigint context may maintain state by being made "permanent".
* It be be later released with a bi_depermanent() and bi_free() call.
*
* It supports the following reduction techniques:
* - Classical
* - Barrett
* - Montgomery
*
* It also implements the following:
* - Karatsuba multiplication
* - Squaring
* - Sliding window exponentiation
* - Chinese Remainder Theorem (implemented in rsa.c).
*
* All the algorithms used are pretty standard, and designed for different
* data bus sizes. Negative numbers are not dealt with at all, so a subtraction
* may need to be tested for negativity.
*
* This library steals some ideas from Jef Poskanzer
* <http://cs.marlboro.edu/term/cs-fall02/algorithms/crypto/RSA/bigint>
* and GMP <http://www.swox.com/gmp>. It gets most of its implementation
* detail from "The Handbook of Applied Cryptography"
* <http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf>
* @{
*/
#include <stdlib.h>
#include <limits.h>
#include <string.h>
#include <stdio.h>
#include <time.h>
#include "os_port.h"
#include "bigint.h"
#define V1 v->comps[v->size-1] /**< v1 for division */
#define V2 v->comps[v->size-2] /**< v2 for division */
#define U(j) tmp_u->comps[tmp_u->size-j-1] /**< uj for division */
#define Q(j) quotient->comps[quotient->size-j-1] /**< qj for division */
static bigint *bi_int_multiply(BI_CTX *ctx, bigint *bi, comp i);
static bigint *bi_int_divide(BI_CTX *ctx, bigint *biR, comp denom);
static bigint *alloc(BI_CTX *ctx, int size);
static bigint *trim(bigint *bi);
static void more_comps(bigint *bi, int n);
#if defined(CONFIG_BIGINT_KARATSUBA) || defined(CONFIG_BIGINT_BARRETT) || \
defined(CONFIG_BIGINT_MONTGOMERY)
static bigint *comp_right_shift(bigint *biR, int num_shifts);
static bigint *comp_left_shift(bigint *biR, int num_shifts);
#endif
#ifdef CONFIG_BIGINT_CHECK_ON
static void check(const bigint *bi);
#else
#define check(A) /**< disappears in normal production mode */
#endif
/**
* @brief Start a new bigint context.
* @return A bigint context.
*/
BI_CTX *bi_initialize(void)
{
/* calloc() sets everything to zero */
BI_CTX *ctx = (BI_CTX *)calloc(1, sizeof(BI_CTX));
/* the radix */
ctx->bi_radix = alloc(ctx, 2);
ctx->bi_radix->comps[0] = 0;
ctx->bi_radix->comps[1] = 1;
bi_permanent(ctx->bi_radix);
return ctx;
}
/**
* @brief Close the bigint context and free any resources.
*
* Free up any used memory - a check is done if all objects were not
* properly freed.
* @param ctx [in] The bigint session context.
*/
void bi_terminate(BI_CTX *ctx)
{
bi_depermanent(ctx->bi_radix);
bi_free(ctx, ctx->bi_radix);
if (ctx->active_count != 0)
{
#ifdef CONFIG_SSL_FULL_MODE
printf("bi_terminate: there were %d un-freed bigints\n",
ctx->active_count);
#endif
abort();
}
bi_clear_cache(ctx);
free(ctx);
}
/**
*@brief Clear the memory cache.
*/
void bi_clear_cache(BI_CTX *ctx)
{
bigint *p, *pn;
if (ctx->free_list == NULL)
return;
for (p = ctx->free_list; p != NULL; p = pn)
{
pn = p->next;
free(p->comps);
free(p);
}
ctx->free_count = 0;
ctx->free_list = NULL;
}
/**
* @brief Increment the number of references to this object.
* It does not do a full copy.
* @param bi [in] The bigint to copy.
* @return A reference to the same bigint.
*/
bigint *bi_copy(bigint *bi)
{
check(bi);
if (bi->refs != PERMANENT)
bi->refs++;
return bi;
}
/**
* @brief Simply make a bigint object "unfreeable" if bi_free() is called on it.
*
* For this object to be freed, bi_depermanent() must be called.
* @param bi [in] The bigint to be made permanent.
*/
void bi_permanent(bigint *bi)
{
check(bi);
if (bi->refs != 1)
{
#ifdef CONFIG_SSL_FULL_MODE
printf("bi_permanent: refs was not 1\n");
#endif
abort();
}
bi->refs = PERMANENT;
}
/**
* @brief Take a permanent object and make it eligible for freedom.
* @param bi [in] The bigint to be made back to temporary.
*/
void bi_depermanent(bigint *bi)
{
check(bi);
if (bi->refs != PERMANENT)
{
#ifdef CONFIG_SSL_FULL_MODE
printf("bi_depermanent: bigint was not permanent\n");
#endif
abort();
}
bi->refs = 1;
}
/**
* @brief Free a bigint object so it can be used again.
*
* The memory itself it not actually freed, just tagged as being available
* @param ctx [in] The bigint session context.
* @param bi [in] The bigint to be freed.
