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Diffstat (limited to 'mpint.c')
| -rw-r--r-- | mpint.c | 2646 |
1 files changed, 0 insertions, 2646 deletions
diff --git a/mpint.c b/mpint.c deleted file mode 100644 index 39f55063..00000000 --- a/mpint.c +++ /dev/null @@ -1,2646 +0,0 @@ -#include <assert.h> -#include <limits.h> -#include <stdio.h> - -#include "defs.h" -#include "misc.h" -#include "puttymem.h" - -#include "mpint.h" -#include "mpint_i.h" - -#define SIZE_T_BITS (CHAR_BIT * sizeof(size_t)) - -/* - * Inline helpers to take min and max of size_t values, used - * throughout this code. - */ -static inline size_t size_t_min(size_t a, size_t b) -{ - return a < b ? a : b; -} -static inline size_t size_t_max(size_t a, size_t b) -{ - return a > b ? a : b; -} - -/* - * Helper to fetch a word of data from x with array overflow checking. - * If x is too short to have that word, 0 is returned. - */ -static inline BignumInt mp_word(mp_int *x, size_t i) -{ - return i < x->nw ? x->w[i] : 0; -} - -/* - * Shift an ordinary C integer by BIGNUM_INT_BITS, in a way that - * avoids writing a shift operator whose RHS is greater or equal to - * the size of the type, because that's undefined behaviour in C. - * - * In fact we must avoid even writing it in a definitely-untaken - * branch of an if, because compilers will sometimes warn about - * that. So you can't just write 'shift too big ? 0 : n >> shift', - * because even if 'shift too big' is a constant-expression - * evaluating to false, you can still get complaints about the - * else clause of the ?:. - * - * So we have to re-check _inside_ that clause, so that the shift - * count is reset to something nonsensical but safe in the case - * where the clause wasn't going to be taken anyway. - */ -static uintmax_t shift_right_by_one_word(uintmax_t n) -{ - bool shift_too_big = BIGNUM_INT_BYTES >= sizeof(n); - return shift_too_big ? 0 : - n >> (shift_too_big ? 0 : BIGNUM_INT_BITS); -} -static uintmax_t shift_left_by_one_word(uintmax_t n) -{ - bool shift_too_big = BIGNUM_INT_BYTES >= sizeof(n); - return shift_too_big ? 0 : - n << (shift_too_big ? 0 : BIGNUM_INT_BITS); -} - -mp_int *mp_make_sized(size_t nw) -{ - mp_int *x = snew_plus(mp_int, nw * sizeof(BignumInt)); - assert(nw); /* we outlaw the zero-word mp_int */ - x->nw = nw; - x->w = snew_plus_get_aux(x); - mp_clear(x); - return x; -} - -mp_int *mp_new(size_t maxbits) -{ - size_t words = (maxbits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS; - return mp_make_sized(words); -} - -mp_int *mp_from_integer(uintmax_t n) -{ - mp_int *x = mp_make_sized( - (sizeof(n) + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES); - for (size_t i = 0; i < x->nw; i++) - x->w[i] = n >> (i * BIGNUM_INT_BITS); - return x; -} - -size_t mp_max_bytes(mp_int *x) -{ - return x->nw * BIGNUM_INT_BYTES; -} - -size_t mp_max_bits(mp_int *x) -{ - return x->nw * BIGNUM_INT_BITS; -} - -void mp_free(mp_int *x) -{ - mp_clear(x); - smemclr(x, sizeof(*x)); - sfree(x); -} - -void mp_dump(FILE *fp, const char *prefix, mp_int *x, const char *suffix) -{ - fprintf(fp, "%s0x", prefix); - for (size_t i = mp_max_bytes(x); i-- > 0 ;) - fprintf(fp, "%02X", mp_get_byte(x, i)); - fputs(suffix, fp); -} - -void mp_copy_into(mp_int *dest, mp_int *src) -{ - size_t copy_nw = size_t_min(dest->nw, src->nw); - memmove(dest->w, src->w, copy_nw * sizeof(BignumInt)); - smemclr(dest->w + copy_nw, (dest->nw - copy_nw) * sizeof(BignumInt)); -} - -void mp_copy_integer_into(mp_int *r, uintmax_t n) -{ - for (size_t i = 0; i < r->nw; i++) { - r->w[i] = n; - n = shift_right_by_one_word(n); - } -} - -/* - * Conditional selection is done by negating 'which', to give a mask - * word which is all 1s if which==1 and all 0s if which==0. Then you - * can select between two inputs a,b without data-dependent control - * flow by XORing them to get their difference; ANDing with the mask - * word to replace that difference with 0 if which==0; and XORing that - * into a, which will either turn it into b or leave it alone. - * - * This trick will be used throughout this code and taken as read the - * rest of the time (or else I'd be here all week typing comments), - * but I felt I ought to explain it in words _once_. - */ -void mp_select_into(mp_int *dest, mp_int *src0, mp_int *src1, - unsigned which) -{ - BignumInt mask = -(BignumInt)(1 & which); - for (size_t i = 0; i < dest->nw; i++) { - BignumInt srcword0 = mp_word(src0, i), srcword1 = mp_word(src1, i); - dest->w[i] = srcword0 ^ ((srcword1 ^ srcword0) & mask); - } -} - -void mp_cond_swap(mp_int *x0, mp_int *x1, unsigned swap) -{ - assert(x0->nw == x1->nw); - volatile BignumInt mask = -(BignumInt)(1 & swap); - for (size_t i = 0; i < x0->nw; i++) { - BignumInt diff = (x0->w[i] ^ x1->w[i]) & mask; - x0->w[i] ^= diff; - x1->w[i] ^= diff; - } -} - -void mp_clear(mp_int *x) -{ - smemclr(x->w, x->nw * sizeof(BignumInt)); -} - -void mp_cond_clear(mp_int *x, unsigned clear) -{ - BignumInt mask = ~-(BignumInt)(1 & clear); - for (size_t i = 0; i < x->nw; i++) - x->w[i] &= mask; -} - -/* - * Common code between mp_from_bytes_{le,be} which reads bytes in an - * arbitrary arithmetic progression. - */ -static mp_int *mp_from_bytes_int(ptrlen bytes, size_t m, size_t c) -{ - size_t nw = (bytes.len + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; - nw = size_t_max(nw, 1); - mp_int *n = mp_make_sized(nw); - for (size_t i = 0; i < bytes.len; i++) - n->w[i / BIGNUM_INT_BYTES] |= - (BignumInt)(((const unsigned char *)bytes.ptr)[m*i+c]) << - (8 * (i % BIGNUM_INT_BYTES)); - return n; -} - -mp_int *mp_from_bytes_le(ptrlen bytes) -{ - return mp_from_bytes_int(bytes, 1, 0); -} - -mp_int *mp_from_bytes_be(ptrlen bytes) -{ - return mp_from_bytes_int(bytes, -1, bytes.len - 1); -} - -static mp_int *mp_from_words(size_t nw, const BignumInt *w) -{ - mp_int *x = mp_make_sized(nw); - memcpy(x->w, w, x->nw * sizeof(BignumInt)); - return x; -} - -/* - * Decimal-to-binary conversion: