2 * Copyright (c) 2007, Cameron Rich
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions are met:
9 * * Redistributions of source code must retain the above copyright notice,
10 * this list of conditions and the following disclaimer.
11 * * Redistributions in binary form must reproduce the above copyright notice,
12 * this list of conditions and the following disclaimer in the documentation
13 * and/or other materials provided with the distribution.
14 * * Neither the name of the axTLS project nor the names of its contributors
15 * may be used to endorse or promote products derived from this software
16 * without specific prior written permission.
18 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
19 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
20 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
21 * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR
22 * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
23 * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
24 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
25 * PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
26 * LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
27 * NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
28 * SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
32 * @defgroup bigint_api Big Integer API
33 * @brief The bigint implementation as used by the axTLS project.
35 * The bigint library is for RSA encryption/decryption as well as signing.
36 * This code tries to minimise use of malloc/free by maintaining a small
37 * cache. A bigint context may maintain state by being made "permanent".
38 * It be be later released with a bi_depermanent() and bi_free() call.
40 * It supports the following reduction techniques:
45 * It also implements the following:
46 * - Karatsuba multiplication
48 * - Sliding window exponentiation
49 * - Chinese Remainder Theorem (implemented in rsa.c).
51 * All the algorithms used are pretty standard, and designed for different
52 * data bus sizes. Negative numbers are not dealt with at all, so a subtraction
53 * may need to be tested for negativity.
55 * This library steals some ideas from Jef Poskanzer
56 * <http://cs.marlboro.edu/term/cs-fall02/algorithms/crypto/RSA/bigint>
57 * and GMP <http://www.swox.com/gmp>. It gets most of its implementation
58 * detail from "The Handbook of Applied Cryptography"
59 * <http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf>
71 #define V1 v->comps[v->size-1] /**< v1 for division */
72 #define V2 v->comps[v->size-2] /**< v2 for division */
73 #define U(j) tmp_u->comps[tmp_u->size-j-1] /**< uj for division */
74 #define Q(j) quotient->comps[quotient->size-j-1] /**< qj for division */
76 static bigint *bi_int_multiply(BI_CTX *ctx, bigint *bi, comp i);
77 static bigint *bi_int_divide(BI_CTX *ctx, bigint *biR, comp denom);
78 static bigint *alloc(BI_CTX *ctx, int size);
79 static bigint *trim(bigint *bi);
80 static void more_comps(bigint *bi, int n);
81 #if defined(CONFIG_BIGINT_KARATSUBA) || defined(CONFIG_BIGINT_BARRETT) || \
82 defined(CONFIG_BIGINT_MONTGOMERY)
83 static bigint *comp_right_shift(bigint *biR, int num_shifts);
84 static bigint *comp_left_shift(bigint *biR, int num_shifts);
87 #ifdef CONFIG_BIGINT_CHECK_ON
88 static void check(const bigint *bi);
90 #define check(A) /**< disappears in normal production mode */
95 * @brief Start a new bigint context.
96 * @return A bigint context.
98 BI_CTX *bi_initialize(void)
100 /* calloc() sets everything to zero */
101 BI_CTX *ctx = (BI_CTX *)calloc(1, sizeof(BI_CTX));
104 ctx->bi_radix = alloc(ctx, 2);
105 ctx->bi_radix->comps[0] = 0;
106 ctx->bi_radix->comps[1] = 1;
107 bi_permanent(ctx->bi_radix);
112 * @brief Close the bigint context and free any resources.
114 * Free up any used memory - a check is done if all objects were not
116 * @param ctx [in] The bigint session context.
118 void bi_terminate(BI_CTX *ctx)
120 bi_depermanent(ctx->bi_radix);
121 bi_free(ctx, ctx->bi_radix);
123 if (ctx->active_count != 0)
125 #ifdef CONFIG_SSL_FULL_MODE
126 printf("bi_terminate: there were %d un-freed bigints\n",
137 *@brief Clear the memory cache.
139 void bi_clear_cache(BI_CTX *ctx)
143 if (ctx->free_list == NULL)
146 for (p = ctx->free_list; p != NULL; p = pn)
154 ctx->free_list = NULL;
158 * @brief Increment the number of references to this object.
159 * It does not do a full copy.
160 * @param bi [in] The bigint to copy.
161 * @return A reference to the same bigint.
163 bigint *bi_copy(bigint *bi)
166 if (bi->refs != PERMANENT)
172 * @brief Simply make a bigint object "unfreeable" if bi_free() is called on it.
174 * For this object to be freed, bi_depermanent() must be called.
175 * @param bi [in] The bigint to be made permanent.