*/
void bi_free(BI_CTX *ctx, bigint *bi)
{
check(bi);
if (bi->refs == PERMANENT)
{
return;
}
if (--bi->refs > 0)
{
return;
}
bi->next = ctx->free_list;
ctx->free_list = bi;
ctx->free_count++;
if (--ctx->active_count < 0)
{
#ifdef CONFIG_SSL_FULL_MODE
printf("bi_free: active_count went negative "
"- double-freed bigint?\n");
#endif
abort();
}
}
/**
* @brief Convert an (unsigned) integer into a bigint.
* @param ctx [in] The bigint session context.
* @param i [in] The (unsigned) integer to be converted.
*
*/
bigint *int_to_bi(BI_CTX *ctx, comp i)
{
bigint *biR = alloc(ctx, 1);
biR->comps[0] = i;
return biR;
}
/**
* @brief Do a full copy of the bigint object.
* @param ctx [in] The bigint session context.
* @param bi [in] The bigint object to be copied.
*/
bigint *bi_clone(BI_CTX *ctx, const bigint *bi)
{
bigint *biR = alloc(ctx, bi->size);
check(bi);
memcpy(biR->comps, bi->comps, bi->size*COMP_BYTE_SIZE);
return biR;
}
/**
* @brief Perform an addition operation between two bigints.
* @param ctx [in] The bigint session context.
* @param bia [in] A bigint.
* @param bib [in] Another bigint.
* @return The result of the addition.
*/
bigint *bi_add(BI_CTX *ctx, bigint *bia, bigint *bib)
{
int n;
comp carry = 0;
comp *pa, *pb;
check(bia);
check(bib);
n = max(bia->size, bib->size);
more_comps(bia, n+1);
more_comps(bib, n);
pa = bia->comps;
pb = bib->comps;
do
{
comp sl, rl, cy1;
sl = *pa + *pb++;
rl = sl + carry;
cy1 = sl < *pa;
carry = cy1 | (rl < sl);
*pa++ = rl;
} while (--n != 0);
*pa = carry; /* do overflow */
bi_free(ctx, bib);
return trim(bia);
}
/**
* @brief Perform a subtraction operation between two bigints.
* @param ctx [in] The bigint session context.
* @param bia [in] A bigint.
* @param bib [in] Another bigint.
* @param is_negative [out] If defined, indicates that the result was negative.
* is_negative may be null.
* @return The result of the subtraction. The result is always positive.
*/
bigint *bi_subtract(BI_CTX *ctx,
bigint *bia, bigint *bib, int *is_negative)
{
int n = bia->size;
comp *pa, *pb, carry = 0;
check(bia);
check(bib);
more_comps(bib, n);
pa = bia->comps;
pb = bib->comps;
do
{
comp sl, rl, cy1;
sl = *pa - *pb++;
rl = sl - carry;
cy1 = sl > *pa;
carry = cy1 | (rl > sl);
*pa++ = rl;
} while (--n != 0);
if (is_negative) /* indicate a negative result */
{
*is_negative = carry;
}
bi_free(ctx, trim(bib)); /* put bib back to the way it was */
return trim(bia);
}
/**
* Perform a multiply between a bigint an an (unsigned) integer
*/
static bigint *bi_int_multiply(BI_CTX *ctx, bigint *bia, comp b)
{
int j = 0, n = bia->size;
bigint *biR = alloc(ctx, n + 1);
comp carry = 0;
comp *r = biR->comps;
comp *a = bia->comps;
check(bia);
/* clear things to start with */
memset(r, 0, ((n+1)*COMP_BYTE_SIZE));
do
{
long_comp tmp = *r + (long_comp)a[j]*b + carry;
*r++ = (comp)tmp; /* downsize */
carry = (comp)(tmp >> COMP_BIT_SIZE);
} while (++j < n);
*r = carry;
bi_free(ctx, bia);
return trim(biR);
}
/**
* @brief Does both division and modulo calculations.
*
* Used extensively when doing classical reduction.
* @param ctx [in] The bigint session context.
* @param u [in] A bigint which is the numerator.
* @param v [in] Either the denominator or the modulus depending on the mode.
* @param is_mod [n] Determines if this is a normal division (0) or a reduction
* (1).
* @return The result of the division/reduction.
