/* * Multi-precision integer library * * Based on XySSL: Copyright (C) 2006-2008 Christophe Devine * * Copyright (C) 2009 Paul Bakker * * 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 names of PolarSSL or XySSL 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. */ /* * This MPI implementation is based on: * * http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf * http://www.stillhq.com/extracted/gnupg-api/mpi/ * http://math.libtomcrypt.com/files/tommath.pdf */ //#include "polarssl/config.h" #include "bn_mul.h" #include #include #include #include "bignum.h" //#include "polarssl/bn_mul.h" #define ciL ((int) sizeof(t_int)) /* chars in limb */ #define biL (ciL << 3) /* bits in limb */ #define biH (ciL << 2) /* half limb size */ /* * Convert between bits/chars and number of limbs */ #define BITS_TO_LIMBS(i) (((i) + biL - 1) / biL) #define CHARS_TO_LIMBS(i) (((i) + ciL - 1) / ciL) /* * Initialize one or more mpi */ void mpi_init(mpi *X, ...) { va_list args; va_start(args, X); while (X != NULL) { X->s = 1; X->n = 0; X->p = NULL; X = va_arg(args, mpi*); } va_end(args); } /* * Unallocate one or more mpi */ void mpi_free(mpi *X, ...) { va_list args; va_start(args, X); while (X != NULL) { if (X->p != NULL) { memset(X->p, 0, X->n * ciL); free(X->p); } X->s = 1; X->n = 0; X->p = NULL; X = va_arg(args, mpi*); } va_end(args); } /* * Enlarge to the specified number of limbs */ int mpi_grow(mpi *X, int nblimbs) { t_int *p; if (X->n < nblimbs) { if ((p = (t_int *)malloc(nblimbs * ciL)) == NULL) return(1); memset(p, 0, nblimbs * ciL); if (X->p != NULL) { memcpy(p, X->p, X->n * ciL); memset(X->p, 0, X->n * ciL); free(X->p); } X->n = nblimbs; X->p = p; } return(0); } /* * Copy the contents of Y into X */ int mpi_copy(mpi *X, mpi *Y) { int ret, i; if (X == Y) return(0); for (i = Y->n - 1; i > 0; i--) if (Y->p[i] != 0) break; i++; X->s = Y->s; MPI_CHK(mpi_grow(X, i)); memset(X->p, 0, X->n * ciL); memcpy(X->p, Y->p, i * ciL); cleanup: return(ret); } /* * Swap the contents of X and Y */ void mpi_swap(mpi *X, mpi *Y) { mpi T; memcpy(&T, X, sizeof(mpi)); memcpy(X, Y, sizeof(mpi)); memcpy(Y, &T, sizeof(mpi)); } /* * Set value from integer */ int mpi_lset(mpi *X, int z) { int ret; MPI_CHK(mpi_grow(X, 1)); memset(X->p, 0, X->n * ciL); X->p[0] = (z < 0) ? -z : z; X->s = (z < 0) ? -1 : 1; cleanup: return(ret); } /* * Return the number of least significant bits */ int mpi_lsb(mpi *X) { int i, j, count = 0; for (i = 0; i < X->n; i++) for (j = 0; j < (int)biL; j++, count++) if (((X->p[i] >> j) & 1) != 0) return(count); return(0); } /* * Return the number of most significant bits */ int mpi_msb(mpi *X) { int i, j; for (i = X->n - 1; i > 0; i--) if (X->p[i] != 0) break; for (j = biL - 1; j >= 0; j--) if (((X->p[i] >> j) & 1) != 0) break; return((i * biL) + j + 1); } /* * Return the total size in bytes */ int mpi_size(mpi *X) { return((mpi_msb(X) + 7) >> 3); } /* * Convert an ASCII character to digit value */ static int mpi_get_digit(t_int *d, int radix, char c) { *d = 255; if (c >= 0x30 && c <= 0x39) *d = c - 0x30; if (c >= 0x41 && c <= 0x46) *d = c - 0x37; if (c >= 0x61 && c <= 0x66) *d = c - 0x57; if (*d >= (t_int)radix) return(POLARSSL_ERR_MPI_INVALID_CHARACTER); return(0); } /* * Import from an ASCII string */ int mpi_read_string(mpi *X, int radix, char *s) { int ret, i, j, n; t_int d; mpi T; if (radix < 2 || radix > 16) return(POLARSSL_ERR_MPI_BAD_INPUT_DATA); mpi_init(&T, NULL); if (radix == 16) { n = BITS_TO_LIMBS(strlen(s) << 2); MPI_CHK(mpi_grow(X, n)); MPI_CHK(mpi_lset(X, 0)); for (i = strlen(s) - 1, j = 0; i >= 0; i--, j++) { if (i == 0 && s[i] == '-') { X->s = -1; break; } MPI_CHK(mpi_get_digit(&d, radix, s[i])); X->p[j / (2 * ciL)] |= d << ((j % (2 * ciL)) << 2); } } else { MPI_CHK(mpi_lset(X, 0)); for (i = 0; i < (int)strlen(s); i++) { if (i == 0 && s[i] == '-') { X->s = -1; continue; } MPI_CHK(mpi_get_digit(&d, radix, s[i])); MPI_CHK(mpi_mul_int(&T, X, radix)); if (X->s == 1) { MPI_CHK(mpi_add_int(X, &T, d)); } else { MPI_CHK(mpi_sub_int(X, &T, d)); } } } cleanup: mpi_free(&T, NULL); return(ret); } /* * Helper to write the digits high-order first */ static int mpi_write_hlp(mpi *X, int radix, char **p) { int ret; t_int r; if (radix < 2 || radix > 16) return(POLARSSL_ERR_MPI_BAD_INPUT_DATA); MPI_CHK(mpi_mod_int(&r, X, radix)); MPI_CHK(mpi_div_int(X, NULL, X, radix)); if (mpi_cmp_int(X, 0) != 0) MPI_CHK(mpi_write_hlp(X, radix, p)); if (r < 10) *(*p)++ = (char)(r + 0x30); else *(*p)++ = (char)(r + 0x37); cleanup: return(ret); } /* * Export into an ASCII string */ int mpi_write_string(mpi *X, int radix, char *s, int *slen) { int ret = 0, n; char *p; mpi T; if (radix < 2 || radix > 16) return(POLARSSL_ERR_MPI_BAD_INPUT_DATA); n = mpi_msb(X); if (radix >= 4) n >>= 1; if (radix >= 16) n >>= 1; n += 3; if (*slen < n) { *slen = n; return(POLARSSL_ERR_MPI_BUFFER_TOO_SMALL); } p = s; mpi_init(&T, NULL); if (X->s == -1) *p++ = '-'; if (radix == 16) { int c, i, j, k; for (i = X->n - 1, k = 0; i >= 0; i--) { for (j = ciL - 1; j >= 0; j--) { c = (X->p[i] >> (j << 3)) & 0xFF; if (c == 0 && k == 0 && (i + j) != 0) continue; p += sprintf(p, "%02X", c); k = 1; } } } else { MPI_CHK(mpi_copy(&T, X)); MPI_CHK(mpi_write_hlp(&T, radix, &p)); } *p++ = '\0'; *slen = p - s; cleanup: mpi_free(&T, NULL); return(ret); } /* * Import X from unsigned binary data, big endian */ int mpi_read_binary(mpi *X, unsigned char *buf, int buflen) { int ret, i, j, n; for (n = 0; n < buflen; n++) if (buf[n] != 0) break; MPI_CHK(mpi_grow(X, CHARS_TO_LIMBS(buflen - n))); MPI_CHK(mpi_lset(X, 0)); for (i = buflen - 1, j = 0; i >= n; i--, j++) X->p[j / ciL] |= ((t_int)buf[i]) << ((j % ciL) << 3); cleanup: return(ret); } /* * Export X into unsigned binary data, big endian */ int mpi_write_binary(mpi *X, unsigned char *buf, int buflen) { int i, j, n; n = mpi_size(X); if (buflen < n) return(POLARSSL_ERR_MPI_BUFFER_TOO_SMALL); memset(buf, 0, buflen); for (i = buflen - 1, j = 0; n > 0; i--, j++, n--) buf[i] = (unsigned char)(X->p[j / ciL] >> ((j % ciL) << 3)); return(0); } /* * Left-shift: X <<= count */ int mpi_shift_l(mpi *X, int count) { int ret, i, v0, t1; t_int r0 = 0, r1; v0 = count / (biL); t1 = count & (biL - 1); i = mpi_msb(X) + count; if (X->n * (int)biL < i) MPI_CHK(mpi_grow(X, BITS_TO_LIMBS(i))); ret = 0; /* * shift by count / limb_size */ if (v0 > 0) { for (i = X->n - 1; i >= v0; i--) X->p[i] = X->p[i - v0]; for (; i >= 0; i--) X->p[i] = 0; } /* * shift by count % limb_size */ if (t1 > 0) { for (i = v0; i < X->n; i++) { r1 = X->p[i] >> (biL - t1); X->p[i] <<= t1; X->p[i] |= r0; r0 = r1; } } cleanup: return(ret); } /* * Right-shift: X >>= count */ int mpi_shift_r(mpi *X, int count) { int i, v0, v1; t_int r0 = 0, r1; v0 = count / biL; v1 = count & (biL - 1); /* * shift by count / limb_size */ if (v0 > 0) { for (i = 0; i < X->n - v0; i++) X->p[i] = X->p[i + v0]; for (; i < X->n; i++) X->p[i] = 0; } /* * shift by count % limb_size */ if (v1 > 0) { for (i = X->n - 1; i >= 0; i--) { r1 = X->p[i] << (biL - v1); X->p[i] >>= v1; X->p[i] |= r0; r0 = r1; } } return(0); } /* * Compare unsigned values */ int mpi_cmp_abs(mpi *X, mpi *Y) { int i, j; for (i = X->n - 1; i >= 0; i--) if (X->p[i] != 0) break; for (j = Y->n - 1; j >= 0; j--) if (Y->p[j] != 0) break; if (i < 0 && j < 0) return(0); if (i > j) return(1); if (j > i) return(-1); for (; i >= 0; i--) { if (X->p[i] > Y->p[i]) return(1); if (X->p[i] < Y->p[i]) return(-1); } return(0); } /* * Compare signed values */ int mpi_cmp_mpi(mpi *X, mpi *Y) { int i, j; for (i = X->n - 1; i >= 0; i--) if (X->p[i] != 0) break; for (j = Y->n - 1; j >= 0; j--) if (Y->p[j] != 0) break; if (i < 0 && j < 0) return(0); if (i > j) return(X->s); if (j > i) return(-X->s); if (X->s > 0 && Y->s < 0) return(1); if (Y->s > 0 && X->s < 0) return(-1); for (; i >= 0; i--) { if (X->p[i] > Y->p[i]) return(X->s); if (X->p[i] < Y->p[i]) return(-X->s); } return(0); } /* * Compare signed values */ int mpi_cmp_int(mpi *X, int z) { mpi Y; t_int p[1]; *p = (z < 0) ? -z : z; Y.s = (z < 0) ? -1 : 1; Y.n = 1; Y.p = p; return(mpi_cmp_mpi(X, &Y)); } /* * Unsigned addition: X = |A| + |B| (HAC 14.7) */ int mpi_add_abs(mpi *X, mpi *A, mpi *B) { int ret, i, j; t_int *o, *p, c; if (X == B) { mpi *T = A; A = X; B = T; } if (X != A) MPI_CHK(mpi_copy(X, A)); for (j = B->n - 1; j >= 0; j--) if (B->p[j] != 0) break; MPI_CHK(mpi_grow(X, j + 1)); o = B->p; p = X->p; c = 0; for (i = 0; i <= j; i++, o++, p++) { *p += c; c = (*p < c); *p += *o; c += (*p < *o); } while (c != 0) { if (i >= X->n) { MPI_CHK(mpi_grow(X, i + 1)); p = X->p + i; } *p += c; c = (*p < c); i++; } cleanup: return(ret); } /* * Helper for mpi substraction */ static void mpi_sub_hlp(int n, t_int *s, t_int *d) { int i; t_int c, z; for (i = c = 0; i < n; i++, s++, d++) { z = (*d < c); *d -= c; c = (*d < *s) + z; *d -= *s; } while (c != 0) { z = (*d < c); *d -= c; c = z; i++; d++; } } /* * Unsigned substraction: X = |A| - |B| (HAC 14.9) */ int mpi_sub_abs(mpi *X, mpi *A, mpi *B) { mpi TB; int ret, n; if (mpi_cmp_abs(A, B) < 0) return(POLARSSL_ERR_MPI_NEGATIVE_VALUE); mpi_init(&TB, NULL); if (X == B) { MPI_CHK(mpi_copy(&TB, B)); B = &TB; } if (X != A) MPI_CHK(mpi_copy(X, A)); ret = 0; for (n = B->n - 1; n >= 0; n--) if (B->p[n] != 0) break; mpi_sub_hlp(n + 1, B->p, X->p); cleanup: mpi_free(&TB, NULL); return(ret); } /* * Signed addition: X = A + B */ int mpi_add_mpi(mpi *X, mpi *A, mpi *B) { int ret, s = A->s; if (A->s * B->s < 0) { if (mpi_cmp_abs(A, B) >= 0) { MPI_CHK(mpi_sub_abs(X, A, B)); X->s = s; } else { MPI_CHK(mpi_sub_abs(X, B, A)); X->s = -s; } } else { MPI_CHK(mpi_add_abs(X, A, B)); X->s = s; } cleanup: return(ret); } /* * Signed substraction: X = A - B */ int mpi_sub_mpi(mpi *X, mpi *A, mpi *B) { int ret, s = A->s; if (A->s * B->s > 0) { if (mpi_cmp_abs(A, B) >= 0) { MPI_CHK(mpi_sub_abs(X, A, B)); X->s = s; } else { MPI_CHK(mpi_sub_abs(X, B, A)); X->s = -s; } } else { MPI_CHK(mpi_add_abs(X, A, B)); X->s = s; } cleanup: return(ret); } /* * Signed addition: X = A + b */ int mpi_add_int(mpi *X, mpi *A, int b) { mpi _B; t_int p[1]; p[0] = (b < 0) ? -b : b; _B.s = (b < 0) ? -1 : 1; _B.n = 1; _B.p = p; return(mpi_add_mpi(X, A, &_B)); } /* * Signed substraction: X = A - b */ int mpi_sub_int(mpi *X, mpi *A, int b) { mpi _B; t_int p[1]; p[0] = (b < 0) ? -b : b; _B.s = (b < 0) ? -1 : 