177 void bi_permanent(bigint *bi)
182 #ifdef CONFIG_SSL_FULL_MODE
183 printf("bi_permanent: refs was not 1\n");
188 bi->refs = PERMANENT;
192 * @brief Take a permanent object and make it eligible for freedom.
193 * @param bi [in] The bigint to be made back to temporary.
195 void bi_depermanent(bigint *bi)
198 if (bi->refs != PERMANENT)
200 #ifdef CONFIG_SSL_FULL_MODE
201 printf("bi_depermanent: bigint was not permanent\n");
210 * @brief Free a bigint object so it can be used again.
212 * The memory itself it not actually freed, just tagged as being available
213 * @param ctx [in] The bigint session context.
214 * @param bi [in] The bigint to be freed.
216 void bi_free(BI_CTX *ctx, bigint *bi)
219 if (bi->refs == PERMANENT)
229 bi->next = ctx->free_list;
233 if (--ctx->active_count < 0)
235 #ifdef CONFIG_SSL_FULL_MODE
236 printf("bi_free: active_count went negative "
237 "- double-freed bigint?\n");
244 * @brief Convert an (unsigned) integer into a bigint.
245 * @param ctx [in] The bigint session context.
246 * @param i [in] The (unsigned) integer to be converted.
249 bigint *int_to_bi(BI_CTX *ctx, comp i)
251 bigint *biR = alloc(ctx, 1);
257 * @brief Do a full copy of the bigint object.
258 * @param ctx [in] The bigint session context.
259 * @param bi [in] The bigint object to be copied.
261 bigint *bi_clone(BI_CTX *ctx, const bigint *bi)
263 bigint *biR = alloc(ctx, bi->size);
265 memcpy(biR->comps, bi->comps, bi->size*COMP_BYTE_SIZE);
270 * @brief Perform an addition operation between two bigints.
271 * @param ctx [in] The bigint session context.
272 * @param bia [in] A bigint.
273 * @param bib [in] Another bigint.
274 * @return The result of the addition.
276 bigint *bi_add(BI_CTX *ctx, bigint *bia, bigint *bib)
285 n = max(bia->size, bib->size);
286 more_comps(bia, n+1);
297 carry = cy1 | (rl < sl);
301 *pa = carry; /* do overflow */
307 * @brief Perform a subtraction operation between two bigints.
308 * @param ctx [in] The bigint session context.
309 * @param bia [in] A bigint.
310 * @param bib [in] Another bigint.
311 * @param is_negative [out] If defined, indicates that the result was negative.
312 * is_negative may be null.
313 * @return The result of the subtraction. The result is always positive.
315 bigint *bi_subtract(BI_CTX *ctx,
316 bigint *bia, bigint *bib, int *is_negative)
319 comp *pa, *pb, carry = 0;
333 carry = cy1 | (rl > sl);
337 if (is_negative) /* indicate a negative result */
339 *is_negative = carry;
342 bi_free(ctx, trim(bib)); /* put bib back to the way it was */
347 * Perform a multiply between a bigint an an (unsigned) integer
349 static bigint *bi_int_multiply(BI_CTX *ctx, bigint *bia, comp b)
351 int j = 0, n = bia->size;
352 bigint *biR = alloc(ctx, n + 1);
354 comp *r = biR->comps;
355 comp *a = bia->comps;
359 /* clear things to start with */
360 memset(r, 0, ((n+1)*COMP_BYTE_SIZE));
364 long_comp tmp = *r + (long_comp)a[j]*b + carry;
365 *r++ = (comp)tmp; /* downsize */
366 carry = (comp)(tmp >> COMP_BIT_SIZE);
375 * @brief Does both division and modulo calculations.
377 * Used extensively when doing classical reduction.
378 * @param ctx [in] The bigint session context.
379 * @param u [in] A bigint which is the numerator.
380 * @param v [in] Either the denominator or the modulus depending on the mode.
381 * @param is_mod [n] Determines if this is a normal division (0) or a reduction
383 * @return The result of the division/reduction.