*/
bigint *bi_divide(BI_CTX *ctx, bigint *u, bigint *v, int is_mod)
{
int n = v->size, m = u->size-n;
int j = 0, orig_u_size = u->size;
uint8_t mod_offset = ctx->mod_offset;
comp d;
bigint *quotient, *tmp_u;
comp q_dash;
check(u);
check(v);
/* if doing reduction and we are < mod, then return mod */
if (is_mod && bi_compare(v, u) > 0)
{
bi_free(ctx, v);
return u;
}
quotient = alloc(ctx, m+1);
tmp_u = alloc(ctx, n+1);
v = trim(v); /* make sure we have no leading 0's */
d = (comp)((long_comp)COMP_RADIX/(V1+1));
/* clear things to start with */
memset(quotient->comps, 0, ((quotient->size)*COMP_BYTE_SIZE));
/* normalise */
if (d > 1)
{
u = bi_int_multiply(ctx, u, d);
if (is_mod)
{
v = ctx->bi_normalised_mod[mod_offset];
}
else
{
v = bi_int_multiply(ctx, v, d);
}
}
if (orig_u_size == u->size) /* new digit position u0 */
{
more_comps(u, orig_u_size + 1);
}
do
{
/* get a temporary short version of u */
memcpy(tmp_u->comps, &u->comps[u->size-n-1-j], (n+1)*COMP_BYTE_SIZE);
/* calculate q' */
if (U(0) == V1)
{
q_dash = COMP_RADIX-1;
}
else
{
q_dash = (comp)(((long_comp)U(0)*COMP_RADIX + U(1))/V1);
if (v->size > 1 && V2)
{
/* we are implementing the following:
if (V2*q_dash > (((U(0)*COMP_RADIX + U(1) -
q_dash*V1)*COMP_RADIX) + U(2))) ... */
comp inner = (comp)((long_comp)COMP_RADIX*U(0) + U(1) -
(long_comp)q_dash*V1);
if ((long_comp)V2*q_dash > (long_comp)inner*COMP_RADIX + U(2))
{
q_dash--;
}
}
}
/* multiply and subtract */
if (q_dash)
{
int is_negative;
tmp_u = bi_subtract(ctx, tmp_u,
bi_int_multiply(ctx, bi_copy(v), q_dash), &is_negative);
more_comps(tmp_u, n+1);
Q(j) = q_dash;
/* add back */
if (is_negative)
{
Q(j)--;
tmp_u = bi_add(ctx, tmp_u, bi_copy(v));
/* lop off the carry */
tmp_u->size--;
v->size--;
}
}
else
{
Q(j) = 0;
}
/* copy back to u */
memcpy(&u->comps[u->size-n-1-j], tmp_u->comps, (n+1)*COMP_BYTE_SIZE);
} while (++j <= m);
bi_free(ctx, tmp_u);
bi_free(ctx, v);
if (is_mod) /* get the remainder */
{
bi_free(ctx, quotient);
return bi_int_divide(ctx, trim(u), d);
}
else /* get the quotient */
{
bi_free(ctx, u);
return trim(quotient);
}
}
/*
* Perform an integer divide on a bigint.
*/
static bigint *bi_int_divide(BI_CTX *ctx, bigint *biR, comp denom)
{
int i = biR->size - 1;
long_comp r = 0;
check(biR);
do
{
r = (r<<COMP_BIT_SIZE) + biR->comps[i];
biR->comps[i] = (comp)(r / denom);
r %= denom;
} while (--i >= 0);
return trim(biR);
}
#ifdef CONFIG_BIGINT_MONTGOMERY
/**
* There is a need for the value of integer N' such that B^-1(B-1)-N^-1N'=1,
* where B^-1(B-1) mod N=1. Actually, only the least significant part of
* N' is needed, hence the definition N0'=N' mod b. We reproduce below the
* simple algorithm from an article by Dusse and Kaliski to efficiently
* find N0' from N0 and b */
static comp modular_inverse(bigint *bim)
{
int i;
comp t = 1;
comp two_2_i_minus_1 = 2; /* 2^(i-1) */
long_comp two_2_i = 4; /* 2^i */
comp N = bim->comps[0];
for (i = 2; i <= COMP_BIT_SIZE; i++)
{
if ((long_comp)N*t%two_2_i >= two_2_i_minus_1)
{
t += two_2_i_minus_1;
}
two_2_i_minus_1 <<= 1;
two_2_i <<= 1;
}
return (comp)(COMP_RADIX-t);
}
#endif
#if defined(CONFIG_BIGINT_KARATSUBA) || defined(CONFIG_BIGINT_BARRETT) || \
defined(CONFIG_BIGINT_MONTGOMERY)
/**
* Take each component and shift down (in terms of components)
*/
static bigint *comp_right_shift(bigint *biR, int num_shifts)
{
int i = biR->size-num_shifts;
comp *x = biR->comps;
comp *y = &biR->comps[num_shifts];
check(biR);
if (i <= 0) /* have we completely right shifted? */
{
biR->comps[0] = 0; /* return 0 */
biR->size = 1;
return biR;
}
do
{
*x++ = *y++;
} while (--i > 0);
biR->size -= num_shifts;
return biR;
}
/**
* Take each component and shift it up (in terms of components)
*/
static bigint *comp_left_shift(bigint *biR, int num_shifts)
{
int i = biR->size-1;
comp *x, *y;
check(biR);
if (num_shifts <= 0)
{
return biR;
}
more_comps(biR, biR->size + num_shifts);
x = &biR->comps[i+num_shifts];
y = &biR->comps[i];
do
{
*x-- = *y--;
} while (i--);
memset(biR->comps, 0, num_shifts*COMP_BYTE_SIZE); /* zero LS comps */
return biR;
}
#endif
/**
* @brief Allow a binary sequence to be imported as a bigint.
* @param ctx [in] The bigint session context.
* @param data [in] The data to be converted.
* @param size [in] The number of bytes of data.
* @return A bigint representing this data.
