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/* mpi-inv.c - MPI functions
* Copyright (C) 1998, 2001, 2002, 2003 Free Software Foundation, Inc.
*
* This file is part of Libgcrypt.
*
* Libgcrypt is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as
* published by the Free Software Foundation; either version 2.1 of
* the License, or (at your option) any later version.
*
* Libgcrypt is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
*/
#include <config.h>
#include <stdio.h>
#include <stdlib.h>
#include "mpi-internal.h"
#include "g10lib.h"
/*
* This uses a modular inversion algorithm designed by Niels Möller
* which was implemented in Nettle. The same algorithm was later also
* adapted to GMP in mpn_sec_invert.
*
* For the description of the algorithm, see Algorithm 5 in Appendix A
* of "Fast Software Polynomial Multiplication on ARM Processors using
* the NEON Engine" by Danilo Câmara, Conrado P. L. Gouvêa, Julio
* López, and Ricardo Dahab:
*
* Note that in the reference above, at the line 2 of Algorithm 5,
* initial value of V was described as V:=1 wrongly. It must be V:=0.
*/
static mpi_ptr_t
mpih_invm_odd (mpi_ptr_t ap, mpi_ptr_t np, mpi_size_t nsize)
{
int secure;
unsigned int iterations;
mpi_ptr_t n1hp;
mpi_ptr_t bp;
mpi_ptr_t up, vp;
secure = _gcry_is_secure (ap);
up = mpi_alloc_limb_space (nsize, secure);
MPN_ZERO (up, nsize);
up[0] = 1;
vp = mpi_alloc_limb_space (nsize, secure);
MPN_ZERO (vp, nsize);
secure = _gcry_is_secure (np);
bp = mpi_alloc_limb_space (nsize, secure);
MPN_COPY (bp, np, nsize);
n1hp = mpi_alloc_limb_space (nsize, secure);
MPN_COPY (n1hp, np, nsize);
_gcry_mpih_rshift (n1hp, n1hp, nsize, 1);
_gcry_mpih_add_1 (n1hp, n1hp, nsize, 1);
iterations = 2 * nsize * BITS_PER_MPI_LIMB;
while (iterations-- > 0)
{
mpi_limb_t odd_a, odd_u, underflow, borrow;
odd_a = ap[0] & 1;
underflow = mpih_sub_n_cond (ap, ap, bp, nsize, odd_a);
mpih_add_n_cond (bp, bp, ap, nsize, underflow);
mpih_abs_cond (ap, ap, nsize, underflow);
mpih_swap_cond (up, vp, nsize, underflow);
_gcry_mpih_rshift (ap, ap, nsize, 1);
borrow = mpih_sub_n_cond (up, up, vp, nsize, odd_a);
mpih_add_n_cond (up, up, np, nsize, borrow);
odd_u = _gcry_mpih_rshift (up, up, nsize, 1) != 0;
mpih_add_n_cond (up, up, n1hp, nsize, odd_u);
}
_gcry_mpi_free_limb_space (n1hp, nsize);
_gcry_mpi_free_limb_space (up, nsize);
if (_gcry_mpih_cmp_ui (bp, nsize, 1) == 0)
{
/* Inverse exists. */
_gcry_mpi_free_limb_space (bp, nsize);
return vp;
}
else
{
_gcry_mpi_free_limb_space (bp, nsize);
_gcry_mpi_free_limb_space (vp, nsize);
return NULL;
}
}
/*
* Calculate the multiplicative inverse X of A mod 2^K
* A must be positive.