385 bigint *bi_divide(BI_CTX *ctx, bigint *u, bigint *v, int is_mod)
387 int n = v->size, m = u->size-n;
388 int j = 0, orig_u_size = u->size;
389 uint8_t mod_offset = ctx->mod_offset;
391 bigint *quotient, *tmp_u;
397 /* if doing reduction and we are < mod, then return mod */
398 if (is_mod && bi_compare(v, u) > 0)
404 quotient = alloc(ctx, m+1);
405 tmp_u = alloc(ctx, n+1);
406 v = trim(v); /* make sure we have no leading 0's */
407 d = (comp)((long_comp)COMP_RADIX/(V1+1));
409 /* clear things to start with */
410 memset(quotient->comps, 0, ((quotient->size)*COMP_BYTE_SIZE));
415 u = bi_int_multiply(ctx, u, d);
419 v = ctx->bi_normalised_mod[mod_offset];
423 v = bi_int_multiply(ctx, v, d);
427 if (orig_u_size == u->size) /* new digit position u0 */
429 more_comps(u, orig_u_size + 1);
434 /* get a temporary short version of u */
435 memcpy(tmp_u->comps, &u->comps[u->size-n-1-j], (n+1)*COMP_BYTE_SIZE);
440 q_dash = COMP_RADIX-1;
444 q_dash = (comp)(((long_comp)U(0)*COMP_RADIX + U(1))/V1);
446 if (v->size > 1 && V2)
448 /* we are implementing the following:
449 if (V2*q_dash > (((U(0)*COMP_RADIX + U(1) -
450 q_dash*V1)*COMP_RADIX) + U(2))) ... */
451 comp inner = (comp)((long_comp)COMP_RADIX*U(0) + U(1) -
452 (long_comp)q_dash*V1);
453 if ((long_comp)V2*q_dash > (long_comp)inner*COMP_RADIX + U(2))
460 /* multiply and subtract */
464 tmp_u = bi_subtract(ctx, tmp_u,
465 bi_int_multiply(ctx, bi_copy(v), q_dash), &is_negative);
466 more_comps(tmp_u, n+1);
474 tmp_u = bi_add(ctx, tmp_u, bi_copy(v));
476 /* lop off the carry */
487 memcpy(&u->comps[u->size-n-1-j], tmp_u->comps, (n+1)*COMP_BYTE_SIZE);
493 if (is_mod) /* get the remainder */
495 bi_free(ctx, quotient);
496 return bi_int_divide(ctx, trim(u), d);
498 else /* get the quotient */
501 return trim(quotient);
506 * Perform an integer divide on a bigint.
508 static bigint *bi_int_divide(BI_CTX *ctx, bigint *biR, comp denom)
510 int i = biR->size - 1;
517 r = (r<<COMP_BIT_SIZE) + biR->comps[i];
518 biR->comps[i] = (comp)(r / denom);
525 #ifdef CONFIG_BIGINT_MONTGOMERY
527 * There is a need for the value of integer N' such that B^-1(B-1)-N^-1N'=1,
528 * where B^-1(B-1) mod N=1. Actually, only the least significant part of
529 * N' is needed, hence the definition N0'=N' mod b. We reproduce below the
530 * simple algorithm from an article by Dusse and Kaliski to efficiently
531 * find N0' from N0 and b */
532 static comp modular_inverse(bigint *bim)
536 comp two_2_i_minus_1 = 2; /* 2^(i-1) */
537 long_comp two_2_i = 4; /* 2^i */
538 comp N = bim->comps[0];
540 for (i = 2; i <= COMP_BIT_SIZE; i++)
542 if ((long_comp)N*t%two_2_i >= two_2_i_minus_1)
544 t += two_2_i_minus_1;
547 two_2_i_minus_1 <<= 1;
551 return (comp)(COMP_RADIX-t);
555 #if defined(CONFIG_BIGINT_KARATSUBA) || defined(CONFIG_BIGINT_BARRETT) || \
556 defined(CONFIG_BIGINT_MONTGOMERY)
558 * Take each component and shift down (in terms of components)
560 static bigint *comp_right_shift(bigint *biR, int num_shifts)
562 int i = biR->size-num_shifts;
563 comp *x = biR->comps;
564 comp *y = &biR->comps[num_shifts];
568 if (i <= 0) /* have we completely right shifted? */
570 biR->comps[0] = 0; /* return 0 */
580 biR->size -= num_shifts;
585 * Take each component and shift it up (in terms of components)
587 static bigint *comp_left_shift(bigint *biR, int num_shifts)
599 more_comps(biR, biR->size + num_shifts);
601 x = &biR->comps[i+num_shifts];
609 memset(biR->comps, 0, num_shifts*COMP_BYTE_SIZE); /* zero LS comps */
615 * @brief Allow a binary sequence to be imported as a bigint.
616 * @param ctx [in] The bigint session context.
617 * @param data [in] The data to be converted.
618 * @param size [in] The number of bytes of data.
619 * @return A bigint representing this data.
621 bigint *bi_import(BI_CTX *ctx, const uint8_t *data, int size)
623 bigint *biR = alloc(ctx, (size+COMP_BYTE_SIZE-1)/COMP_BYTE_SIZE);
624 int i, j = 0, offset = 0;
626 memset(biR->comps, 0, biR->size*COMP_BYTE_SIZE);
628 for (i = size-1; i >= 0; i--)
630 biR->comps[offset] += data[i] << (j*8);
632 if (++j == COMP_BYTE_SIZE)
642 #ifdef CONFIG_SSL_FULL_MODE
644 * @brief The testharness uses this code to import text hex-streams and
645 * convert them into bigints.