*/
bigint *bi_import(BI_CTX *ctx, const uint8_t *data, int size)
{
bigint *biR = alloc(ctx, (size+COMP_BYTE_SIZE-1)/COMP_BYTE_SIZE);
int i, j = 0, offset = 0;
memset(biR->comps, 0, biR->size*COMP_BYTE_SIZE);
for (i = size-1; i >= 0; i--)
{
biR->comps[offset] += data[i] << (j*8);
if (++j == COMP_BYTE_SIZE)
{
j = 0;
offset ++;
}
}
return trim(biR);
}
#ifdef CONFIG_SSL_FULL_MODE
/**
* @brief The testharness uses this code to import text hex-streams and
* convert them into bigints.
* @param ctx [in] The bigint session context.
* @param data [in] A string consisting of hex characters. The characters must
* be in upper case.
* @return A bigint representing this data.
*/
bigint *bi_str_import(BI_CTX *ctx, const char *data)
{
int size = strlen(data);
bigint *biR = alloc(ctx, (size+COMP_NUM_NIBBLES-1)/COMP_NUM_NIBBLES);
int i, j = 0, offset = 0;
memset(biR->comps, 0, biR->size*COMP_BYTE_SIZE);
for (i = size-1; i >= 0; i--)
{
int num = (data[i] <= '9') ? (data[i] - '0') : (data[i] - 'A' + 10);
biR->comps[offset] += num << (j*4);
if (++j == COMP_NUM_NIBBLES)
{
j = 0;
offset ++;
}
}
return biR;
}
void bi_print(const char *label, bigint *x)
{
int i, j;
if (x == NULL)
{
printf("%s: (null)\n", label);
return;
}
printf("%s: (size %d)\n", label, x->size);
for (i = x->size-1; i >= 0; i--)
{
for (j = COMP_NUM_NIBBLES-1; j >= 0; j--)
{
comp mask = 0x0f << (j*4);
comp num = (x->comps[i] & mask) >> (j*4);
putc((num <= 9) ? (num + '0') : (num + 'A' - 10), stdout);
}
}
printf("\n");
}
#endif
/**
* @brief Take a bigint and convert it into a byte sequence.
*
* This is useful after a decrypt operation.
* @param ctx [in] The bigint session context.
* @param x [in] The bigint to be converted.
* @param data [out] The converted data as a byte stream.
* @param size [in] The maximum size of the byte stream. Unused bytes will be
* zeroed.
*/
void bi_export(BI_CTX *ctx, bigint *x, uint8_t *data, int size)
{
int i, j, k = size-1;
check(x);
memset(data, 0, size); /* ensure all leading 0's are cleared */
for (i = 0; i < x->size; i++)
{
for (j = 0; j < COMP_BYTE_SIZE; j++)
{
comp mask = 0xff << (j*8);
int num = (x->comps[i] & mask) >> (j*8);
data[k--] = num;
if (k < 0)
{
goto buf_done;
}
}
}
buf_done:
bi_free(ctx, x);
}
/**
* @brief Pre-calculate some of the expensive steps in reduction.
*
* This function should only be called once (normally when a session starts).
* When the session is over, bi_free_mod() should be called. bi_mod_power()
* relies on this function being called.
* @param ctx [in] The bigint session context.
* @param bim [in] The bigint modulus that will be used.
* @param mod_offset [in] There are three moduluii that can be stored - the
* standard modulus, and its two primes p and q. This offset refers to which
* modulus we are referring to.
* @see bi_free_mod(), bi_mod_power().
*/
void bi_set_mod(BI_CTX *ctx, bigint *bim, int mod_offset)
{
int k = bim->size;
comp d = (comp)((long_comp)COMP_RADIX/(bim->comps[k-1]+1));
#ifdef CONFIG_BIGINT_MONTGOMERY
bigint *R, *R2;
#endif
ctx->bi_mod[mod_offset] = bim;
bi_permanent(ctx->bi_mod[mod_offset]);
ctx->bi_normalised_mod[mod_offset] = bi_int_multiply(ctx, bim, d);
bi_permanent(ctx->bi_normalised_mod[mod_offset]);
#if defined(CONFIG_BIGINT_MONTGOMERY)
/* set montgomery variables */
R = comp_left_shift(bi_clone(ctx, ctx->bi_radix), k-1); /* R */
R2 = comp_left_shift(bi_clone(ctx, ctx->bi_radix), k*2-1); /* R^2 */
ctx->bi_RR_mod_m[mod_offset] = bi_mod(ctx, R2); /* R^2 mod m */
ctx->bi_R_mod_m[mod_offset] = bi_mod(ctx, R); /* R mod m */
bi_permanent(ctx->bi_RR_mod_m[mod_offset]);
bi_permanent(ctx->bi_R_mod_m[mod_offset]);
ctx->N0_dash[mod_offset] = modular_inverse(ctx->bi_mod[mod_offset]);
#elif defined (CONFIG_BIGINT_BARRETT)
ctx->bi_mu[mod_offset] =
bi_divide(ctx, comp_left_shift(
bi_clone(ctx, ctx->bi_radix), k*2-1), ctx->bi_mod[mod_offset], 0);
bi_permanent(ctx->bi_mu[mod_offset]);
#endif
}
/**
* @brief Used when cleaning various bigints at the end of a session.
* @param ctx [in] The bigint session context.
* @param mod_offset [in] The offset to use.
* @see bi_set_mod().