1; _B.n = 1; _B.p = p; return(mpi_sub_mpi(X, A, &_B)); } /* * Helper for mpi multiplication */ static void mpi_mul_hlp(int i, t_int *s, t_int *d, t_int b) { t_int c = 0, t = 0; #if defined(MULADDC_HUIT) for (; i >= 8; i -= 8) { MULADDC_INIT MULADDC_HUIT MULADDC_STOP } for (; i > 0; i--) { MULADDC_INIT MULADDC_CORE MULADDC_STOP } #else for (; i >= 16; i -= 16) { MULADDC_INIT MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_STOP } for (; i >= 8; i -= 8) { MULADDC_INIT MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_STOP } for (; i > 0; i--) { MULADDC_INIT MULADDC_CORE MULADDC_STOP } #endif t++; do { *d += c; c = (*d < c); d++; } while (c != 0); } /* * Baseline multiplication: X = A * B (HAC 14.12) */ int mpi_mul_mpi(mpi *X, mpi *A, mpi *B) { int ret, i, j; mpi TA, TB; mpi_init(&TA, &TB, NULL); if (X == A) { MPI_CHK(mpi_copy(&TA, A)); A = &TA; } if (X == B) { MPI_CHK(mpi_copy(&TB, B)); B = &TB; } for (i = A->n - 1; i >= 0; i--) if (A->p[i] != 0) break; for (j = B->n - 1; j >= 0; j--) if (B->p[j] != 0) break; MPI_CHK(mpi_grow(X, i + j + 2)); MPI_CHK(mpi_lset(X, 0)); for (i++; j >= 0; j--) mpi_mul_hlp(i, A->p, X->p + j, B->p[j]); X->s = A->s * B->s; cleanup: mpi_free(&TB, &TA, NULL); return(ret); } /* * Baseline multiplication: X = A * b */ int mpi_mul_int(mpi *X, mpi *A, t_int b) { mpi _B; t_int p[1]; _B.s = 1; _B.n = 1; _B.p = p; p[0] = b; return(mpi_mul_mpi(X, A, &_B)); } /* * Division by mpi: A = Q * B + R (HAC 14.20) */ int mpi_div_mpi(mpi *Q, mpi *R, mpi *A, mpi *B) { int ret, i, n, t, k; mpi X, Y, Z, T1, T2; if (mpi_cmp_int(B, 0) == 0) return(POLARSSL_ERR_MPI_DIVISION_BY_ZERO); mpi_init(&X, &Y, &Z, &T1, &T2, NULL); if (mpi_cmp_abs(A, B) < 0) { if (Q != NULL) MPI_CHK(mpi_lset(Q, 0)); if (R != NULL) MPI_CHK(mpi_copy(R, A)); return(0); } MPI_CHK(mpi_copy(&X, A)); MPI_CHK(mpi_copy(&Y, B)); X.s = Y.s = 1; MPI_CHK(mpi_grow(&Z, A->n + 2)); MPI_CHK(mpi_lset(&Z, 0)); MPI_CHK(mpi_grow(&T1, 2)); MPI_CHK(mpi_grow(&T2, 3)); k = mpi_msb(&Y) % biL; if (k < (int)biL - 1) { k = biL - 1 - k; MPI_CHK(mpi_shift_l(&X, k)); MPI_CHK(mpi_shift_l(&Y, k)); } else k = 0; n = X.n - 1; t = Y.n - 1; mpi_shift_l(&Y, biL * (n - t)); while (mpi_cmp_mpi(&X, &Y) >= 0) { Z.p[n - t]++; mpi_sub_mpi(&X, &X, &Y); } mpi_shift_r(&Y, biL * (n - t)); for (i = n; i > t; i--) { if (X.p[i] >= Y.p[t]) Z.p[i - t - 1] = ~0; else { #if defined(POLARSSL_HAVE_LONGLONG) t_dbl r; r = (t_dbl)X.p[i] << biL; r |= (t_dbl)X.p[i - 1]; r /= Y.p[t]; if (r > ((t_dbl)1 << biL) - 1) r = ((t_dbl)1 << biL) - 1; Z.p[i - t - 1] = (t_int)r; #else /* * __udiv_qrnnd_c, from gmp/longlong.h */ t_int q0, q1, r0, r1; t_int d0, d1, d, m; d = Y.p[t]; d0 = (d << biH) >> biH; d1 = (d >> biH); q1 = X.p[i] / d1; r1 = X.p[i] - d1 * q1; r1 <<= biH; r1 |= (X.p[i - 1] >> biH); m = q1 * d0; if (r1 < m) { q1--, r1 += d; while (r1 >= d && r1 < m) q1--, r1 += d; } r1 -= m; q0 = r1 / d1; r0 = r1 - d1 * q0; r0 <<= biH; r0 |= (X.p[i - 1] << biH) >> biH; m = q0 * d0; if (r0 < m) { q0--, r0 += d; while (r0 >= d && r0 < m) q0--, r0 += d; } r0 -= m; Z.p[i - t - 1] = (q1 << biH) | q0; #endif } Z.p[i - t - 1]++; do { Z.p[i - t - 1]--; MPI_CHK(mpi_lset(&T1, 0)); T1.p[0] = (t < 1) ? 0 : Y.p[t - 1]; T1.p[1] = Y.p[t]; MPI_CHK(mpi_mul_int(&T1, &T1, Z.p[i - t - 1])); MPI_CHK(mpi_lset(&T2, 0)); T2.p[0] = (i < 2) ? 0 : X.p[i - 2]; T2.p[1] = (i < 1) ? 