*
* See section 7 in "A New Algorithm for Inversion mod p^k" by Çetin
*/
static mpi_ptr_t
mpih_invm_pow2 (mpi_ptr_t ap, mpi_size_t asize, unsigned int k)
{
int secure = _gcry_is_secure (ap);
mpi_size_t i;
unsigned int iterations;
mpi_ptr_t xp, wp, up, vp;
mpi_size_t usize;
if (!(ap[0] & 1))
return NULL;
iterations = ((k + BITS_PER_MPI_LIMB - 1) / BITS_PER_MPI_LIMB)
* BITS_PER_MPI_LIMB;
usize = iterations / BITS_PER_MPI_LIMB;
up = mpi_alloc_limb_space (usize, secure);
MPN_ZERO (up, usize);
up[0] = 1;
vp = mpi_alloc_limb_space (usize, secure);
for (i = 0; i < (usize < asize ? usize : asize); i++)
vp[i] = ap[i];
for (; i < usize; i++)
vp[i] = 0;
if ((k % BITS_PER_MPI_LIMB))
for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)
vp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);
wp = mpi_alloc_limb_space (usize, secure);
MPN_COPY (wp, up, usize);
xp = mpi_alloc_limb_space (usize, secure);
MPN_ZERO (xp, usize);
/*
* It can be considered that overflow at _gcry_mpih_sub_n results
* adding 2^(USIZE*BITS_PER_MPI_LIMB), which is no problem in modulo
* 2^K computation.
*/
for (i = 0; i < iterations; i++)
{
int b0 = (up[0] & 1);
xp[i/BITS_PER_MPI_LIMB] |= ((mpi_limb_t)b0<<(i%BITS_PER_MPI_LIMB));
_gcry_mpih_sub_n (wp, up, vp, usize);
mpih_set_cond (up, wp, usize, b0);
_gcry_mpih_rshift (up, up, usize, 1);
}
if ((k % BITS_PER_MPI_LIMB))
for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)
xp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);
_gcry_mpi_free_limb_space (up, usize);
_gcry_mpi_free_limb_space (vp, usize);
_gcry_mpi_free_limb_space (wp, usize);
return xp;
}
/****************
* Calculate the multiplicative inverse X of A mod N
* That is: Find the solution x for
* 1 = (a*x) mod n
*/
static int
mpi_invm_generic (gcry_mpi_t x, gcry_mpi_t a, gcry_mpi_t n)
{
int is_gcd_one;
#if 0
/* Extended Euclid's algorithm (See TAOCP Vol II, 4.5.2, Alg X) */
gcry_mpi_t u, v, u1, u2, u3, v1, v2, v3, q, t1, t2, t3;
u = mpi_copy(a);
v = mpi_copy(n);
u1 = mpi_alloc_set_ui(1);
u2 = mpi_alloc_set_ui(0);
u3 = mpi_copy(u);
v1 = mpi_alloc_set_ui(0);
v2 = mpi_alloc_set_ui(1);
v3 = mpi_copy(v);
q = mpi_alloc( mpi_get_nlimbs(u)+1 );
t1 = mpi_alloc( mpi_get_nlimbs(u)+1 );
t2 = mpi_alloc( mpi_get_nlimbs(u)+1 );
t3 = mpi_alloc( mpi_get_nlimbs(u)+1 );
while( mpi_cmp_ui( v3, 0 ) ) {
mpi_fdiv_q( q, u3, v3 );
mpi_mul(t1, v1, q); mpi_mul(t2, v2, q); mpi_mul(t3, v3, q);
mpi_sub(t1, u1, t1); mpi_sub(t2, u2, t2); mpi_sub(t3, u3, t3);
mpi_set(u1, v1); mpi_set(u2, v2); mpi_set(u3, v3);
mpi_set(v1, t1); mpi_set(v2, t2); mpi_set(v3, t3);
}
/* log_debug("result:\n");
log_mpidump("q =", q );
log_mpidump("u1=", u1);
log_mpidump("u2=", u2);
log_mpidump("u3=", u3);
log_mpidump("v1=", v1);
log_mpidump("v2=", v2); */
mpi_set(x, u1);
is_gcd_one = (mpi_cmp_ui (u3, 1) == 0);
mpi_free(u1);
mpi_free(u2);
mpi_free(u3);
mpi_free(v1);
mpi_free(v2);
mpi_free(v3);
mpi_free(q);
mpi_free(t1);
mpi_free(t2);
mpi_free(t3);
mpi_free(u);
mpi_free(v);
#elif 0
/* Extended Euclid's algorithm (See TAOCP Vol II, 4.5.2, Alg X)
* modified according to Michael Penk's solution for Exercise 35
* (in the first edition)
* In the third edition, it's Exercise 39, and it is described in
* page 646 of ANSWERS TO EXERCISES chapter.