646 * @param ctx [in] The bigint session context.
647 * @param data [in] A string consisting of hex characters. The characters must
649 * @return A bigint representing this data.
651 bigint *bi_str_import(BI_CTX *ctx, const char *data)
653 int size = strlen(data);
654 bigint *biR = alloc(ctx, (size+COMP_NUM_NIBBLES-1)/COMP_NUM_NIBBLES);
655 int i, j = 0, offset = 0;
656 memset(biR->comps, 0, biR->size*COMP_BYTE_SIZE);
658 for (i = size-1; i >= 0; i--)
660 int num = (data[i] <= '9') ? (data[i] - '0') : (data[i] - 'A' + 10);
661 biR->comps[offset] += num << (j*4);
663 if (++j == COMP_NUM_NIBBLES)
673 void bi_print(const char *label, bigint *x)
679 printf("%s: (null)\n", label);
683 printf("%s: (size %d)\n", label, x->size);
684 for (i = x->size-1; i >= 0; i--)
686 for (j = COMP_NUM_NIBBLES-1; j >= 0; j--)
688 comp mask = 0x0f << (j*4);
689 comp num = (x->comps[i] & mask) >> (j*4);
690 putc((num <= 9) ? (num + '0') : (num + 'A' - 10), stdout);
699 * @brief Take a bigint and convert it into a byte sequence.
701 * This is useful after a decrypt operation.
702 * @param ctx [in] The bigint session context.
703 * @param x [in] The bigint to be converted.
704 * @param data [out] The converted data as a byte stream.
705 * @param size [in] The maximum size of the byte stream. Unused bytes will be
708 void bi_export(BI_CTX *ctx, bigint *x, uint8_t *data, int size)
710 int i, j, k = size-1;
713 memset(data, 0, size); /* ensure all leading 0's are cleared */
715 for (i = 0; i < x->size; i++)
717 for (j = 0; j < COMP_BYTE_SIZE; j++)
719 comp mask = 0xff << (j*8);
720 int num = (x->comps[i] & mask) >> (j*8);
735 * @brief Pre-calculate some of the expensive steps in reduction.
737 * This function should only be called once (normally when a session starts).
738 * When the session is over, bi_free_mod() should be called. bi_mod_power()
739 * relies on this function being called.
740 * @param ctx [in] The bigint session context.
741 * @param bim [in] The bigint modulus that will be used.
742 * @param mod_offset [in] There are three moduluii that can be stored - the
743 * standard modulus, and its two primes p and q. This offset refers to which
744 * modulus we are referring to.
745 * @see bi_free_mod(), bi_mod_power().
747 void bi_set_mod(BI_CTX *ctx, bigint *bim, int mod_offset)
750 comp d = (comp)((long_comp)COMP_RADIX/(bim->comps[k-1]+1));
751 #ifdef CONFIG_BIGINT_MONTGOMERY
755 ctx->bi_mod[mod_offset] = bim;
756 bi_permanent(ctx->bi_mod[mod_offset]);
757 ctx->bi_normalised_mod[mod_offset] = bi_int_multiply(ctx, bim, d);
758 bi_permanent(ctx->bi_normalised_mod[mod_offset]);
760 #if defined(CONFIG_BIGINT_MONTGOMERY)
761 /* set montgomery variables */
762 R = comp_left_shift(bi_clone(ctx, ctx->bi_radix), k-1); /* R */
763 R2 = comp_left_shift(bi_clone(ctx, ctx->bi_radix), k*2-1); /* R^2 */
764 ctx->bi_RR_mod_m[mod_offset] = bi_mod(ctx, R2); /* R^2 mod m */
765 ctx->bi_R_mod_m[mod_offset] = bi_mod(ctx, R); /* R mod m */
767 bi_permanent(ctx->bi_RR_mod_m[mod_offset]);
768 bi_permanent(ctx->bi_R_mod_m[mod_offset]);
770 ctx->N0_dash[mod_offset] = modular_inverse(ctx->bi_mod[mod_offset]);
772 #elif defined (CONFIG_BIGINT_BARRETT)
773 ctx->bi_mu[mod_offset] =
774 bi_divide(ctx, comp_left_shift(
775 bi_clone(ctx, ctx->bi_radix), k*2-1), ctx->bi_mod[mod_offset], 0);
776 bi_permanent(ctx->bi_mu[mod_offset]);
781 * @brief Used when cleaning various bigints at the end of a session.
782 * @param ctx [in] The bigint session context.
783 * @param mod_offset [in] The offset to use.