0 : X.p[i - 1]; T2.p[2] = X.p[i]; } while (mpi_cmp_mpi(&T1, &T2) > 0); MPI_CHK(mpi_mul_int(&T1, &Y, Z.p[i - t - 1])); MPI_CHK(mpi_shift_l(&T1, biL * (i - t - 1))); MPI_CHK(mpi_sub_mpi(&X, &X, &T1)); if (mpi_cmp_int(&X, 0) < 0) { MPI_CHK(mpi_copy(&T1, &Y)); MPI_CHK(mpi_shift_l(&T1, biL * (i - t - 1))); MPI_CHK(mpi_add_mpi(&X, &X, &T1)); Z.p[i - t - 1]--; } } if (Q != NULL) { mpi_copy(Q, &Z); Q->s = A->s * B->s; } if (R != NULL) { mpi_shift_r(&X, k); mpi_copy(R, &X); R->s = A->s; if (mpi_cmp_int(R, 0) == 0) R->s = 1; } cleanup: mpi_free(&X, &Y, &Z, &T1, &T2, NULL); return(ret); } /* * Division by int: A = Q * b + R * * Returns 0 if successful * 1 if memory allocation failed * POLARSSL_ERR_MPI_DIVISION_BY_ZERO if b == 0 */ int mpi_div_int(mpi *Q, mpi *R, mpi *A, int b) { mpi _B; t_int p[1]; p[0] = (b < 0) ? -b : b; _B.s = (b < 0) ? -1 : 1; _B.n = 1; _B.p = p; return(mpi_div_mpi(Q, R, A, &_B)); } /* * Modulo: R = A mod B */ int mpi_mod_mpi(mpi *R, mpi *A, mpi *B) { int ret; MPI_CHK(mpi_div_mpi(NULL, R, A, B)); while (mpi_cmp_int(R, 0) < 0) MPI_CHK(mpi_add_mpi(R, R, B)); while (mpi_cmp_mpi(R, B) >= 0) MPI_CHK(mpi_sub_mpi(R, R, B)); cleanup: return(ret); } /* * Modulo: r = A mod b */ int mpi_mod_int(t_int *r, mpi *A, int b) { int i; t_int x, y, z; if (b == 0) return(POLARSSL_ERR_MPI_DIVISION_BY_ZERO); if (b < 0) b = -b; /* * handle trivial cases */ if (b == 1) { *r = 0; return(0); } if (b == 2) { *r = A->p[0] & 1; return(0); } /* * general case */ for (i = A->n - 1, y = 0; i >= 0; i--) { x = A->p[i]; y = (y << biH) | (x >> biH); z = y / b; y -= z * b; x <<= biH; y = (y << biH) | (x >> biH); z = y / b; y -= z * b; } *r = y; return(0); } /* * Fast Montgomery initialization (thanks to Tom St Denis) */ static void mpi_montg_init(t_int *mm, mpi *N) { t_int x, m0 = N->p[0]; x = m0; x += ((m0 + 2) & 4) << 1; x *= (2 - (m0 * x)); if (biL >= 16) x *= (2 - (m0 * x)); if (biL >= 32) x *= (2 - (m0 * x)); if (biL >= 64) x *= (2 - (m0 * x)); *mm = ~x + 1; } /* * Montgomery multiplication: A = A * B * R^-1 mod N (HAC 14.36) */ static void mpi_montmul(mpi *A, mpi *B, mpi *N, t_int mm, mpi *T) { int i, n, m; t_int u0, u1, *d; memset(T->p, 0, T->n * ciL); d = T->p; n = N->n; m = (B->n < n) ? B->n : n; for (i = 0; i < n; i++) { /* * T = (T + u0*B + u1*N) / 2^biL */ u0 = A->p[i]; u1 = (d[0] + u0 * B->p[0]) * mm; mpi_mul_hlp(m, B->p, d, u0); mpi_mul_hlp(n, N->p, d, u1); *d++ = u0; d[n + 1] = 0; } memcpy(A->p, d, (n + 1) * ciL); if (mpi_cmp_abs(A, N) >= 0) mpi_sub_hlp(n, N->p, A->p); else /* prevent timing attacks */ mpi_sub_hlp(n, A->p, T->p); } /* * Montgomery reduction: A = A * R^-1 mod N */ static void mpi_montred(mpi *A, mpi *N, t_int mm, mpi *T) { t_int z = 1; mpi U; U.n = U.s = z; U.p = &z; mpi_montmul(A, &U, N, mm, T); } /* * Sliding-window exponentiation: X = A^E mod N (HAC 14.85) */ int mpi_exp_mod(mpi *X, mpi *A, mpi *E, mpi *N, mpi *_RR) { int ret, i, j, wsize, wbits; int bufsize, nblimbs, nbits; t_int ei, mm, state; mpi RR, T, W[64]; if (mpi_cmp_int(N, 0) < 0 || (N->p[0] & 1) == 0) return(POLARSSL_ERR_MPI_BAD_INPUT_DATA); /* * Init temps and window size */ mpi_montg_init(&mm, N); mpi_init(&RR, &T, NULL); memset(W, 0, sizeof(W)); i = mpi_msb(E); wsize = (i > 671) ? 6 : (i > 239) ? 5 : (i > 79) ? 4 : (i > 23) ? 