just go through the input string - * adding on the decimal value of each digit, and then multiplying the - * number so far by 10. - */ -mp_int *mp_from_decimal_pl(ptrlen decimal) -{ - /* 196/59 is an upper bound (and also a continued-fraction - * convergent) for log2(10), so this conservatively estimates the - * number of bits that will be needed to store any number that can - * be written in this many decimal digits. */ - assert(decimal.len < (~(size_t)0) / 196); - size_t bits = 196 * decimal.len / 59; - - /* Now round that up to words. */ - size_t words = bits / BIGNUM_INT_BITS + 1; - - mp_int *x = mp_make_sized(words); - for (size_t i = 0; i < decimal.len; i++) { - mp_add_integer_into(x, x, ((const char *)decimal.ptr)[i] - '0'); - - if (i+1 == decimal.len) - break; - - mp_mul_integer_into(x, x, 10); - } - return x; -} - -mp_int *mp_from_decimal(const char *decimal) -{ - return mp_from_decimal_pl(ptrlen_from_asciz(decimal)); -} - -/* - * Hex-to-binary conversion: _algorithmically_ simpler than decimal - * (none of those multiplications by 10), but there's some fiddly - * bit-twiddling needed to process each hex digit without diverging - * control flow depending on whether it's a letter or a number. - */ -mp_int *mp_from_hex_pl(ptrlen hex) -{ - assert(hex.len <= (~(size_t)0) / 4); - size_t bits = hex.len * 4; - size_t words = (bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS; - words = size_t_max(words, 1); - mp_int *x = mp_make_sized(words); - for (size_t nibble = 0; nibble < hex.len; nibble++) { - BignumInt digit = ((const char *)hex.ptr)[hex.len-1 - nibble]; - - BignumInt lmask = ~-((BignumInt)((digit-'a')|('f'-digit)) - >> (BIGNUM_INT_BITS-1)); - BignumInt umask = ~-((BignumInt)((digit-'A')|('F'-digit)) - >> (BIGNUM_INT_BITS-1)); - - BignumInt digitval = digit - '0'; - digitval ^= (digitval ^ (digit - 'a' + 10)) & lmask; - digitval ^= (digitval ^ (digit - 'A' + 10)) & umask; - digitval &= 0xF; /* at least be _slightly_ nice about weird input */ - - size_t word_idx = nibble / (BIGNUM_INT_BYTES*2); - size_t nibble_within_word = nibble % (BIGNUM_INT_BYTES*2); - x->w[word_idx] |= digitval << (nibble_within_word * 4); - } - return x; -} - -mp_int *mp_from_hex(const char *hex) -{ - return mp_from_hex_pl(ptrlen_from_asciz(hex)); -} - -mp_int *mp_copy(mp_int *x) -{ - return mp_from_words(x->nw, x->w); -} - -uint8_t mp_get_byte(mp_int *x, size_t byte) -{ - return 0xFF & (mp_word(x, byte / BIGNUM_INT_BYTES) >> - (8 * (byte % BIGNUM_INT_BYTES))); -} - -unsigned mp_get_bit(mp_int *x, size_t bit) -{ - return 1 & (mp_word(x, bit / BIGNUM_INT_BITS) >> - (bit % BIGNUM_INT_BITS)); -} - -uintmax_t mp_get_integer(mp_int *x) -{ - uintmax_t toret = 0; - for (size_t i = x->nw; i-- > 0 ;) - toret = shift_left_by_one_word(toret) | x->w[i]; - return toret; -} - -void mp_set_bit(mp_int *x, size_t bit, unsigned val) -{ - size_t word = bit / BIGNUM_INT_BITS; - assert(word < x->nw); - - unsigned shift = (bit % BIGNUM_INT_BITS); - - x->w[word] &= ~((BignumInt)1 << shift); - x->w[word] |= (BignumInt)(val & 1) << shift; -} - -/* - * Helper function used here and there to normalise any nonzero input - * value to 1. - */ -static inline unsigned normalise_to_1(BignumInt n) -{ - n = (n >> 1) | (n & 1); /* ensure top bit is clear */ - n = (BignumInt)(-n) >> (BIGNUM_INT_BITS - 1); /* normalise to 0 or 1 */ - return n; -} -static inline unsigned normalise_to_1_u64(uint64_t n) -{ - n = (n >> 1) | (n & 1); /* ensure top bit is clear */ - n = (-n) >> 63; /* normalise to 0 or 1 */ - return n; -} - -/* - * Find the highest nonzero word in a number. Returns the index of the - * word in x->w, and also a pair of output uint64_t in which that word - * appears in the high one shifted left by 'shift_wanted' bits, the - * words immediately below it occupy the space to the right, and the - * words below _that_ fill up the low one. - * - * If there is no nonzero word at all, the passed-by-reference output - * variables retain their original values. - */ -static inline void mp_find_highest_nonzero_word_pair( - mp_int *x, size_t shift_wanted, size_t *index, - uint64_t *hi, uint64_t *lo) -{ - uint64_t curr_hi = 0, curr_lo = 0; - - for (size_t curr_index = 0; curr_index < x->nw; curr_index++) { - BignumInt curr_word = x->w[curr_index]; - unsigned indicator = normalise_to_1(curr_word); - - curr_lo = (BIGNUM_INT_BITS < 64 ? (curr_lo >> BIGNUM_INT_BITS) : 0) | - (curr_hi << (64 - BIGNUM_INT_BITS)); - curr_hi = (BIGNUM_INT_BITS < 64 ? (curr_hi >> BIGNUM_INT_BITS) : 0) | - ((uint64_t)curr_word << shift_wanted); - - if (hi) *hi ^= (curr_hi ^ *hi ) & -(uint64_t)indicator; - if (lo) *lo ^= (curr_lo ^ *lo ) & -(uint64_t)indicator; - if (index) *index ^= (curr_index ^ *index) & -(size_t) indicator; - } -} - -size_t mp_get_nbits(mp_int *x) -{ - /* Sentinel values in case there are no bits set at all: we - * imagine that there's a word at position -1 (i.e. the topmost - * fraction word) which is all 1s, because that way, we handle a - * zero input by considering its highest set bit to be the top one - * of that word, i.e. just below the units digit, i.e. at bit - * index -1, i.e. so we'll return 0 on output. */ - size_t hiword_index = -(size_t)1; - uint64_t hiword64 = ~(BignumInt)0; - - /* - * Find the highest nonzero word and its index. - */ - mp_find_highest_nonzero_word_pair(x, 0, &hiword_index, &hiword64, NULL); - BignumInt hiword = hiword64; /* in case BignumInt is a narrower type */ - - /* - * Find the index of the highest set bit within hiword. - */ - BignumInt hibit_index = 0; - for (size_t i = (1 << (BIGNUM_INT_BITS_BITS-1)); i != 0; i >>= 1) { - BignumInt shifted_word = hiword >> i; - BignumInt indicator = - (BignumInt)(-shifted_word) >> (BIGNUM_INT_BITS-1); - hiword ^= (shifted_word ^ hiword ) & -indicator; - hibit_index += i & -(size_t)indicator; - } - - /* - * Put together the result. - */ - return (hiword_index << BIGNUM_INT_BITS_BITS) + hibit_index + 