*/
void bi_free_mod(BI_CTX *ctx, int mod_offset)
{
bi_depermanent(ctx->bi_mod[mod_offset]);
bi_free(ctx, ctx->bi_mod[mod_offset]);
#if defined (CONFIG_BIGINT_MONTGOMERY)
bi_depermanent(ctx->bi_RR_mod_m[mod_offset]);
bi_depermanent(ctx->bi_R_mod_m[mod_offset]);
bi_free(ctx, ctx->bi_RR_mod_m[mod_offset]);
bi_free(ctx, ctx->bi_R_mod_m[mod_offset]);
#elif defined(CONFIG_BIGINT_BARRETT)
bi_depermanent(ctx->bi_mu[mod_offset]);
bi_free(ctx, ctx->bi_mu[mod_offset]);
#endif
bi_depermanent(ctx->bi_normalised_mod[mod_offset]);
bi_free(ctx, ctx->bi_normalised_mod[mod_offset]);
}
/**
* Perform a standard multiplication between two bigints.
*
* Barrett reduction has no need for some parts of the product, so ignore bits
* of the multiply. This routine gives Barrett its big performance
* improvements over Classical/Montgomery reduction methods.
*/
static bigint *regular_multiply(BI_CTX *ctx, bigint *bia, bigint *bib,
int inner_partial, int outer_partial)
{
int i = 0, j;
int n = bia->size;
int t = bib->size;
bigint *biR = alloc(ctx, n + t);
comp *sr = biR->comps;
comp *sa = bia->comps;
comp *sb = bib->comps;
check(bia);
check(bib);
/* clear things to start with */
memset(biR->comps, 0, ((n+t)*COMP_BYTE_SIZE));
do
{
long_comp tmp;
comp carry = 0;
int r_index = i;
j = 0;
if (outer_partial && outer_partial-i > 0 && outer_partial < n)
{
r_index = outer_partial-1;
j = outer_partial-i-1;
}
do
{
if (inner_partial && r_index >= inner_partial)
{
break;
}
tmp = sr[r_index] + ((long_comp)sa[j])*sb[i] + carry;
sr[r_index++] = (comp)tmp; /* downsize */
carry = tmp >> COMP_BIT_SIZE;
} while (++j < n);
sr[r_index] = carry;
} while (++i < t);
bi_free(ctx, bia);
bi_free(ctx, bib);
return trim(biR);
}
#ifdef CONFIG_BIGINT_KARATSUBA
/*
* Karatsuba improves on regular multiplication due to only 3 multiplications
* being done instead of 4. The additional additions/subtractions are O(N)
* rather than O(N^2) and so for big numbers it saves on a few operations
*/
static bigint *karatsuba(BI_CTX *ctx, bigint *bia, bigint *bib, int is_square)
{
bigint *x0, *x1;
bigint *p0, *p1, *p2;
int m;
if (is_square)
{
m = (bia->size + 1)/2;
}
else
{
m = (max(bia->size, bib->size) + 1)/2;
}
x0 = bi_clone(ctx, bia);
x0->size = m;
x1 = bi_clone(ctx, bia);
comp_right_shift(x1, m);
bi_free(ctx, bia);
/* work out the 3 partial products */
if (is_square)
{
p0 = bi_square(ctx, bi_copy(x0));
p2 = bi_square(ctx, bi_copy(x1));
p1 = bi_square(ctx, bi_add(ctx, x0, x1));
}
else /* normal multiply */
{
bigint *y0, *y1;
y0 = bi_clone(ctx, bib);
y0->size = m;
y1 = bi_clone(ctx, bib);
comp_right_shift(y1, m);
bi_free(ctx, bib);
p0 = bi_multiply(ctx, bi_copy(x0), bi_copy(y0));
p2 = bi_multiply(ctx, bi_copy(x1), bi_copy(y1));
p1 = bi_multiply(ctx, bi_add(ctx, x0, x1), bi_add(ctx, y0, y1));
}
p1 = bi_subtract(ctx,
bi_subtract(ctx, p1, bi_copy(p2), NULL), bi_copy(p0), NULL);
comp_left_shift(p1, m);
comp_left_shift(p2, 2*m);
return bi_add(ctx, p1, bi_add(ctx, p0, p2));
}
#endif
/**
* @brief Perform a multiplication operation between two bigints.
* @param ctx [in] The bigint session context.
* @param bia [in] A bigint.
* @param bib [in] Another bigint.
* @return The result of the multiplication.
*/
bigint *bi_multiply(BI_CTX *ctx, bigint *bia, bigint *bib)
{
check(bia);
check(bib);
#ifdef CONFIG_BIGINT_KARATSUBA
if (min(bia->size, bib->size) < MUL_KARATSUBA_THRESH)
{
return regular_multiply(ctx, bia, bib, 0, 0);
}
return karatsuba(ctx, bia, bib, 0);
#else
return regular_multiply(ctx, bia, bib, 0, 0);
#endif
}
#ifdef CONFIG_BIGINT_SQUARE
/*
* Perform the actual square operion. It takes into account overflow.