*/
/* FIXME: we can simplify this in most cases (see Knuth) */
gcry_mpi_t u, v, u1, u2, u3, v1, v2, v3, t1, t2, t3;
unsigned k;
int sign;
u = mpi_copy(a);
v = mpi_copy(n);
for(k=0; !mpi_test_bit(u,0) && !mpi_test_bit(v,0); k++ ) {
mpi_rshift(u, u, 1);
mpi_rshift(v, v, 1);
}
u1 = mpi_alloc_set_ui(1);
u2 = mpi_alloc_set_ui(0);
u3 = mpi_copy(u);
v1 = mpi_copy(v); /* !-- used as const 1 */
v2 = mpi_alloc( mpi_get_nlimbs(u) ); mpi_sub( v2, u1, u );
v3 = mpi_copy(v);
if( mpi_test_bit(u, 0) ) { /* u is odd */
t1 = mpi_alloc_set_ui(0);
t2 = mpi_alloc_set_ui(1); t2->sign = 1;
t3 = mpi_copy(v); t3->sign = !t3->sign;
goto Y4;
}
else {
t1 = mpi_alloc_set_ui(1);
t2 = mpi_alloc_set_ui(0);
t3 = mpi_copy(u);
}
do {
do {
if( mpi_test_bit(t1, 0) || mpi_test_bit(t2, 0) ) { /* one is odd */
mpi_add(t1, t1, v);
mpi_sub(t2, t2, u);
}
mpi_rshift(t1, t1, 1);
mpi_rshift(t2, t2, 1);
mpi_rshift(t3, t3, 1);
Y4:
;
} while( !mpi_test_bit( t3, 0 ) ); /* while t3 is even */
if( !t3->sign ) {
mpi_set(u1, t1);
mpi_set(u2, t2);
mpi_set(u3, t3);
}
else {
mpi_sub(v1, v, t1);
sign = u->sign; u->sign = !u->sign;
mpi_sub(v2, u, t2);
u->sign = sign;
sign = t3->sign; t3->sign = !t3->sign;
mpi_set(v3, t3);
t3->sign = sign;
}
mpi_sub(t1, u1, v1);
mpi_sub(t2, u2, v2);
mpi_sub(t3, u3, v3);
if( t1->sign ) {
mpi_add(t1, t1, v);
mpi_sub(t2, t2, u);
}
} while( mpi_cmp_ui( t3, 0 ) ); /* while t3 != 0 */
/* mpi_lshift( u3, u3, k ); */
is_gcd_one = (k == 0 && mpi_cmp_ui (u3, 1) == 0);
mpi_set(x, u1);
mpi_free(u1);
mpi_free(u2);
mpi_free(u3);
mpi_free(v1);
mpi_free(v2);
mpi_free(v3);
mpi_free(t1);
mpi_free(t2);
mpi_free(t3);
#else
/* Extended Euclid's algorithm (See TAOCP Vol II, 4.5.2, Alg X)
* modified according to Michael Penk's solution for Exercise 35
* with further enhancement */
/* The reference in the comment above is for the first edition.
* In the third edition, it's Exercise 39, and it is described in
* page 646 of ANSWERS TO EXERCISES chapter.