1; -} - -/* - * Shared code between the hex and decimal output functions to get rid - * of leading zeroes on the output string. The idea is that we wrote - * out a fixed number of digits and a trailing \0 byte into 'buf', and - * now we want to shift it all left so that the first nonzero digit - * moves to buf[0] (or, if there are no nonzero digits at all, we move - * up by 'maxtrim', so that we return 0 as "0" instead of ""). - */ -static void trim_leading_zeroes(char *buf, size_t bufsize, size_t maxtrim) -{ - size_t trim = maxtrim; - - /* - * Look for the first character not equal to '0', to find the - * shift count. - */ - if (trim > 0) { - for (size_t pos = trim; pos-- > 0 ;) { - uint8_t diff = buf[pos] ^ '0'; - size_t mask = -((((size_t)diff) - 1) >> (SIZE_T_BITS - 1)); - trim ^= (trim ^ pos) & ~mask; - } - } - - /* - * Now do the shift, in log n passes each of which does a - * conditional shift by 2^i bytes if bit i is set in the shift - * count. - */ - uint8_t *ubuf = (uint8_t *)buf; - for (size_t logd = 0; bufsize >> logd; logd++) { - uint8_t mask = -(uint8_t)((trim >> logd) & 1); - size_t d = (size_t)1 << logd; - for (size_t i = 0; i+d < bufsize; i++) { - uint8_t diff = mask & (ubuf[i] ^ ubuf[i+d]); - ubuf[i] ^= diff; - ubuf[i+d] ^= diff; - } - } -} - -/* - * Binary to decimal conversion. Our strategy here is to extract each - * decimal digit by finding the input number's residue mod 10, then - * subtract that off to give an exact multiple of 10, which then means - * you can safely divide by 10 by means of shifting right one bit and - * then multiplying by the inverse of 5 mod 2^n. - */ -char *mp_get_decimal(mp_int *x_orig) -{ - mp_int *x = mp_copy(x_orig), *y = mp_make_sized(x->nw); - - /* - * The inverse of 5 mod 2^lots is 0xccccccccccccccccccccd, for an - * appropriate number of 'c's. Manually construct an integer the - * right size. - */ - mp_int *inv5 = mp_make_sized(x->nw); - assert(BIGNUM_INT_BITS % 8 == 0); - for (size_t i = 0; i < inv5->nw; i++) - inv5->w[i] = BIGNUM_INT_MASK / 5 * 4; - inv5->w[0]++; - - /* - * 146/485 is an upper bound (and also a continued-fraction - * convergent) of log10(2), so this is a conservative estimate of - * the number of decimal digits needed to store a value that fits - * in this many binary bits. - */ - assert(x->nw < (~(size_t)1) / (146 * BIGNUM_INT_BITS)); - size_t bufsize = size_t_max(x->nw * (146 * BIGNUM_INT_BITS) / 485, 1) + 2; - char *outbuf = snewn(bufsize, char); - outbuf[bufsize - 1] = '\0'; - - /* - * Loop over the number generating digits from the least - * significant upwards, so that we write to outbuf in reverse - * order. - */ - for (size_t pos = bufsize - 1; pos-- > 0 ;) { - /* - * Find the current residue mod 10. We do this by first - * summing the bytes of the number, with all but the lowest - * one multiplied by 6 (because 256^i == 6 mod 10 for all - * i>0). That gives us a single word congruent mod 10 to the - * input number, and then we reduce it further by manual - * multiplication and shifting, just in case the compiler - * target implements the C division operator in a way that has - * input-dependent timing. - */ - uint32_t low_digit = 0, maxval = 0, mult = 1; - for (size_t i = 0; i < x->nw; i++) { - for (unsigned j = 0; j < BIGNUM_INT_BYTES; j++) { - low_digit += mult * (0xFF & (x->w[i] >> (8*j))); - maxval += mult * 0xFF; - mult = 6; - } - /* - * For _really_ big numbers, prevent overflow of t by - * periodically folding the top half of the accumulator - * into the bottom half, using the same rule 'multiply by - * 6 when shifting down by one or more whole bytes'. - */ - if (maxval > UINT32_MAX - (6 * 0xFF * BIGNUM_INT_BYTES)) { - low_digit = (low_digit & 0xFFFF) + 6 * (low_digit >> 16); - maxval = (maxval & 0xFFFF) + 6 * (maxval >> 16); - } - } - - /* - * Final reduction of low_digit. We multiply by 2^32 / 10 - * (that's the constant 0x19999999) to get a 64-bit value - * whose top 32 bits are the approximate quotient - * low_digit/10; then we subtract off 10 times that; and - * finally we do one last trial subtraction of 10 by adding 6 - * (which sets bit 4 if the number was just over 10) and then - * testing bit 4. - */ - low_digit -= 10 * ((0x19999999ULL * low_digit) >> 32); - low_digit -= 10 * ((low_digit + 6) >> 4); - - assert(low_digit < 10); /* make sure we did reduce fully */ - outbuf[pos] = '0' + low_digit; - - /* - * Now subtract off that digit, divide by 2 (using a right - * shift) and by 5 (using the modular inverse), to get the - * next output digit into the units position. - */ - mp_sub_integer_into(x, x, low_digit); - mp_rshift_fixed_into(y, x, 1); - mp_mul_into(x, y, inv5); - } - - mp_free(x); - mp_free(y); - mp_free(inv5); - - trim_leading_zeroes(outbuf, bufsize, bufsize - 2); - return outbuf; -} - -/* - * Binary to hex conversion. Reasonably simple (only a spot of bit - * twiddling to choose whether to output a digit or a letter for each - * nibble). - */ -static char *mp_get_hex_internal(mp_int *x, uint8_t letter_offset) -{ - size_t nibbles = x->nw * BIGNUM_INT_BYTES * 2; - size_t bufsize = nibbles + 1; - char *outbuf = snewn(bufsize, char); - outbuf[nibbles] = '\0'; - - for (size_t nibble = 0; nibble < nibbles; nibble++) { - size_t word_idx = nibble / (BIGNUM_INT_BYTES*2); - size_t nibble_within_word = nibble % (BIGNUM_INT_BYTES*2); - uint8_t digitval = 0xF & (x->w[word_idx] >> (nibble_within_word * 4)); - - uint8_t mask = -((digitval + 6) >> 4); - char digit = digitval + '0' + (letter_offset & mask); - outbuf[nibbles-1 - nibble] = digit; - } - - trim_leading_zeroes(outbuf, bufsize, nibbles - 1); - return outbuf; -} - -char *mp_get_hex(mp_int *x) -{ - return mp_get_hex_internal(x, 'a' - ('0'+10)); -} - -char *mp_get_hex_uppercase(mp_int *x) -{ - return mp_get_hex_internal(x, 'A' - ('0'+10)); -} - -/* - * Routines for reading and writing the SSH-1 and SSH-2 wire formats - * for multiprecision integers, declared in marshal.h. - * - * These