786 void bi_free_mod(BI_CTX *ctx, int mod_offset)
788 bi_depermanent(ctx->bi_mod[mod_offset]);
789 bi_free(ctx, ctx->bi_mod[mod_offset]);
790 #if defined (CONFIG_BIGINT_MONTGOMERY)
791 bi_depermanent(ctx->bi_RR_mod_m[mod_offset]);
792 bi_depermanent(ctx->bi_R_mod_m[mod_offset]);
793 bi_free(ctx, ctx->bi_RR_mod_m[mod_offset]);
794 bi_free(ctx, ctx->bi_R_mod_m[mod_offset]);
795 #elif defined(CONFIG_BIGINT_BARRETT)
796 bi_depermanent(ctx->bi_mu[mod_offset]);
797 bi_free(ctx, ctx->bi_mu[mod_offset]);
799 bi_depermanent(ctx->bi_normalised_mod[mod_offset]);
800 bi_free(ctx, ctx->bi_normalised_mod[mod_offset]);
804 * Perform a standard multiplication between two bigints.
806 * Barrett reduction has no need for some parts of the product, so ignore bits
807 * of the multiply. This routine gives Barrett its big performance
808 * improvements over Classical/Montgomery reduction methods.
810 static bigint *regular_multiply(BI_CTX *ctx, bigint *bia, bigint *bib,
811 int inner_partial, int outer_partial)
816 bigint *biR = alloc(ctx, n + t);
817 comp *sr = biR->comps;
818 comp *sa = bia->comps;
819 comp *sb = bib->comps;
824 /* clear things to start with */
825 memset(biR->comps, 0, ((n+t)*COMP_BYTE_SIZE));
834 if (outer_partial && outer_partial-i > 0 && outer_partial < n)
836 r_index = outer_partial-1;
837 j = outer_partial-i-1;
842 if (inner_partial && r_index >= inner_partial)
847 tmp = sr[r_index] + ((long_comp)sa[j])*sb[i] + carry;
848 sr[r_index++] = (comp)tmp; /* downsize */
849 carry = tmp >> COMP_BIT_SIZE;
860 #ifdef CONFIG_BIGINT_KARATSUBA
862 * Karatsuba improves on regular multiplication due to only 3 multiplications
863 * being done instead of 4. The additional additions/subtractions are O(N)
864 * rather than O(N^2) and so for big numbers it saves on a few operations
866 static bigint *karatsuba(BI_CTX *ctx, bigint *bia, bigint *bib, int is_square)
869 bigint *p0, *p1, *p2;
874 m = (bia->size + 1)/2;
878 m = (max(bia->size, bib->size) + 1)/2;
881 x0 = bi_clone(ctx, bia);
883 x1 = bi_clone(ctx, bia);
884 comp_right_shift(x1, m);
887 /* work out the 3 partial products */
890 p0 = bi_square(ctx, bi_copy(x0));
891 p2 = bi_square(ctx, bi_copy(x1));
892 p1 = bi_square(ctx, bi_add(ctx, x0, x1));
894 else /* normal multiply */
897 y0 = bi_clone(ctx, bib);
899 y1 = bi_clone(ctx, bib);
900 comp_right_shift(y1, m);
903 p0 = bi_multiply(ctx, bi_copy(x0), bi_copy(y0));
904 p2 = bi_multiply(ctx, bi_copy(x1), bi_copy(y1));
905 p1 = bi_multiply(ctx, bi_add(ctx, x0, x1), bi_add(ctx, y0, y1));
908 p1 = bi_subtract(ctx,
909 bi_subtract(ctx, p1, bi_copy(p2), NULL), bi_copy(p0), NULL);
911 comp_left_shift(p1, m);
912 comp_left_shift(p2, 2*m);
913 return bi_add(ctx, p1, bi_add(ctx, p0, p2));
918 * @brief Perform a multiplication operation between two bigints.
919 * @param ctx [in] The bigint session context.
920 * @param bia [in] A bigint.
921 * @param bib [in] Another bigint.
922 * @return The result of the multiplication.
924 bigint *bi_multiply(BI_CTX *ctx, bigint *bia, bigint *bib)
929 #ifdef CONFIG_BIGINT_KARATSUBA
930 if (min(bia->size, bib->size) < MUL_KARATSUBA_THRESH)
932 return regular_multiply(ctx, bia, bib, 0, 0);
935 return karatsuba(ctx, bia, bib, 0);
937 return regular_multiply(ctx, bia, bib, 0, 0);
941 #ifdef CONFIG_BIGINT_SQUARE
943 * Perform the actual square operion. It takes into account overflow.