*/
gcry_mpi_t u, v, u1, u2=NULL, u3, v1, v2=NULL, v3, t1, t2=NULL, t3;
unsigned k;
int sign;
int odd ;
u = mpi_copy(a);
v = mpi_copy(n);
for(k=0; !mpi_test_bit(u,0) && !mpi_test_bit(v,0); k++ ) {
mpi_rshift(u, u, 1);
mpi_rshift(v, v, 1);
}
odd = mpi_test_bit(v,0);
u1 = mpi_alloc_set_ui(1);
if( !odd )
u2 = mpi_alloc_set_ui(0);
u3 = mpi_copy(u);
v1 = mpi_copy(v);
if( !odd ) {
v2 = mpi_alloc( mpi_get_nlimbs(u) );
mpi_sub( v2, u1, u ); /* U is used as const 1 */
}
v3 = mpi_copy(v);
if( mpi_test_bit(u, 0) ) { /* u is odd */
t1 = mpi_alloc_set_ui(0);
if( !odd ) {
t2 = mpi_alloc_set_ui(1); t2->sign = 1;
}
t3 = mpi_copy(v); t3->sign = !t3->sign;
goto Y4;
}
else {
t1 = mpi_alloc_set_ui(1);
if( !odd )
t2 = mpi_alloc_set_ui(0);
t3 = mpi_copy(u);
}
do {
do {
if( !odd ) {
if( mpi_test_bit(t1, 0) || mpi_test_bit(t2, 0) ) { /* one is odd */
mpi_add(t1, t1, v);
mpi_sub(t2, t2, u);
}
mpi_rshift(t1, t1, 1);
mpi_rshift(t2, t2, 1);
mpi_rshift(t3, t3, 1);
}
else {
if( mpi_test_bit(t1, 0) )
mpi_add(t1, t1, v);
mpi_rshift(t1, t1, 1);
mpi_rshift(t3, t3, 1);
}
Y4:
;
} while( !mpi_test_bit( t3, 0 ) ); /* while t3 is even */
if( !t3->sign ) {
mpi_set(u1, t1);
if( !odd )
mpi_set(u2, t2);
mpi_set(u3, t3);
}
else {
mpi_sub(v1, v, t1);
sign = u->sign; u->sign = !u->sign;
if( !odd )
mpi_sub(v2, u, t2);
u->sign = sign;
sign = t3->sign; t3->sign = !t3->sign;
mpi_set(v3, t3);
t3->sign = sign;
}
mpi_sub(t1, u1, v1);
if( !odd )
mpi_sub(t2, u2, v2);
mpi_sub(t3, u3, v3);
if( t1->sign ) {
mpi_add(t1, t1, v);
if( !odd )
mpi_sub(t2, t2, u);
}
} while( mpi_cmp_ui( t3, 0 ) ); /* while t3 != 0 */
/* mpi_lshift( u3, u3, k ); */
is_gcd_one = (k == 0 && mpi_cmp_ui (u3, 1) == 0);
mpi_set(x, u1);
mpi_free(u1);
mpi_free(v1);
mpi_free(t1);
if( !odd ) {
mpi_free(u2);
mpi_free(v2);
mpi_free(t2);
}
mpi_free(u3);
mpi_free(v3);
mpi_free(t3);
mpi_free(u);
mpi_free(v);
#endif
return is_gcd_one;
}
/*
* Set X to the multiplicative inverse of A mod M. Return true if the
* inverse exists.