3 : 1; j = N->n + 1; MPI_CHK(mpi_grow(X, j)); MPI_CHK(mpi_grow(&W[1], j)); MPI_CHK(mpi_grow(&T, j * 2)); /* * If 1st call, pre-compute R^2 mod N */ if (_RR == NULL || _RR->p == NULL) { MPI_CHK(mpi_lset(&RR, 1)); MPI_CHK(mpi_shift_l(&RR, N->n * 2 * biL)); MPI_CHK(mpi_mod_mpi(&RR, &RR, N)); if (_RR != NULL) memcpy(_RR, &RR, sizeof(mpi)); } else memcpy(&RR, _RR, sizeof(mpi)); /* * W[1] = A * R^2 * R^-1 mod N = A * R mod N */ if (mpi_cmp_mpi(A, N) >= 0) mpi_mod_mpi(&W[1], A, N); else mpi_copy(&W[1], A); mpi_montmul(&W[1], &RR, N, mm, &T); /* * X = R^2 * R^-1 mod N = R mod N */ MPI_CHK(mpi_copy(X, &RR)); mpi_montred(X, N, mm, &T); if (wsize > 1) { /* * W[1 << (wsize - 1)] = W[1] ^ (wsize - 1) */ j = 1 << (wsize - 1); MPI_CHK(mpi_grow(&W[j], N->n + 1)); MPI_CHK(mpi_copy(&W[j], &W[1])); for (i = 0; i < wsize - 1; i++) mpi_montmul(&W[j], &W[j], N, mm, &T); /* * W[i] = W[i - 1] * W[1] */ for (i = j + 1; i < (1 << wsize); i++) { MPI_CHK(mpi_grow(&W[i], N->n + 1)); MPI_CHK(mpi_copy(&W[i], &W[i - 1])); mpi_montmul(&W[i], &W[1], N, mm, &T); } } nblimbs = E->n; bufsize = 0; nbits = 0; wbits = 0; state = 0; while (1) { if (bufsize == 0) { if (nblimbs-- == 0) break; bufsize = sizeof(t_int) << 3; } bufsize--; ei = (E->p[nblimbs] >> bufsize) & 1; /* * skip leading 0s */ if (ei == 0 && state == 0) continue; if (ei == 0 && state == 1) { /* * out of window, square X */ mpi_montmul(X, X, N, mm, &T); continue; } /* * add ei to current window */ state = 2; nbits++; wbits |= (ei << (wsize - nbits)); if (nbits == wsize) { /* * X = X^wsize R^-1 mod N */ for (i = 0; i < wsize; i++) mpi_montmul(X, X, N, mm, &T); /* * X = X * W[wbits] R^-1 mod N */ mpi_montmul(X, &W[wbits], N, mm, &T); state--; nbits = 0; wbits = 0; } } /* * process the remaining bits */ for (i = 0; i < nbits; i++) { mpi_montmul(X, X, N, mm, &T); wbits <<= 1; if ((wbits & (1 << wsize)) != 0) mpi_montmul(X, &W[1], N, mm, &T); } /* * X = A^E * R * R^-1 mod N = A^E mod N */ mpi_montred(X, N, mm, &T); cleanup: for (i = (1 << (wsize - 1)); i < (1 << wsize); i++) mpi_free(&W[i], NULL); if (_RR != NULL) mpi_free(&W[1], &T, NULL); else mpi_free(&W[1], &T, &RR, NULL); return(ret); } /* * Greatest common divisor: G = gcd(A, B) (HAC 14.54) */ int mpi_gcd(mpi *G, mpi *A, mpi *B) { int ret, lz, lzt; mpi TGcd, TA, TB; mpi_init(&TGcd, &TA, &TB, NULL); MPI_CHK(mpi_copy(&TA, A)); MPI_CHK(mpi_copy(&TB, B)); lz = mpi_lsb(&TA); lzt = mpi_lsb(&TB); if (lzt < lz) lz = lzt; MPI_CHK(mpi_shift_r(&TA, lz)); MPI_CHK(mpi_shift_r(&TB, lz)); TA.s = TB.s = 1; while (mpi_cmp_int(&TA, 0) != 0) { MPI_CHK(mpi_shift_r(&TA, mpi_lsb(&TA))); MPI_CHK(mpi_shift_r(&TB, mpi_lsb(&TB))); if (mpi_cmp_mpi(&TA, &TB) >= 0) { MPI_CHK(mpi_sub_abs(&TA, &TA, &TB)); MPI_CHK(mpi_shift_r(&TA, 1)); } else { MPI_CHK(mpi_sub_abs(&TB, &TB, &TA)); MPI_CHK(mpi_shift_r(&TB, 1)); } } MPI_CHK(mpi_shift_l(&TB, lz)); MPI_CHK(mpi_copy(G, &TB)); cleanup: mpi_free(&TB, &TA, &TGcd, NULL); return(ret); } #if defined(POLARSSL_GENPRIME) /* * Modular inverse: X = A^-1 mod N (HAC 14.61 / 14.64) */ int mpi_inv_mod(mpi *X, mpi *A, mpi *N) { int ret; mpi G, TA, TU, U1, U2, TB, TV, V1, V2; if (mpi_cmp_int(N, 0) <= 0) return(POLARSSL_ERR_MPI_BAD_INPUT_DATA); mpi_init(&TA, &TU, &U1, &U2, &G, &TB, &TV, &V1, &V2, NULL); MPI_CHK(mpi_gcd(&G, A, N)); if (mpi_cmp_int(&G, 1) != 0) { ret = POLARSSL_ERR_MPI_NOT_ACCEPTABLE; goto cleanup; } MPI_CHK(mpi_mod_mpi(&TA, A, N)); MPI_CHK(mpi_copy(&TU, &TA)); MPI_CHK(mpi_copy(&TB, N)); MPI_CHK(mpi_copy(&TV, N)); MPI_CHK(mpi_lset(&U1, 1)); MPI_CHK(mpi_lset(&U2, 0)); MPI_CHK(mpi_lset(&V1, 0)); MPI_CHK(mpi_lset(&V2, 1)); do { while ((TU.p[0] & 1) == 0) { MPI_CHK(mpi_shift_r(&TU, 1)); if ((U1.p[0] & 1) != 0 || (U2.p[0] & 1) != 