*/
static bigint *regular_square(BI_CTX *ctx, bigint *bi)
{
int t = bi->size;
int i = 0, j;
bigint *biR = alloc(ctx, t*2+1);
comp *w = biR->comps;
comp *x = bi->comps;
long_comp carry;
memset(w, 0, biR->size*COMP_BYTE_SIZE);
do
{
long_comp tmp = w[2*i] + (long_comp)x[i]*x[i];
w[2*i] = (comp)tmp;
carry = tmp >> COMP_BIT_SIZE;
for (j = i+1; j < t; j++)
{
uint8_t c = 0;
long_comp xx = (long_comp)x[i]*x[j];
if ((COMP_MAX-xx) < xx)
c = 1;
tmp = (xx<<1);
if ((COMP_MAX-tmp) < w[i+j])
c = 1;
tmp += w[i+j];
if ((COMP_MAX-tmp) < carry)
c = 1;
tmp += carry;
w[i+j] = (comp)tmp;
carry = tmp >> COMP_BIT_SIZE;
if (c)
carry += COMP_RADIX;
}
tmp = w[i+t] + carry;
w[i+t] = (comp)tmp;
w[i+t+1] = tmp >> COMP_BIT_SIZE;
} while (++i < t);
bi_free(ctx, bi);
return trim(biR);
}
/**
* @brief Perform a square operation on a bigint.
* @param ctx [in] The bigint session context.
* @param bia [in] A bigint.
* @return The result of the multiplication.
*/
bigint *bi_square(BI_CTX *ctx, bigint *bia)
{
check(bia);
#ifdef CONFIG_BIGINT_KARATSUBA
if (bia->size < SQU_KARATSUBA_THRESH)
{
return regular_square(ctx, bia);
}
return karatsuba(ctx, bia, NULL, 1);
#else
return regular_square(ctx, bia);
#endif
}
#endif
/**
* @brief Compare two bigints.
* @param bia [in] A bigint.
* @param bib [in] Another bigint.
* @return -1 if smaller, 1 if larger and 0 if equal.
*/
int bi_compare(bigint *bia, bigint *bib)
{
int r, i;
check(bia);
check(bib);
if (bia->size > bib->size)
r = 1;
else if (bia->size < bib->size)
r = -1;
else
{
comp *a = bia->comps;
comp *b = bib->comps;
/* Same number of components. Compare starting from the high end
* and working down. */
r = 0;
i = bia->size - 1;
do
{
if (a[i] > b[i])
{
r = 1;
break;
}
else if (a[i] < b[i])
{
r = -1;
break;
}
} while (--i >= 0);
}
return r;
}
/*
* Allocate and zero more components. Does not consume bi.
*/
static void more_comps(bigint *bi, int n)
{
if (n > bi->max_comps)
{
bi->max_comps = max(bi->max_comps * 2, n);
bi->comps = (comp*)realloc(bi->comps, bi->max_comps * COMP_BYTE_SIZE);
}
if (n > bi->size)
{
memset(&bi->comps[bi->size], 0, (n-bi->size)*COMP_BYTE_SIZE);
}
bi->size = n;
}
/*
* Make a new empty bigint. It may just use an old one if one is available.
* Otherwise get one off the heap.
*/
static bigint *alloc(BI_CTX *ctx, int size)
{
bigint *biR;
/* Can we recycle an old bigint? */
if (ctx->free_list != NULL)
{
biR = ctx->free_list;
ctx->free_list = biR->next;
ctx->free_count--;
if (biR->refs != 0)
{
#ifdef CONFIG_SSL_FULL_MODE
printf("alloc: refs was not 0\n");
#endif
abort(); /* create a stack trace from a core dump */
}
more_comps(biR, size);
}
else
{
/* No free bigints available - create a new one. */
biR = (bigint *)malloc(sizeof(bigint));
biR->comps = (comp*)malloc(size * COMP_BYTE_SIZE);
biR->max_comps = size; /* give some space to spare */
}
biR->size = size;
biR->refs = 1;
biR->next = NULL;
ctx->active_count++;
return biR;
}
/*
* Work out the highest '1' bit in an exponent. Used when doing sliding-window
* exponentiation.
*/
static int find_max_exp_index(bigint *biexp)
{
int i = COMP_BIT_SIZE-1;
comp shift = COMP_RADIX/2;
comp test = biexp->comps[biexp->size-1]; /* assume no leading zeroes */
check(biexp);
do
{
if (test & shift)
{
return i+(biexp->size-1)*COMP_BIT_SIZE;
}
shift >>= 1;
} while (i-- != 0);
return -1; /* error - must have been a leading 0 */
}
/*
* Is a particular bit is an exponent 1 or 0? Used when doing sliding-window
* exponentiation.
*/
static int exp_bit_is_one(bigint *biexp, int offset)
{
comp test = biexp->comps[offset / COMP_BIT_SIZE];
int num_shifts = offset % COMP_BIT_SIZE;
comp shift = 1;
int i;
check(biexp);
for (i = 0; i < num_shifts; i++)
{
shift <<= 1;
}
return (test & shift) != 0;
}
#ifdef CONFIG_BIGINT_CHECK_ON
/*
* Perform a sanity check on bi.