945 static bigint *regular_square(BI_CTX *ctx, bigint *bi)
949 bigint *biR = alloc(ctx, t*2+1);
950 comp *w = biR->comps;
953 memset(w, 0, biR->size*COMP_BYTE_SIZE);
957 long_comp tmp = w[2*i] + (long_comp)x[i]*x[i];
959 carry = tmp >> COMP_BIT_SIZE;
961 for (j = i+1; j < t; j++)
964 long_comp xx = (long_comp)x[i]*x[j];
965 if ((COMP_MAX-xx) < xx)
970 if ((COMP_MAX-tmp) < w[i+j])
975 if ((COMP_MAX-tmp) < carry)
980 carry = tmp >> COMP_BIT_SIZE;
986 tmp = w[i+t] + carry;
988 w[i+t+1] = tmp >> COMP_BIT_SIZE;
996 * @brief Perform a square operation on a bigint.
997 * @param ctx [in] The bigint session context.
998 * @param bia [in] A bigint.
999 * @return The result of the multiplication.
1001 bigint *bi_square(BI_CTX *ctx, bigint *bia)
1005 #ifdef CONFIG_BIGINT_KARATSUBA
1006 if (bia->size < SQU_KARATSUBA_THRESH)
1008 return regular_square(ctx, bia);
1011 return karatsuba(ctx, bia, NULL, 1);
1013 return regular_square(ctx, bia);
1019 * @brief Compare two bigints.
1020 * @param bia [in] A bigint.
1021 * @param bib [in] Another bigint.
1022 * @return -1 if smaller, 1 if larger and 0 if equal.
1024 int bi_compare(bigint *bia, bigint *bib)
1031 if (bia->size > bib->size)
1033 else if (bia->size < bib->size)
1037 comp *a = bia->comps;
1038 comp *b = bib->comps;
1040 /* Same number of components. Compare starting from the high end
1041 * and working down. */
1052 else if (a[i] < b[i])
1064 * Allocate and zero more components. Does not consume bi.
1066 static void more_comps(bigint *bi, int n)
1068 if (n > bi->max_comps)
1070 bi->max_comps = max(bi->max_comps * 2, n);
1071 bi->comps = (comp*)realloc(bi->comps, bi->max_comps * COMP_BYTE_SIZE);
1076 memset(&bi->comps[bi->size], 0, (n-bi->size)*COMP_BYTE_SIZE);
1083 * Make a new empty bigint. It may just use an old one if one is available.
1084 * Otherwise get one off the heap.
1086 static bigint *alloc(BI_CTX *ctx, int size)
1090 /* Can we recycle an old bigint? */
1091 if (ctx->free_list != NULL)
1093 biR = ctx->free_list;
1094 ctx->free_list = biR->next;
1099 #ifdef CONFIG_SSL_FULL_MODE
1100 printf("alloc: refs was not 0\n");
1102 abort(); /* create a stack trace from a core dump */
1105 more_comps(biR, size);
1109 /* No free bigints available - create a new one. */
1110 biR = (bigint *)malloc(sizeof(bigint));
1111 biR->comps = (comp*)malloc(size * COMP_BYTE_SIZE);
1112 biR->max_comps = size; /* give some space to spare */
1118 ctx->active_count++;
1123 * Work out the highest '1' bit in an exponent. Used when doing sliding-window
1126 static int find_max_exp_index(bigint *biexp)
1128 int i = COMP_BIT_SIZE-1;
1129 comp shift = COMP_RADIX/2;
1130 comp test = biexp->comps[biexp->size-1]; /* assume no leading zeroes */
1138 return i+(biexp->size-1)*COMP_BIT_SIZE;
1144 return -1; /* error - must have been a leading 0 */
1148 * Is a particular bit is an exponent 1 or 0? Used when doing sliding-window
1151 static int exp_bit_is_one(bigint *biexp, int offset)
1153 comp test = biexp->comps[offset / COMP_BIT_SIZE];
1154 int num_shifts = offset % COMP_BIT_SIZE;
1160 for (i = 0; i < num_shifts; i++)
1165 return (test & shift) != 0;
1168 #ifdef CONFIG_BIGINT_CHECK_ON
1170 * Perform a sanity check on bi.
1172 static void check(const bigint *bi)
1176 printf("check: zero or negative refs in bigint\n");
1180 if (bi->next != NULL)
1182 printf("check: attempt to use a bigint from "
1190 * Delete any leading 0's (and allow for 0).
1192 static bigint *trim(bigint *bi)
1196 while (bi->comps[bi->size-1] == 0 && bi->size > 1)
1204 #if defined(CONFIG_BIGINT_MONTGOMERY)
1206 * @brief Perform a single montgomery reduction.
1207 * @param ctx [in] The bigint session context.
1208 * @param bixy [in] A bigint.
1209 * @return The result of the montgomery reduction.