can't avoid having control flow dependent on the true bit - * size of the number, because the wire format requires the number of - * output bytes to depend on that. - */ -void BinarySink_put_mp_ssh1(BinarySink *bs, mp_int *x) -{ - size_t bits = mp_get_nbits(x); - size_t bytes = (bits + 7) / 8; - - assert(bits < 0x10000); - put_uint16(bs, bits); - for (size_t i = bytes; i-- > 0 ;) - put_byte(bs, mp_get_byte(x, i)); -} - -void BinarySink_put_mp_ssh2(BinarySink *bs, mp_int *x) -{ - size_t bytes = (mp_get_nbits(x) + 8) / 8; - - put_uint32(bs, bytes); - for (size_t i = bytes; i-- > 0 ;) - put_byte(bs, mp_get_byte(x, i)); -} - -mp_int *BinarySource_get_mp_ssh1(BinarySource *src) -{ - unsigned bitc = get_uint16(src); - ptrlen bytes = get_data(src, (bitc + 7) / 8); - if (get_err(src)) { - return mp_from_integer(0); - } else { - mp_int *toret = mp_from_bytes_be(bytes); - /* SSH-1.5 spec says that it's OK for the prefix uint16 to be - * _greater_ than the actual number of bits */ - if (mp_get_nbits(toret) > bitc) { - src->err = BSE_INVALID; - mp_free(toret); - toret = mp_from_integer(0); - } - return toret; - } -} - -mp_int *BinarySource_get_mp_ssh2(BinarySource *src) -{ - ptrlen bytes = get_string(src); - if (get_err(src)) { - return mp_from_integer(0); - } else { - const unsigned char *p = bytes.ptr; - if ((bytes.len > 0 && - ((p[0] & 0x80) || - (p[0] == 0 && (bytes.len <= 1 || !(p[1] & 0x80)))))) { - src->err = BSE_INVALID; - return mp_from_integer(0); - } - return mp_from_bytes_be(bytes); - } -} - -/* - * Make an mp_int structure whose words array aliases a subinterval of - * some other mp_int. This makes it easy to read or write just the low - * or high words of a number, e.g. to add a number starting from a - * high bit position, or to reduce mod 2^{n*BIGNUM_INT_BITS}. - * - * The convention throughout this code is that when we store an mp_int - * directly by value, we always expect it to be an alias of some kind, - * so its words array won't ever need freeing. Whereas an 'mp_int *' - * has an owner, who knows whether it needs freeing or whether it was - * created by address-taking an alias. - */ -static mp_int mp_make_alias(mp_int *in, size_t offset, size_t len) -{ - /* - * Bounds-check the offset and length so that we always return - * something valid, even if it's not necessarily the length the - * caller asked for. - */ - if (offset > in->nw) - offset = in->nw; - if (len > in->nw - offset) - len = in->nw - offset; - - mp_int toret; - toret.nw = len; - toret.w = in->w + offset; - return toret; -} - -/* - * A special case of mp_make_alias: in some cases we preallocate a - * large mp_int to use as scratch space (to avoid pointless - * malloc/free churn in recursive or iterative work). - * - * mp_alloc_from_scratch creates an alias of size 'len' to part of - * 'pool', and adjusts 'pool' itself so that further allocations won't - * overwrite that space. - * - * There's no free function to go with this. Typically you just copy - * the pool mp_int by value, allocate from the copy, and when you're - * done with those allocations, throw the copy away and go back to the - * original value of pool. (A mark/release system.) - */ -static mp_int mp_alloc_from_scratch(mp_int *pool, size_t len) -{ - assert(len <= pool->nw); - mp_int toret = mp_make_alias(pool, 0, len); - *pool = mp_make_alias(pool, len, pool->nw); - return toret; -} - -/* - * Internal component common to lots of assorted add/subtract code. - * Reads words from a,b; writes into w_out (which might be NULL if the - * output isn't even needed). Takes an input carry flag in 'carry', - * and returns the output carry. Each word read from b is ANDed with - * b_and and then XORed with b_xor. - * - * So you can implement addition by setting b_and to all 1s and b_xor - * to 0; you can subtract by making b_xor all 1s too (effectively - * bit-flipping b) and also passing 1 as the input carry (to turn - * one's complement into two's complement). And you can do conditional - * add/subtract by choosing b_and to be all 1s or all 0s based on a - * condition, because the value of b will be totally ignored if b_and - * == 0. - */ -static BignumCarry mp_add_masked_into( - BignumInt *w_out, size_t rw, mp_int *a, mp_int *b, - BignumInt b_and, BignumInt b_xor, BignumCarry carry) -{ - for (size_t i = 0; i < rw; i++) { - BignumInt aword = mp_word(a, i), bword = mp_word(b, i), out; - bword = (bword & b_and) ^ b_xor; - BignumADC(out, carry, aword, bword, carry); - if (w_out) - w_out[i] = out; - } - return carry; -} - -/* - * Like the public mp_add_into except that it returns the output carry. - */ -static inline BignumCarry mp_add_into_internal(mp_int *r, mp_int *a, mp_int *b) -{ - return mp_add_masked_into(r->w, r->nw, a, b, ~(BignumInt)0, 0, 0); -} - -void mp_add_into(mp_int *r, mp_int *a, mp_int *b) -{ - mp_add_into_internal(r, a, b); -} - -void mp_sub_into(mp_int *r, mp_int *a, mp_int *b) -{ - mp_add_masked_into(r->w, r->nw, a, b, ~(BignumInt)0, ~(BignumInt)0, 1); -} - -void mp_and_into(mp_int *r, mp_int *a, mp_int *b) -{ - for (size_t i = 0; i < r->nw; i++) { - BignumInt aword = mp_word(a, i), bword = mp_word(b, i); - r->w[i] = aword & bword; - } -} - -void mp_or_into(mp_int *r, mp_int *a, mp_int *b) -{ - for (size_t i = 0; i < r->nw; i++) { - BignumInt aword = mp_word(a, i), bword = mp_word(b, i); - r->w[i] = aword | bword; - } -} - -void mp_xor_into(mp_int *r, mp_int *a, mp_int *b) -{ - for (size_t i = 0; i < r->nw; i++) { - BignumInt aword = mp_word(a, i), bword = mp_word(b, i); - r->w[i] = aword ^ bword; - } -} - -void mp_bic_into(mp_int *r, mp_int *a, mp_int *b) -{ - for (size_t i = 0; i < r->nw; i++) { - BignumInt aword = mp_word(a, i), bword = mp_word(b, i); - r->w[i] = aword & ~bword; - } -} - -static void mp_cond_negate(mp_int *r, mp_int *x, unsigned yes) -{ - BignumCarry carry = yes; - BignumInt flip = -(BignumInt)yes; - for (size_t i = 0; i < r->nw; i++) { - BignumInt xword = mp_word(x, i); - xword ^= flip; - BignumADC(r->w[i], carry, 0, xword, carry); - } -} - -/* - * Similar to mp_add_masked_into, but takes a C integer instead of an - * mp_int as the masked operand. - */ -static BignumCarry mp_add_masked_integer_into( - BignumInt *w_out, size_t rw, mp_int *a, uintmax_t b, - BignumInt b_and, BignumInt b_xor, BignumCarry carry) -{ - for (size_t i = 0; i < rw; i++) { - BignumInt aword = mp_word(a, i); - BignumInt bword = b; - b = shift_right_by_one_word(b); - BignumInt out; - bword = (bword ^ b_xor) & b_and; - BignumADC(out, carry, aword, bword, carry); - if (w_out) - w_out[i] = out; - } - return carry; -} - -void mp_add_integer_into(mp_int *r, mp_int *a, uintmax_t n) -{ - mp_add_masked_integer_into(r->w, r->nw, a, n, ~(BignumInt)0, 0, 0); -} - -void mp_sub_integer_into(mp_int *r, mp_int *a, uintmax_t n) -{ - mp_add_masked_integer_into(r->w, r->nw, a, n, - ~(BignumInt)0, ~(BignumInt)0, 1); -} - -/* - * Sets r to a + n << (word_index * BIGNUM_INT_BITS), treating - * word_index as secret data. - */ -static void mp_add_integer_into_shifted_by_words( - mp_int *r, mp_int *a, uintmax_t n, size_t word_index) -{ - unsigned indicator = 0; - BignumCarry carry = 0; - - for (size_t i = 0; i < r->nw; i++) { - /* indicator becomes 1 when we reach the index that the least - * significant bits of n want to be placed at, and it stays 1 - * thereafter. */ - indicator |= 1 ^ normalise_to_1(i ^ word_index); - - /* If indicator is 1, we add the low bits of n into r, and - * shift n down. If it's 0, we add zero bits into r, and - * leave n alone. */ - BignumInt bword = n & -(BignumInt)indicator; - uintmax_t new_n = shift_right_by_one_word(n); - n ^= (n ^ new_n) & -(uintmax_t)indicator; - - BignumInt aword = mp_word(a, i); - BignumInt out; - BignumADC(out, carry, aword, bword, carry); - r->w[i] = out; - } -} - -void mp_mul_integer_into(mp_int *r, mp_int *a, uint16_t n) -{ - BignumInt carry = 0, mult = n; - for (size_t i = 0; i < r->nw; i++) { - BignumInt aword = mp_word(a, i); - BignumMULADD(carry, r->w[i], aword, mult, carry); - } - assert(!carry); -} - -void mp_cond_add_into(mp_int *r, mp_int *a, mp_int *b, unsigned yes) -{ - BignumInt mask = -(BignumInt)(yes & 1); - mp_add_masked_into(r->w, r->nw, a, b, mask, 0, 0); -} - -void mp_cond_sub_into(mp_int *r, mp_int *a, mp_int *b, unsigned yes) -{ - BignumInt mask = -(BignumInt)(yes & 1); - mp_add_masked_into(r->w, r->nw, a, b, mask, mask, 1 & mask); -} - -/* - * Ordered comparison between unsigned numbers is done by subtracting - * one from the other and looking at the output carry. - */ -unsigned mp_cmp_hs(mp_int *a, mp_int *b) -{ - size_t rw = size_t_max(a->nw, b->nw); - return mp_add_masked_into(NULL, rw, a, b, ~(BignumInt)0, ~(BignumInt)0, 1); -} - -unsigned mp_hs_integer(mp_int *x, uintmax_t n) -{ - BignumInt carry = 1; - size_t nwords = sizeof(n)/BIGNUM_INT_BYTES; - for (size_t i = 0, e = size_t_max(x->nw, nwords); i < e; i++) { - BignumInt nword = n; - n = shift_right_by_one_word(n); - BignumInt dummy_out; - BignumADC(dummy_out, carry, mp_word(x, i), ~nword, carry); - (void)dummy_out; - } - return carry; -} - -/* - * Equality comparison is done by bitwise XOR of the input numbers, - * ORing together all the output words, and normalising the result - * using our careful normalise_to_1 helper function. - */ -unsigned mp_cmp_eq(mp_int *a, mp_int *b) -{ - BignumInt diff = 0; - for (size_t i = 0, limit = size_t_max(a->nw, b->nw); i < limit; i++) - diff |= mp_word(a, i) ^ mp_word(b, i); - return 1 ^ normalise_to_1(diff); /* return 1 if diff _is_ zero */ -} - -unsigned mp_eq_integer(mp_int *x, uintmax_t n) -{ - BignumInt diff = 0; - size_t nwords = sizeof(n)/BIGNUM_INT_BYTES; - for (size_t i = 0, e = size_t_max(x->nw, nwords); i < e; i++) { - BignumInt nword = n; - n = shift_right_by_one_word(n); - diff |= mp_word(x, i) ^ nword; - } - return 1 ^ normalise_to_1(diff); /* return 1 if diff _is_ zero */ -} - -static void mp_neg_into(mp_int *r, mp_int *a) -{ - mp_int zero; - zero.nw = 0; - mp_sub_into(r, &zero, a); -} - -mp_int *mp_add(mp_int *x, mp_int *y) -{ - mp_int *r = mp_make_sized(size_t_max(x->nw, y->nw) + 1); - mp_add_into(r, x, y); - return r; -} - -mp_int *mp_sub(mp_int *x, mp_int *y) -{ - mp_int *r = mp_make_sized(size_t_max(x->nw, y->nw)); - mp_sub_into(r, x, y); - return r; -} - -/* - * Internal routine: multiply and accumulate in the trivial O(N^2) - * way. Sets r <- r + a*b. - */ -static void mp_mul_add_simple(mp_int *r, mp_int *a, mp_int *b) -{ - BignumInt *aend = a->w + a->nw, *bend = b->w + b->nw, *rend = r->w + r->nw; - - for (BignumInt *ap = a->w, *rp = r->w; - ap < aend && rp < rend; ap++, rp++) { - - BignumInt adata = *ap, carry = 0, *rq = rp; - - for (BignumInt *bp = b->w; bp < bend && rq < rend; bp++, rq++) { - BignumInt bdata = bp < bend ? *bp : 0; - BignumMULADD2(carry, *rq, adata, bdata, *rq, carry); - } - - for (; rq < rend; rq++) - BignumADC(*rq, carry, carry, *rq, 0); - } -} - -#ifndef KARATSUBA_THRESHOLD /* allow redefinition via -D for testing */ -#define KARATSUBA_THRESHOLD 24 -#endif - -static inline size_t mp_mul_scratchspace_unary(size_t n) -{ - /* - * Simplistic and overcautious bound on the amount of scratch - * space that the recursive multiply function will need. - * - * The rationale is: on the main Karatsuba branch of - * mp_mul_internal, which is the most space-intensive one, we - * allocate space for (a0+a1) and (b0+b1) (each just over half the - * input length n) and their product (the sum of those sizes, i.e. - * just over n itself). Then in order to actually compute the - * product, we do a recursive multiplication of size just over n. - * - * If all those 'just over' weren't there, and everything was - * _exactly_ half the length, you'd get the amount of space for a - * size-n multiply defined by the recurrence M(n) = 2n + M(n/2), - * which is satisfied by M(n) = 4n. But instead it's (2n plus a - * word or two) and M(n/2 plus a word or