0) { MPI_CHK(mpi_add_mpi(&U1, &U1, &TB)); MPI_CHK(mpi_sub_mpi(&U2, &U2, &TA)); } MPI_CHK(mpi_shift_r(&U1, 1)); MPI_CHK(mpi_shift_r(&U2, 1)); } while ((TV.p[0] & 1) == 0) { MPI_CHK(mpi_shift_r(&TV, 1)); if ((V1.p[0] & 1) != 0 || (V2.p[0] & 1) != 0) { MPI_CHK(mpi_add_mpi(&V1, &V1, &TB)); MPI_CHK(mpi_sub_mpi(&V2, &V2, &TA)); } MPI_CHK(mpi_shift_r(&V1, 1)); MPI_CHK(mpi_shift_r(&V2, 1)); } if (mpi_cmp_mpi(&TU, &TV) >= 0) { MPI_CHK(mpi_sub_mpi(&TU, &TU, &TV)); MPI_CHK(mpi_sub_mpi(&U1, &U1, &V1)); MPI_CHK(mpi_sub_mpi(&U2, &U2, &V2)); } else { MPI_CHK(mpi_sub_mpi(&TV, &TV, &TU)); MPI_CHK(mpi_sub_mpi(&V1, &V1, &U1)); MPI_CHK(mpi_sub_mpi(&V2, &V2, &U2)); } } while (mpi_cmp_int(&TU, 0) != 0); while (mpi_cmp_int(&V1, 0) < 0) MPI_CHK(mpi_add_mpi(&V1, &V1, N)); while (mpi_cmp_mpi(&V1, N) >= 0) MPI_CHK(mpi_sub_mpi(&V1, &V1, N)); MPI_CHK(mpi_copy(X, &V1)); cleanup: mpi_free(&V2, &V1, &TV, &TB, &G, &U2, &U1, &TU, &TA, NULL); return(ret); } static const int small_prime[] = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, -103 }; /* * Miller-Rabin primality test (HAC 4.24) */ int mpi_is_prime(mpi *X, int(*f_rng)(void *), void *p_rng) { int ret, i, j, n, s, xs; mpi W, R, T, A, RR; unsigned char *p; if (mpi_cmp_int(X, 0) == 0) return(0); mpi_init(&W, &R, &T, &A, &RR, NULL); xs = X->s; X->s = 1; /* * test trivial factors first */ if ((X->p[0] & 1) == 0) return(POLARSSL_ERR_MPI_NOT_ACCEPTABLE); for (i = 0; small_prime[i] > 0; i++) { t_int r; if (mpi_cmp_int(X, small_prime[i]) <= 0) return(0); MPI_CHK(mpi_mod_int(&r, X, small_prime[i])); if (r == 0) return(POLARSSL_ERR_MPI_NOT_ACCEPTABLE); } /* * W = |X| - 1 * R = W >> lsb( W ) */ s = mpi_lsb(&W); MPI_CHK(mpi_sub_int(&W, X, 1)); MPI_CHK(mpi_copy(&R, &W)); MPI_CHK(mpi_shift_r(&R, s)); i = mpi_msb(X); /* * HAC, table 4.4 */ n = ((i >= 1300) ? 2 : (i >= 850) ? 3 : (i >= 650) ? 4 : (i >= 350) ? 8 : (i >= 250) ? 12 : (i >= 150) ? 18 : 27); for (i = 0; i < n; i++) { /* * pick a random A, 1 < A < |X| - 1 */ MPI_CHK(mpi_grow(&A, X->n)); p = (unsigned char *)A.p; for (j = 0; j < A.n * ciL; j++) *p++ = (unsigned char)f_rng(p_rng); j = mpi_msb(&A) - mpi_msb(&W); MPI_CHK(mpi_shift_r(&A, j + 1)); A.p[0] |= 3; /* * A = A^R mod |X| */ MPI_CHK(mpi_exp_mod(&A, &A, &R, X, &RR)); if (mpi_cmp_mpi(&A, &W) == 0 || mpi_cmp_int(&A, 1) == 0) continue; j = 1; while (j < s && mpi_cmp_mpi(&A, &W) != 0) { /* * A = A * A mod |X| */ MPI_CHK(mpi_mul_mpi(&T, &A, &A)); MPI_CHK(mpi_mod_mpi(&A, &T, X)); if (mpi_cmp_int(&A, 1) == 0) break; j++; } /* * not prime if A != |X| - 1 or A == 1 */ if (mpi_cmp_mpi(&A, &W) != 0 || mpi_cmp_int(&A, 1) == 0) { ret = POLARSSL_ERR_MPI_NOT_ACCEPTABLE; break; } } cleanup: X->s = xs; mpi_free(&RR, &A, &T, &R, &W, NULL); return(ret); } /* * Prime number generation */ int mpi_gen_prime(mpi *X, int nbits, int dh_flag, int(*f_rng)(void *), void *p_rng) { int ret, k, n; unsigned char *p; mpi Y; if (nbits < 3) return(POLARSSL_ERR_MPI_BAD_INPUT_DATA); mpi_init(&Y, NULL); n = BITS_TO_LIMBS(nbits); MPI_CHK(mpi_grow(X, n)); MPI_CHK(mpi_lset(X, 0)); p = (unsigned char *)X->p; for (k = 0; k < X->n * ciL; k++) *p++ = (unsigned char)f_rng(p_rng); k = mpi_msb(X); if (k < nbits) MPI_CHK(mpi_shift_l(X, nbits - k)); if (k > nbits) MPI_CHK(mpi_shift_r(X, k - nbits)); X->p[0] |= 3; if (dh_flag == 0) { while ((ret = mpi_is_prime(X, f_rng, p_rng)) != 0) { if (ret != POLARSSL_ERR_MPI_NOT_ACCEPTABLE) goto cleanup; MPI_CHK(mpi_add_int(X, X, 2)); } } else { MPI_CHK(mpi_sub_int(&Y, X, 1)); MPI_CHK(mpi_shift_r(&Y, 