1211 bigint *bi_mont(BI_CTX *ctx, bigint *bixy)
1214 uint8_t mod_offset = ctx->mod_offset;
1215 bigint *bim = ctx->bi_mod[mod_offset];
1216 comp mod_inv = ctx->N0_dash[mod_offset];
1220 if (ctx->use_classical) /* just use classical instead */
1222 return bi_mod(ctx, bixy);
1229 bixy = bi_add(ctx, bixy, comp_left_shift(
1230 bi_int_multiply(ctx, bim, bixy->comps[i]*mod_inv), i));
1233 comp_right_shift(bixy, n);
1235 if (bi_compare(bixy, bim) >= 0)
1237 bixy = bi_subtract(ctx, bixy, bim, NULL);
1243 #elif defined(CONFIG_BIGINT_BARRETT)
1245 * Stomp on the most significant components to give the illusion of a "mod base
1248 static bigint *comp_mod(bigint *bi, int mod)
1261 * @brief Perform a single Barrett reduction.
1262 * @param ctx [in] The bigint session context.
1263 * @param bi [in] A bigint.
1264 * @return The result of the Barrett reduction.
1266 bigint *bi_barrett(BI_CTX *ctx, bigint *bi)
1269 bigint *q1, *q2, *q3, *r1, *r2, *r;
1270 uint8_t mod_offset = ctx->mod_offset;
1271 bigint *bim = ctx->bi_mod[mod_offset];
1277 /* use Classical method instead - Barrett cannot help here */
1281 return bi_mod(ctx, bi);
1283 bigint* a = bi_clone(ctx, bi);
1284 q1 = comp_right_shift(a, k-1);
1286 /* do outer partial multiply */
1287 q2 = regular_multiply(ctx, q1, ctx->bi_mu[mod_offset], 0, k-1);
1288 q3 = comp_right_shift(q2, k+1);
1289 r1 = comp_mod(bi, k+1);
1291 /* do inner partial multiply */
1292 r2 = comp_mod(regular_multiply(ctx, q3, bim, k+1, 0), k+1);
1293 r = bi_subtract(ctx, r1, r2, NULL);
1295 /* if (r >= m) r = r - m; */
1296 if (bi_compare(r, bim) >= 0)
1299 r = bi_subtract(ctx, r, bim, NULL);
1304 #endif /* CONFIG_BIGINT_BARRETT */
1306 #ifdef CONFIG_BIGINT_SLIDING_WINDOW
1308 * Work out g1, g3, g5, g7... etc for the sliding-window algorithm
1310 static void precompute_slide_window(BI_CTX *ctx, int window, bigint *g1)
1315 for (i = 0; i < window-1; i++) /* compute 2^(window-1) */
1320 ctx->g = (bigint **)malloc(k*sizeof(bigint *));
1321 ctx->g[0] = bi_clone(ctx, g1);
1322 bi_permanent(ctx->g[0]);
1323 g2 = bi_residue(ctx, bi_square(ctx, ctx->g[0])); /* g^2 */
1325 for (i = 1; i < k; i++)
1327 ctx->g[i] = bi_residue(ctx, bi_multiply(ctx, ctx->g[i-1], bi_copy(g2)));
1328 bi_permanent(ctx->g[i]);
1337 * @brief Perform a modular exponentiation.
1339 * This function requires bi_set_mod() to have been called previously. This is
1340 * one of the optimisations used for performance.
1341 * @param ctx [in] The bigint session context.
1342 * @param bi [in] The bigint on which to perform the mod power operation.
1343 * @param biexp [in] The bigint exponent.
1344 * @return The result of the mod exponentiation operation
1345 * @see bi_set_mod().