*/
static void check(const bigint *bi)
{
if (bi->refs <= 0)
{
printf("check: zero or negative refs in bigint\n");
abort();
}
if (bi->next != NULL)
{
printf("check: attempt to use a bigint from "
"the free list\n");
abort();
}
}
#endif
/*
* Delete any leading 0's (and allow for 0).
*/
static bigint *trim(bigint *bi)
{
check(bi);
while (bi->comps[bi->size-1] == 0 && bi->size > 1)
{
bi->size--;
}
return bi;
}
#if defined(CONFIG_BIGINT_MONTGOMERY)
/**
* @brief Perform a single montgomery reduction.
* @param ctx [in] The bigint session context.
* @param bixy [in] A bigint.
* @return The result of the montgomery reduction.
*/
bigint *bi_mont(BI_CTX *ctx, bigint *bixy)
{
int i = 0, n;
uint8_t mod_offset = ctx->mod_offset;
bigint *bim = ctx->bi_mod[mod_offset];
comp mod_inv = ctx->N0_dash[mod_offset];
check(bixy);
if (ctx->use_classical) /* just use classical instead */
{
return bi_mod(ctx, bixy);
}
n = bim->size;
do
{
bixy = bi_add(ctx, bixy, comp_left_shift(
bi_int_multiply(ctx, bim, bixy->comps[i]*mod_inv), i));
} while (++i < n);
comp_right_shift(bixy, n);
if (bi_compare(bixy, bim) >= 0)
{
bixy = bi_subtract(ctx, bixy, bim, NULL);
}
return bixy;
}
#elif defined(CONFIG_BIGINT_BARRETT)
/*
* Stomp on the most significant components to give the illusion of a "mod base
* radix" operation
*/
static bigint *comp_mod(bigint *bi, int mod)
{
check(bi);
if (bi->size > mod)
{
bi->size = mod;
}
return bi;
}
/**
* @brief Perform a single Barrett reduction.
* @param ctx [in] The bigint session context.
* @param bi [in] A bigint.
* @return The result of the Barrett reduction.
*/
bigint *bi_barrett(BI_CTX *ctx, bigint *bi)
{
bigint *q1, *q2, *q3, *r1, *r2, *r;
uint8_t mod_offset = ctx->mod_offset;
bigint *bim = ctx->bi_mod[mod_offset];
int k = bim->size;
check(bi);
check(bim);
/* use Classical method instead - Barrett cannot help here */
if (bi->size > k*2)
{
return bi_mod(ctx, bi);
}
q1 = comp_right_shift(bi_clone(ctx, bi), k-1);
/* do outer partial multiply */
q2 = regular_multiply(ctx, q1, ctx->bi_mu[mod_offset], 0, k-1);
q3 = comp_right_shift(q2, k+1);
r1 = comp_mod(bi, k+1);
/* do inner partial multiply */
r2 = comp_mod(regular_multiply(ctx, q3, bim, k+1, 0), k+1);
r = bi_subtract(ctx, r1, r2, NULL);
/* if (r >= m) r = r - m; */
if (bi_compare(r, bim) >= 0)
{
r = bi_subtract(ctx, r, bim, NULL);
}
return r;
}
#endif /* CONFIG_BIGINT_BARRETT */
#ifdef CONFIG_BIGINT_SLIDING_WINDOW
/*
* Work out g1, g3, g5, g7... etc for the sliding-window algorithm
*/
static void precompute_slide_window(BI_CTX *ctx, int window, bigint *g1)
{
int k = 1, i;
bigint *g2;
for (i = 0; i < window-1; i++) /* compute 2^(window-1) */
{
k <<= 1;
}
ctx->g = (bigint **)malloc(k*sizeof(bigint *));
ctx->g[0] = bi_clone(ctx, g1);
bi_permanent(ctx->g[0]);
g2 = bi_residue(ctx, bi_square(ctx, ctx->g[0])); /* g^2 */
for (i = 1; i < k; i++)
{
ctx->g[i] = bi_residue(ctx, bi_multiply(ctx, ctx->g[i-1], bi_copy(g2)));
bi_permanent(ctx->g[i]);
}
bi_free(ctx, g2);
ctx->window = k;
}
#endif
/**
* @brief Perform a modular exponentiation.
*
* This function requires bi_set_mod() to have been called previously. This is
* one of the optimisations used for performance.
* @param ctx [in] The bigint session context.
* @param bi [in] The bigint on which to perform the mod power operation.
* @param biexp [in] The bigint exponent.
* @return The result of the mod exponentiation operation
* @see bi_set_mod().