two). On the assumption - * that there's still some constant k such that M(n) <= kn, this - * gives us kn = 2n + w + k(n/2 + w), where w is a small constant - * (one or two words). That simplifies to kn/2 = 2n + (k+1)w, and - * since we don't even _start_ needing scratch space until n is at - * least 50, we can bound 2n + (k+1)w above by 3n, giving k=6. - * - * So I claim that 6n words of scratch space will suffice, and I - * check that by assertion at every stage of the recursion. - */ - return n * 6; -} - -static size_t mp_mul_scratchspace(size_t rw, size_t aw, size_t bw) -{ - size_t inlen = size_t_min(rw, size_t_max(aw, bw)); - return mp_mul_scratchspace_unary(inlen); -} - -static void mp_mul_internal(mp_int *r, mp_int *a, mp_int *b, mp_int scratch) -{ - size_t inlen = size_t_min(r->nw, size_t_max(a->nw, b->nw)); - assert(scratch.nw >= mp_mul_scratchspace_unary(inlen)); - - mp_clear(r); - - if (inlen < KARATSUBA_THRESHOLD || a->nw == 0 || b->nw == 0) { - /* - * The input numbers are too small to bother optimising. Go - * straight to the simple primitive approach. - */ - mp_mul_add_simple(r, a, b); - return; - } - - /* - * Karatsuba divide-and-conquer algorithm. We cut each input in - * half, so that it's expressed as two big 'digits' in a giant - * base D: - * - * a = a_1 D + a_0 - * b = b_1 D + b_0 - * - * Then the product is of course - * - * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 - * - * and we compute the three coefficients by recursively calling - * ourself to do half-length multiplications. - * - * The clever bit that makes this worth doing is that we only need - * _one_ half-length multiplication for the central coefficient - * rather than the two that it obviouly looks like, because we can - * use a single multiplication to compute - * - * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0 - * - * and then we subtract the other two coefficients (a_1 b_1 and - * a_0 b_0) which we were computing anyway. - * - * Hence we get to multiply two numbers of length N in about three - * times as much work as it takes to multiply numbers of length - * N/2, which is obviously better than the four times as much work - * it would take if we just did a long conventional multiply. - */ - - /* Break up the input as botlen + toplen, with botlen >= toplen. - * The 'base' D is equal to 2^{botlen * BIGNUM_INT_BITS}. */ - size_t toplen = inlen / 2; - size_t botlen = inlen - toplen; - - /* Alias bignums that address the two halves of a,b, and useful - * pieces of r. */ - mp_int a0 = mp_make_alias(a, 0, botlen); - mp_int b0 = mp_make_alias(b, 0, botlen); - mp_int a1 = mp_make_alias(a, botlen, toplen); - mp_int b1 = mp_make_alias(b, botlen, toplen); - mp_int r0 = mp_make_alias(r, 0, botlen*2); - mp_int r1 = mp_make_alias(r, botlen, r->nw); - mp_int r2 = mp_make_alias(r, botlen*2, r->nw); - - /* Recurse to compute a0*b0 and a1*b1, in their correct positions - * in the output bignum. They can't overlap. */ - mp_mul_internal(&r0, &a0, &b0, scratch); - mp_mul_internal(&r2, &a1, &b1, scratch); - - if (r->nw < inlen*2) { - /* - * The output buffer isn't large enough to require the whole - * product, so some of a1*b1 won't have been stored. In that - * case we won't try to do the full Karatsuba optimisation; - * we'll just recurse again to compute a0*b1 and a1*b0 - or at - * least as much of them as the output buffer size requires - - * and add each one in. - */ - mp_int s = mp_alloc_from_scratch( - &scratch, size_t_min(botlen+toplen, r1.nw)); - - mp_mul_internal(&s, &a0, &b1, scratch); - mp_add_into(&r1, &r1, &s); - mp_mul_internal(&s, &a1, &b0, scratch); - mp_add_into(&r1, &r1, &s); - return; - } - - /* a0+a1 and b0+b1 */ - mp_int asum = mp_alloc_from_scratch(&scratch, botlen+1); - mp_int bsum = mp_alloc_from_scratch(&scratch, botlen+1); - mp_add_into(&asum, &a0, &a1); - mp_add_into(&bsum, &b0, &b1); - - /* Their product */ - mp_int product = mp_alloc_from_scratch(&scratch, botlen*2+1); - mp_mul_internal(&product, &asum, &bsum, scratch); - - /* Subtract off the outer terms we already have */ - mp_sub_into(&product, &product, &r0); - mp_sub_into(&product, &product, &r2); - - /* And add it in with the right offset. */ - mp_add_into(&r1, &r1, &product); -} - -void mp_mul_into(mp_int *r, mp_int *a, mp_int *b) -{ - mp_int *scratch = mp_make_sized(mp_mul_scratchspace(r->nw, a->nw, b->nw)); - mp_mul_internal(r, a, b, *scratch); - mp_free(scratch); -} - -mp_int *mp_mul(mp_int *x, mp_int *y) -{ - mp_int *r = mp_make_sized(x->nw + y->nw); - mp_mul_into(r, x, y); - return r; -} - -void mp_lshift_fixed_into(mp_int *r, mp_int *a, size_t bits) -{ - size_t words = bits / BIGNUM_INT_BITS; - size_t bitoff = bits % BIGNUM_INT_BITS; - - for (size_t i = r->nw; i-- > 0 ;) { - if (i < words) { - r->w[i] = 0; - } else { - r->w[i] = mp_word(a, i - words); - if (bitoff != 0) { - r->w[i] <<= bitoff; - if (i > words) - r->w[i] |= mp_word(a, i - words - 1) >> - (BIGNUM_INT_BITS - bitoff); - } - } - } -} - -void mp_rshift_fixed_into(mp_int *r, mp_int *a, size_t bits) -{ - size_t words = bits / BIGNUM_INT_BITS; - size_t bitoff = bits % BIGNUM_INT_BITS; - - for (size_t i = 0; i < r->nw; i++) { - r->w[i] = mp_word(a, i + words); - if (bitoff != 0) { - r->w[i] >>= bitoff; - r->w[i] |= mp_word(a, i + words + 1) << (BIGNUM_INT_BITS - bitoff); - } - } -} - -mp_int *mp_lshift_fixed(mp_int *x, size_t bits) -{ - size_t words = (bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS; - mp_int *r = mp_make_sized(x->nw + words); - mp_lshift_fixed_into(r, x, bits); - return r; -} - -mp_int *mp_rshift_fixed(mp_int *x, size_t bits) -{ - size_t words = bits / BIGNUM_INT_BITS; - size_t nw = x->nw - size_t_min(x->nw, words); - mp_int *r = mp_make_sized(size_t_max(nw, 1)); - mp_rshift_fixed_into(r, x, bits); - return r; -} - -/* - * Safe right shift is done using the same technique as - * trim_leading_zeroes above: you make an n-word left shift by - * composing an appropriate