1)); while (1) { if ((ret = mpi_is_prime(X, f_rng, p_rng)) == 0) { if ((ret = mpi_is_prime(&Y, f_rng, p_rng)) == 0) break; if (ret != POLARSSL_ERR_MPI_NOT_ACCEPTABLE) goto cleanup; } if (ret != POLARSSL_ERR_MPI_NOT_ACCEPTABLE) goto cleanup; MPI_CHK(mpi_add_int(&Y, X, 1)); MPI_CHK(mpi_add_int(X, X, 2)); MPI_CHK(mpi_shift_r(&Y, 1)); } } cleanup: mpi_free(&Y, NULL); return(ret); } #endif #if defined(POLARSSL_SELF_TEST) #define GCD_PAIR_COUNT 3 static const int gcd_pairs[GCD_PAIR_COUNT][3] = { { 693, 609, 21 }, { 1764, 868, 28 }, { 768454923, 542167814, 1 } }; /* * Checkup routine */ int mpi_self_test(int verbose) { int ret, i; mpi A, E, N, X, Y, U, V; mpi_init(&A, &E, &N, &X, &Y, &U, &V, NULL); MPI_CHK(mpi_read_string(&A, 16, "EFE021C2645FD1DC586E69184AF4A31E" \ "D5F53E93B5F123FA41680867BA110131" \ "944FE7952E2517337780CB0DB80E61AA" \ "E7C8DDC6C5C6AADEB34EB38A2F40D5E6")); MPI_CHK(mpi_read_string(&E, 16, "B2E7EFD37075B9F03FF989C7C5051C20" \ "34D2A323810251127E7BF8625A4F49A5" \ "F3E27F4DA8BD59C47D6DAABA4C8127BD" \ "5B5C25763222FEFCCFC38B832366C29E")); MPI_CHK(mpi_read_string(&N, 16, "0066A198186C18C10B2F5ED9B522752A" \ "9830B69916E535C8F047518A889A43A5" \ "94B6BED27A168D31D4A52F88925AA8F5")); MPI_CHK(mpi_mul_mpi(&X, &A, &N)); MPI_CHK(mpi_read_string(&U, 16, "602AB7ECA597A3D6B56FF9829A5E8B85" \ "9E857EA95A03512E2BAE7391688D264A" \ "A5663B0341DB9CCFD2C4C5F421FEC814" \ "8001B72E848A38CAE1C65F78E56ABDEF" \ "E12D3C039B8A02D6BE593F0BBBDA56F1" \ "ECF677152EF804370C1A305CAF3B5BF1" \ "30879B56C61DE584A0F53A2447A51E")); if (verbose != 0) printf(" MPI test #1 (mul_mpi): "); if (mpi_cmp_mpi(&X, &U) != 0) { if (verbose != 0) printf("failed\n"); return(1); } if (verbose != 0) printf("passed\n"); MPI_CHK(mpi_div_mpi(&X, &Y, &A, &N)); MPI_CHK(mpi_read_string(&U, 16, "256567336059E52CAE22925474705F39A94")); MPI_CHK(mpi_read_string(&V, 16, "6613F26162223DF488E9CD48CC132C7A" \ "0AC93C701B001B092E4E5B9F73BCD27B" \ "9EE50D0657C77F374E903CDFA4C642")); if (verbose != 0) printf(" MPI test #2 (div_mpi): "); if (mpi_cmp_mpi(&X, &U) != 0 || mpi_cmp_mpi(&Y, &V) != 0) { if (verbose != 0) printf("failed\n"); return(1); } if (verbose != 0) printf("passed\n"); MPI_CHK(mpi_exp_mod(&X, &A, &E, &N, NULL)); MPI_CHK(mpi_read_string(&U, 16, "36E139AEA55215609D2816998ED020BB" \ "BD96C37890F65171D948E9BC7CBAA4D9" \ "325D24D6A3C12710F10A09FA08AB87")); if (verbose != 0) printf(" MPI test #3 (exp_mod): "); if (mpi_cmp_mpi(&X, &U) != 0) { if (verbose != 0) printf("failed\n"); return(1); } if (verbose != 0) printf("passed\n"); MPI_CHK(mpi_inv_mod(&X, &A, &N)); MPI_CHK(mpi_read_string(&U, 16, "003A0AAEDD7E784FC07D8F9EC6E3BFD5" \ "C3DBA76456363A10869622EAC2DD84EC" \ "C5B8A74DAC4D09E03B5E0BE779F2DF61")); if (verbose != 0) printf(" MPI test #4 (inv_mod): "); if (mpi_cmp_mpi(&X, &U) != 0) { if (verbose != 0) printf("failed\n"); return(1); } if (verbose != 0) printf("passed\n"); if (verbose != 0) printf(" MPI test #5 (simple gcd): "); for (i = 0; i < GCD_PAIR_COUNT; i++) { MPI_CHK(mpi_lset(&X, gcd_pairs[i][0])); MPI_CHK(mpi_lset(&Y, gcd_pairs[i][1])); MPI_CHK(mpi_gcd(&A, &X, &Y)); if (mpi_cmp_int(&A, gcd_pairs[i][2]) != 0) { if (verbose != 0) printf("failed at %d\n", i); return(1); } } if (verbose != 0) printf("passed\n"); cleanup: if (ret != 0 && verbose != 0) printf("Unexpected error, return code = %08X\n", ret); mpi_free(&V, &U, &Y, &X, &N, &E, &A, NULL); if (verbose != 0) printf("\n"); return(ret); } #endif