*/
bigint *bi_mod_power(BI_CTX *ctx, bigint *bi, bigint *biexp)
{
int i = find_max_exp_index(biexp), j, window_size = 1;
bigint *biR = int_to_bi(ctx, 1);
#if defined(CONFIG_BIGINT_MONTGOMERY)
uint8_t mod_offset = ctx->mod_offset;
if (!ctx->use_classical)
{
/* preconvert */
bi = bi_mont(ctx,
bi_multiply(ctx, bi, ctx->bi_RR_mod_m[mod_offset])); /* x' */
bi_free(ctx, biR);
biR = ctx->bi_R_mod_m[mod_offset]; /* A */
}
#endif
check(bi);
check(biexp);
#ifdef CONFIG_BIGINT_SLIDING_WINDOW
for (j = i; j > 32; j /= 5) /* work out an optimum size */
window_size++;
/* work out the slide constants */
precompute_slide_window(ctx, window_size, bi);
#else /* just one constant */
ctx->g = (bigint **)malloc(sizeof(bigint *));
ctx->g[0] = bi_clone(ctx, bi);
ctx->window = 1;
bi_permanent(ctx->g[0]);
#endif
/* if sliding-window is off, then only one bit will be done at a time and
* will reduce to standard left-to-right exponentiation */
do
{
if (exp_bit_is_one(biexp, i))
{
int l = i-window_size+1;
int part_exp = 0;
if (l < 0) /* LSB of exponent will always be 1 */
l = 0;
else
{
while (exp_bit_is_one(biexp, l) == 0)
l++; /* go back up */
}
/* build up the section of the exponent */
for (j = i; j >= l; j--)
{
biR = bi_residue(ctx, bi_square(ctx, biR));
if (exp_bit_is_one(biexp, j))
part_exp++;
if (j != l)
part_exp <<= 1;
}
part_exp = (part_exp-1)/2; /* adjust for array */
biR = bi_residue(ctx, bi_multiply(ctx, biR, ctx->g[part_exp]));
i = l-1;
}
else /* square it */
{
biR = bi_residue(ctx, bi_square(ctx, biR));
i--;
}
} while (i >= 0);
/* cleanup */
for (i = 0; i < ctx->window; i++)
{
bi_depermanent(ctx->g[i]);
bi_free(ctx, ctx->g[i]);
}
free(ctx->g);
bi_free(ctx, bi);
bi_free(ctx, biexp);
#if defined CONFIG_BIGINT_MONTGOMERY
return ctx->use_classical ? biR : bi_mont(ctx, biR); /* convert back */
#else /* CONFIG_BIGINT_CLASSICAL or CONFIG_BIGINT_BARRETT */
return biR;
#endif
}
#ifdef CONFIG_SSL_CERT_VERIFICATION
/**
* @brief Perform a modular exponentiation using a temporary modulus.
*
* We need this function to check the signatures of certificates. The modulus
* of this function is temporary as it's just used for authentication.
* @param ctx [in] The bigint session context.
* @param bi [in] The bigint to perform the exp/mod.
* @param bim [in] The temporary modulus.
* @param biexp [in] The bigint exponent.
* @return The result of the mod exponentiation operation
* @see bi_set_mod().
*/
bigint *bi_mod_power2(BI_CTX *ctx, bigint *bi, bigint *bim, bigint *biexp)
{
bigint *biR, *tmp_biR;
/* Set up a temporary bigint context and transfer what we need between
* them. We need to do this since we want to keep the original modulus
* which is already in this context. This operation is only called when
* doing peer verification, and so is not expensive :-) */
BI_CTX *tmp_ctx = bi_initialize();
bi_set_mod(tmp_ctx, bi_clone(tmp_ctx, bim), BIGINT_M_OFFSET);
tmp_biR = bi_mod_power(tmp_ctx,
bi_clone(tmp_ctx, bi),
bi_clone(tmp_ctx, biexp));
biR = bi_clone(ctx, tmp_biR);
bi_free(tmp_ctx, tmp_biR);
bi_free_mod(tmp_ctx, BIGINT_M_OFFSET);
bi_terminate(tmp_ctx);
bi_free(ctx, bi);
bi_free(ctx, bim);
bi_free(ctx, biexp);
return biR;
}
#endif
#ifdef CONFIG_BIGINT_CRT
/**
* @brief Use the Chinese Remainder Theorem to quickly perform RSA decrypts.
*
* @param ctx [in] The bigint session context.
* @param bi [in] The bigint to perform the exp/mod.
* @param dP [in] CRT's dP bigint
* @param dQ [in] CRT's dQ bigint
* @param p [in] CRT's p bigint
* @param q [in] CRT's q bigint
* @param qInv [in] CRT's qInv bigint
* @return The result of the CRT operation
*/
bigint *bi_crt(BI_CTX *ctx, bigint *bi,
bigint *dP, bigint *dQ,
bigint *p, bigint *q, bigint *qInv)
{
bigint *m1, *m2, *h;
/* Montgomery has a condition the 0 < x, y < m and these products violate
* that condition. So disable Montgomery when using CRT */
#if defined(CONFIG_BIGINT_MONTGOMERY)
ctx->use_classical = 1;
#endif
ctx->mod_offset = BIGINT_P_OFFSET;
m1 = bi_mod_power(ctx, bi_copy(bi), dP);
ctx->mod_offset = BIGINT_Q_OFFSET;
m2 = bi_mod_power(ctx, bi, dQ);
h = bi_subtract(ctx, bi_add(ctx, m1, p), bi_copy(m2), NULL);
h = bi_multiply(ctx, h, qInv);
ctx->mod_offset = BIGINT_P_OFFSET;
h = bi_residue(ctx, h);
#if defined(CONFIG_BIGINT_MONTGOMERY)
ctx->use_classical = 0; /* reset for any further operation */
#endif
return bi_add(ctx, m2, bi_multiply(ctx, q, h));
}
#endif
/** @} */
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