subset of power-of-2-sized shifts, so it - * takes log_2(n) loop iterations each of which does a different shift - * by a power of 2 words, using the usual bit twiddling to make the - * whole shift conditional on the appropriate bit of n. - */ -static void mp_rshift_safe_in_place(mp_int *r, size_t bits) -{ - size_t wordshift = bits / BIGNUM_INT_BITS; - size_t bitshift = bits % BIGNUM_INT_BITS; - - unsigned clear = (r->nw - wordshift) >> (CHAR_BIT * sizeof(size_t) - 1); - mp_cond_clear(r, clear); - - for (unsigned bit = 0; r->nw >> bit; bit++) { - size_t word_offset = (size_t)1 << bit; - BignumInt mask = -(BignumInt)((wordshift >> bit) & 1); - for (size_t i = 0; i < r->nw; i++) { - BignumInt w = mp_word(r, i + word_offset); - r->w[i] ^= (r->w[i] ^ w) & mask; - } - } - - /* - * That's done the shifting by words; now we do the shifting by - * bits. - */ - for (unsigned bit = 0; bit < BIGNUM_INT_BITS_BITS; bit++) { - unsigned shift = 1 << bit, upshift = BIGNUM_INT_BITS - shift; - BignumInt mask = -(BignumInt)((bitshift >> bit) & 1); - for (size_t i = 0; i < r->nw; i++) { - BignumInt w = ((r->w[i] >> shift) | (mp_word(r, i+1) << upshift)); - r->w[i] ^= (r->w[i] ^ w) & mask; - } - } -} - -mp_int *mp_rshift_safe(mp_int *x, size_t bits) -{ - mp_int *r = mp_copy(x); - mp_rshift_safe_in_place(r, bits); - return r; -} - -void mp_rshift_safe_into(mp_int *r, mp_int *x, size_t bits) -{ - mp_copy_into(r, x); - mp_rshift_safe_in_place(r, bits); -} - -static void mp_lshift_safe_in_place(mp_int *r, size_t bits) -{ - size_t wordshift = bits / BIGNUM_INT_BITS; - size_t bitshift = bits % BIGNUM_INT_BITS; - - /* - * Same strategy as mp_rshift_safe_in_place, but of course the - * other way up. - */ - - unsigned clear = (r->nw - wordshift) >> (CHAR_BIT * sizeof(size_t) - 1); - mp_cond_clear(r, clear); - - for (unsigned bit = 0; r->nw >> bit; bit++) { - size_t word_offset = (size_t)1 << bit; - BignumInt mask = -(BignumInt)((wordshift >> bit) & 1); - for (size_t i = r->nw; i-- > 0 ;) { - BignumInt w = mp_word(r, i - word_offset); - r->w[i] ^= (r->w[i] ^ w) & mask; - } - } - - size_t downshift = BIGNUM_INT_BITS - bitshift; - size_t no_shift = (downshift >> BIGNUM_INT_BITS_BITS); - downshift &= ~-(size_t)no_shift; - BignumInt downshifted_mask = ~-(BignumInt)no_shift; - - for (size_t i = r->nw; i-- > 0 ;) { - r->w[i] = (r->w[i] << bitshift) | - ((mp_word(r, i-1) >> downshift) & downshifted_mask); - } -} - -void mp_lshift_safe_into(mp_int *r, mp_int *x, size_t bits) -{ - mp_copy_into(r, x); - mp_lshift_safe_in_place(r, bits); -} - -void mp_reduce_mod_2to(mp_int *x, size_t p) -{ - size_t word = p / BIGNUM_INT_BITS; - size_t mask = ((size_t)1 << (p % BIGNUM_INT_BITS)) - 1; - for (; word < x->nw; word++) { - x->w[word] &= mask; - mask = 0; - } -} - -/* - * Inverse mod 2^n is computed by an iterative technique which doubles - * the number of bits at each step. - */ -mp_int *mp_invert_mod_2to(mp_int *x, size_t p) -{ - /* Input checks: x must be coprime to the modulus, i.e. odd, and p - * can't be zero */ - assert(x->nw > 0); - assert(x->w[0] & 1); - assert(p > 0); - - size_t rw = (p + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS; - rw = size_t_max(rw, 1); - mp_int *r = mp_make_sized(rw); - - size_t mul_scratchsize = mp_mul_scratchspace(2*rw, rw, rw); - mp_int *scratch_orig = mp_make_sized(6 * rw + mul_scratchsize); - mp_int scratch_per_iter = *scratch_orig; - mp_int mul_scratch = mp_alloc_from_scratch( - &scratch_per_iter, mul_scratchsize); - - r->w[0] = 1; - - for (size_t b = 1; b < p; b <<= 1) { - /* - * In each step of this iteration, we have the inverse of x - * mod 2^b, and we want the inverse of x mod 2^{2b}. - * - * Write B = 2^b for convenience, so we want x^{-1} mod B^2. - * Let x = x_0 + B x_1 + k B^2, with 0 <= x_0,x_1 < B. - * - * We want to find r_0 and r_1 such that - * (r_1 B + r_0) (x_1 B + x_0) == 1 (mod B^2) - * - * To begin with, we know r_0 must be the inverse mod B of - * x_0, i.e. of x, i.e. it is the inverse we computed in the - * previous iteration. So now all we need is r_1. - * - * Multiplying out, neglecting multiples of B^2, and writing - * x_0 r_0 = K B + 1, we have - * - * r_1 x_0 B + r_0 x_1 B + K B == 0 (mod B^2) - * => r_1 x_0 B == - r_0 x_1 B - K B (mod B^2) - * => r_1 x_0 == - r_0 x_1 - K (mod B) - * => r_1 == r_0 (- r_0 x_1 - K) (mod B) - * - * (the last step because we multiply through by the inverse - * of x_0, which we already know is r_0). - */ - - mp_int scratch_this_iter = scratch_per_iter; - size_t Bw = (b + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS; - size_t B2w = (2*b + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS; - - /* Start by finding K: multiply x_0 by r_0, and shift down. */ - mp_int x0 = mp_alloc_from_scratch(&scratch_this_iter, Bw); - mp_copy_into(&x0, x); - mp_reduce_mod_2to(&x0, b); - mp_int r0 = mp_make_alias(r, 0, Bw); - mp_int Kshift = mp_alloc_from_scratch(&scratch_this_iter, B2w); - mp_mul_internal(&Kshift, &x0, &r0, mul_scratch); - mp_int K = mp_alloc_from_scratch(&scratch_this_iter, Bw); - mp_rshift_fixed_into(&K, &Kshift, b); - - /* Now compute the product r_0 x_1, reusing the space of Kshift. */ - mp_int x1 = mp_alloc_from_scratch(&scratch_this_iter, Bw); - mp_rshift_fixed_into(&x1, x, b); - mp_reduce_mod_2to(&x1, b); - mp_int r0x1 = mp_make_alias(&Kshift, 0, Bw); - mp_mul_internal(&r0x1, &r0, &x1, mul_scratch); - - /* Add K to that. */ - mp_add_into(&r0x1, &r0x1, &K); - - /* Negate it. */ - mp_neg_into(&r0x1, &r0x1); - - /* Multiply by r_0. */ - mp_int r1 = mp_alloc_from_scratch(&scratch_this_iter, Bw); - mp_mul_internal(&r1, &r0, &r0x1, mul_scratch); - mp_reduce_mod_2to(&r1, b); - - /* That's our r_1, so add it on to r_0 to get the full inverse - * output from this iteration. */ - mp_lshift_fixed_into(&K, &r1, (b % BIGNUM_INT_BITS)); - size_t Bpos = b / BIGNUM_INT_BITS; - mp_int r1_position = mp_make_alias(r, Bpos, B2w-Bpos); - mp_add_into(&r1_position, &r1_position, &K); - } - - /* Finally, reduce