1347 bigint *bi_mod_power(BI_CTX *ctx, bigint *bi, bigint *biexp)
1349 int i = find_max_exp_index(biexp), j, window_size = 1;
1350 bigint *biR = int_to_bi(ctx, 1);
1353 #if defined(CONFIG_BIGINT_MONTGOMERY)
1354 uint8_t mod_offset = ctx->mod_offset;
1355 if (!ctx->use_classical)
1359 bi_multiply(ctx, bi, ctx->bi_RR_mod_m[mod_offset])); /* x' */
1361 biR = ctx->bi_R_mod_m[mod_offset]; /* A */
1368 #ifdef CONFIG_BIGINT_SLIDING_WINDOW
1369 for (j = i; j > 32; j /= 5) /* work out an optimum size */
1372 /* work out the slide constants */
1373 precompute_slide_window(ctx, window_size, bi);
1374 #else /* just one constant */
1375 ctx->g = (bigint **)malloc(sizeof(bigint *));
1376 ctx->g[0] = bi_clone(ctx, bi);
1378 bi_permanent(ctx->g[0]);
1381 /* if sliding-window is off, then only one bit will be done at a time and
1382 * will reduce to standard left-to-right exponentiation */
1385 if (exp_bit_is_one(biexp, i))
1387 int l = i-window_size+1;
1390 if (l < 0) /* LSB of exponent will always be 1 */
1394 while (exp_bit_is_one(biexp, l) == 0)
1395 l++; /* go back up */
1397 /* build up the section of the exponent */
1398 for (j = i; j >= l; j--)
1400 biR = bi_residue(ctx, bi_square(ctx, biR));
1401 if (exp_bit_is_one(biexp, j))
1407 part_exp = (part_exp-1)/2; /* adjust for array */
1408 bigint* a = bi_multiply(ctx, biR, ctx->g[part_exp]);
1409 biR = bi_residue(ctx, a);
1412 else /* square it */
1414 biR = bi_residue(ctx, bi_square(ctx, biR));
1421 for (i = 0; i < ctx->window; i++)
1423 bi_depermanent(ctx->g[i]);
1424 bi_free(ctx, ctx->g[i]);
1429 bi_free(ctx, biexp);
1430 #if defined CONFIG_BIGINT_MONTGOMERY
1431 return ctx->use_classical ? biR : bi_mont(ctx, biR); /* convert back */
1432 #else /* CONFIG_BIGINT_CLASSICAL or CONFIG_BIGINT_BARRETT */
1437 #ifdef CONFIG_SSL_CERT_VERIFICATION
1439 * @brief Perform a modular exponentiation using a temporary modulus.
1441 * We need this function to check the signatures of certificates. The modulus
1442 * of this function is temporary as it's just used for authentication.
1443 * @param ctx [in] The bigint session context.
1444 * @param bi [in] The bigint to perform the exp/mod.
1445 * @param bim [in] The temporary modulus.
1446 * @param biexp [in] The bigint exponent.
1447 * @return The result of the mod exponentiation operation
1448 * @see bi_set_mod().
1450 bigint *bi_mod_power2(BI_CTX *ctx, bigint *bi, bigint *bim, bigint *biexp)
1452 bigint *biR, *tmp_biR;
1454 /* Set up a temporary bigint context and transfer what we need between
1455 * them. We need to do this since we want to keep the original modulus
1456 * which is already in this context. This operation is only called when
1457 * doing peer verification, and so is not expensive :-) */
1458 BI_CTX *tmp_ctx = bi_initialize();
1459 bi_set_mod(tmp_ctx, bi_clone(tmp_ctx, bim), BIGINT_M_OFFSET);
1460 tmp_biR = bi_mod_power(tmp_ctx,
1461 bi_clone(tmp_ctx, bi),
1462 bi_clone(tmp_ctx, biexp));
1463 biR = bi_clone(ctx, tmp_biR);
1464 bi_free(tmp_ctx, tmp_biR);
1465 bi_free_mod(tmp_ctx, BIGINT_M_OFFSET);
1466 bi_terminate(tmp_ctx);
1470 bi_free(ctx, biexp);
1475 #ifdef CONFIG_BIGINT_CRT
1477 * @brief Use the Chinese Remainder Theorem to quickly perform RSA decrypts.
1479 * @param ctx [in] The bigint session context.
1480 * @param bi [in] The bigint to perform the exp/mod.
1481 * @param dP [in] CRT's dP bigint
1482 * @param dQ [in] CRT's dQ bigint
1483 * @param p [in] CRT's p bigint
1484 * @param q [in] CRT's q bigint
1485 * @param qInv [in] CRT's qInv bigint
1486 * @return The result of the CRT operation
1488 bigint *bi_crt(BI_CTX *ctx, bigint *bi,
1489 bigint *dP, bigint *dQ,
1490 bigint *p, bigint *q, bigint *qInv)
1492 bigint *m1, *m2, *h;
1494 /* Montgomery has a condition the 0 < x, y < m and these products violate
1495 * that condition. So disable Montgomery when using CRT */
1496 #if defined(CONFIG_BIGINT_MONTGOMERY)
1497 ctx->use_classical = 1;
1499 ctx->mod_offset = BIGINT_P_OFFSET;
1500 m1 = bi_mod_power(ctx, bi_copy(bi), dP);
1502 ctx->mod_offset = BIGINT_Q_OFFSET;
1503 m2 = bi_mod_power(ctx, bi, dQ);
1505 h = bi_subtract(ctx, bi_add(ctx, m1, p), bi_copy(m2), NULL);
1506 h = bi_multiply(ctx, h, qInv);
1507 ctx->mod_offset = BIGINT_P_OFFSET;
1508 h = bi_residue(ctx, h);
1509 #if defined(CONFIG_BIGINT_MONTGOMERY)
1510 ctx->use_classical = 0; /* reset for any further operation */
1512 return bi_add(ctx, m2, bi_multiply(ctx, q, h));