*/
int
_gcry_mpi_invm (gcry_mpi_t x, gcry_mpi_t a, gcry_mpi_t n)
{
mpi_ptr_t ap, xp;
if (!mpi_cmp_ui (a, 0))
return 0; /* Inverse does not exists. */
if (!mpi_cmp_ui (n, 1))
return 0; /* Inverse does not exists. */
if (mpi_test_bit (n, 0))
{
if (a->nlimbs <= n->nlimbs)
{
ap = mpi_alloc_limb_space (n->nlimbs, _gcry_is_secure (a->d));
MPN_ZERO (ap, n->nlimbs);
MPN_COPY (ap, a->d, a->nlimbs);
}
else
ap = _gcry_mpih_mod (a->d, a->nlimbs, n->d, n->nlimbs);
xp = mpih_invm_odd (ap, n->d, n->nlimbs);
_gcry_mpi_free_limb_space (ap, n->nlimbs);
if (xp)
{
_gcry_mpi_assign_limb_space (x, xp, n->nlimbs);
x->nlimbs = n->nlimbs;
return 1;
}
else
return 0; /* Inverse does not exists. */
}
else if (!a->sign && !n->sign)
{
unsigned int k = mpi_trailing_zeros (n);
mpi_size_t x1size = ((k + BITS_PER_MPI_LIMB - 1) / BITS_PER_MPI_LIMB);
mpi_size_t hsize;
gcry_mpi_t q;
mpi_ptr_t x1p, x2p, q_invp, hp, diffp;
mpi_size_t i;
if (k == _gcry_mpi_get_nbits (n) - 1)
{
x1p = mpih_invm_pow2 (a->d, a->nlimbs, k);
if (x1p)
{
_gcry_mpi_assign_limb_space (x, x1p, x1size);
x->nlimbs = x1size;
return 1;
}
else
return 0; /* Inverse does not exists. */
}
/* N can be expressed as P * Q, where P = 2^K. P and Q are coprime. */
/*
* Compute X1 = invm (A, P) and X2 = invm (A, Q), and combine
* them by Garner's formula, to get X = invm (A, P*Q).
* A special case of Chinese Remainder Theorem.
*/
/* X1 = invm (A, P) */
x1p = mpih_invm_pow2 (a->d, a->nlimbs, k);
if (!x1p)
return 0; /* Inverse does not exists. */
/* Q = N / P */
q = mpi_new (0);
mpi_rshift (q, n, k);
/* X2 = invm (A%Q, Q) */
ap = _gcry_mpih_mod (a->d, a->nlimbs, q->d, q->nlimbs);
x2p = mpih_invm_odd (ap, q->d, q->nlimbs);
_gcry_mpi_free_limb_space (ap, q->nlimbs);
if (!x2p)
{
_gcry_mpi_free_limb_space (x1p, x1size);
mpi_free (q);
return 0; /* Inverse does not exists. */
}
/* Q_inv = Q^(-1) = invm (Q, P) */
q_invp = mpih_invm_pow2 (q->d, q->nlimbs, k);
/* H = (X1 - X2) * Q_inv % P */
diffp = mpi_alloc_limb_space (x1size, _gcry_is_secure (a->d));
if (x1size >= q->nlimbs)
_gcry_mpih_sub (diffp, x1p, x1size, x2p, q->nlimbs);
else
_gcry_mpih_sub_n (diffp, x1p, x2p, x1size);
_gcry_mpi_free_limb_space (x1p, x1size);
if ((k % BITS_PER_MPI_LIMB))
for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)
diffp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);
hsize = x1size * 2;
hp = mpi_alloc_limb_space (hsize, _gcry_is_secure (a->d));
_gcry_mpih_mul_n (hp, diffp, q_invp, x1size);
_gcry_mpi_free_limb_space (diffp, x1size);
_gcry_mpi_free_limb_space (q_invp, x1size);
for (i = x1size; i < hsize; i++)
hp[i] = 0;
if ((k % BITS_PER_MPI_LIMB))
for (i = k % BITS_PER_MPI_LIMB; i < BITS_PER_MPI_LIMB; i++)
hp[k/BITS_PER_MPI_LIMB] &= ~(((mpi_limb_t)1) << i);
xp = mpi_alloc_limb_space (x1size + q->nlimbs, _gcry_is_secure (a->d));
if (x1size >= q->nlimbs)
_gcry_mpih_mul (xp, hp, x1size, q->d, q->nlimbs);
else
_gcry_mpih_mul (xp, q->d, q->nlimbs, hp, x1size);
_gcry_mpi_free_limb_space (hp, hsize);
_gcry_mpih_add (xp, xp, x1size + q->nlimbs, x2p, q->nlimbs);
_gcry_mpi_free_limb_space (x2p, q->nlimbs);
_gcry_mpi_assign_limb_space (x, xp, x1size + q->nlimbs);
x->nlimbs = x1size + q->nlimbs;
mpi_free (q);
return 1;
}
else
return mpi_invm_generic (x, a, n);
}