mod the precise desired number of bits. */ - mp_reduce_mod_2to(r, p); - - mp_free(scratch_orig); - return r; -} - -static size_t monty_scratch_size(MontyContext *mc) -{ - return 3*mc->rw + mc->pw + mp_mul_scratchspace(mc->pw, mc->rw, mc->rw); -} - -MontyContext *monty_new(mp_int *modulus) -{ - MontyContext *mc = snew(MontyContext); - - mc->rw = modulus->nw; - mc->rbits = mc->rw * BIGNUM_INT_BITS; - mc->pw = mc->rw * 2 + 1; - - mc->m = mp_copy(modulus); - - mc->minus_minv_mod_r = mp_invert_mod_2to(mc->m, mc->rbits); - mp_neg_into(mc->minus_minv_mod_r, mc->minus_minv_mod_r); - - mp_int *r = mp_make_sized(mc->rw + 1); - r->w[mc->rw] = 1; - mc->powers_of_r_mod_m[0] = mp_mod(r, mc->m); - mp_free(r); - - for (size_t j = 1; j < lenof(mc->powers_of_r_mod_m); j++) - mc->powers_of_r_mod_m[j] = mp_modmul( - mc->powers_of_r_mod_m[0], mc->powers_of_r_mod_m[j-1], mc->m); - - mc->scratch = mp_make_sized(monty_scratch_size(mc)); - - return mc; -} - -void monty_free(MontyContext *mc) -{ - mp_free(mc->m); - for (size_t j = 0; j < 3; j++) - mp_free(mc->powers_of_r_mod_m[j]); - mp_free(mc->minus_minv_mod_r); - mp_free(mc->scratch); - smemclr(mc, sizeof(*mc)); - sfree(mc); -} - -/* - * The main Montgomery reduction step. - */ -static mp_int monty_reduce_internal(MontyContext *mc, mp_int *x, mp_int scratch) -{ - /* - * The trick with Montgomery reduction is that on the one hand we - * want to reduce the size of the input by a factor of about r, - * and on the other hand, the two numbers we just multiplied were - * both stored with an extra factor of r multiplied in. So we - * computed ar*br = ab r^2, but we want to return abr, so we need - * to divide by r - and if we can do that by _actually dividing_ - * by r then this also reduces the size of the number. - * - * But we can only do that if the number we're dividing by r is a - * multiple of r. So first we must add an adjustment to it which - * clears its bottom 'rbits' bits. That adjustment must be a - * multiple of m in order to leave the residue mod n unchanged, so - * the question is, what multiple of m can we add to x to make it - * congruent to 0 mod r? And the answer is, x * (-m)^{-1} mod r. - */ - - /* x mod r */ - mp_int x_lo = mp_make_alias(x, 0, mc->rbits); - - /* x * (-m)^{-1}, i.e. the number we want to multiply by m */ - mp_int k = mp_alloc_from_scratch(&scratch, mc->rw); - mp_mul_internal(&k, &x_lo, mc->minus_minv_mod_r, scratch); - - /* m times that, i.e. the number we want to add to x */ - mp_int mk = mp_alloc_from_scratch(&scratch, mc->pw); - mp_mul_internal(&mk, mc->m, &k, scratch); - - /* Add it to x */ - mp_add_into(&mk, x, &mk); - - /* Reduce mod r, by simply making an alias to the upper words of x */ - mp_int toret = mp_make_alias(&mk, mc->rw, mk.nw - mc->rw); - - /* - * We'll generally be doing this after a multiplication of two - * fully reduced values. So our input could be anything up to m^2, - * and then we added up to rm to it. Hence, the maximum value is - * rm+m^2, and after dividing by r, that becomes r + m(m/r) < 2r. - * So a single trial-subtraction will finish reducing to the - * interval [0,m). - */ - mp_cond_sub_into(&toret, &toret, mc->m, mp_cmp_hs(&toret, mc->m)); - return toret; -} - -void monty_mul_into(MontyContext *mc, mp_int *r, mp_int *x, mp_int *y) -{ - assert(x->nw <= mc->rw); - assert(y->nw <= mc->rw); - - mp_int scratch = *mc->scratch; - mp_int tmp = mp_alloc_from_scratch(&scratch, 2*mc->rw); - mp_mul_into(&tmp, x, y); - mp_int reduced = monty_reduce_internal(mc, &tmp, scratch); - mp_copy_into(r, &reduced); - mp_clear(mc->scratch); -} - -mp_int *monty_mul(MontyContext *mc, mp_int *x, mp_int *y) -{ - mp_int *toret = mp_make_sized(mc->rw); - monty_mul_into(mc, toret, x, y); - return toret; -} - -mp_int *monty_modulus(MontyContext *mc) -{ - return mc->m; -} - -mp_int *monty_identity(MontyContext *mc) -{ - return mc->powers_of_r_mod_m[0]; -} - -mp_int *monty_invert(MontyContext *mc, mp_int *x) -{ - /* Given xr, we want to return x^{-1}r = (xr)^{-1} r^2 = - * monty_reduce((xr)^{-1} r^3) */ - mp_int *tmp = mp_invert(x, mc->m); - mp_int *toret = monty_mul(mc, tmp, mc->powers_of_r_mod_m[2]); - mp_free(tmp); - return toret; -} - -/* - * Importing a number into Montgomery representation involves - * multiplying it by r and reducing mod m. We use the general-purpose - * mp_modmul for this, in case the input number is out of range. - */ -mp_int *monty_import(MontyContext *mc, mp_int *x) -{ - return mp_modmul(x, mc->powers_of_r_mod_m[0], mc->m); -} - -void monty_import_into(MontyContext *mc, mp_int *r, mp_int *x) -{ - mp_int *imported = monty_import(mc, x); - mp_copy_into(r, imported); - mp_free(imported); -} - -/* - * Exporting a number means multiplying it by r^{-1}, which is exactly - * what monty_reduce does anyway, so we just do that. - */ -void monty_export_into(MontyContext *mc, mp_int *r, mp_int *x) -{ - assert(x->nw <= 2*mc->rw); - mp_int reduced = monty_reduce_internal(mc, x, *mc->scratch); - mp_copy_into(r, &reduced); - mp_clear(mc->scratch); -} - -mp_int *monty_export(MontyContext *mc, mp_int *x) -{ - mp_int *toret = mp_make_sized(mc->rw); - monty_export_into(mc, toret, x); - return toret; -} - -static void monty_reduce(MontyContext *mc, mp_int *x) -{ - mp_int reduced = monty_reduce_internal(mc, x, *mc->scratch); - mp_copy_into(x, &reduced); - mp_clear(mc->scratch); -} - -mp_int *monty_pow(MontyContext *mc, mp_int *base, mp_int *exponent) -{ - /* square builds up powers of the form base^{2^i}. */ - mp_int *square = mp_copy(base); - size_t i = 0; - - /* out accumulates the output value. Starts at 1 (in Montgomery - * representation) and we multiply in each base^{2^i}. */ - mp_int *out = mp_copy(mc->powers_of_r_mod_m[0]); - - /* tmp holds each product we compute and reduce. */ - mp_int *tmp = mp_make_sized(mc->rw * 2); - - while (true) { - mp_mul_into(tmp, out, square); - monty_reduce(mc, tmp); - mp_select_into(out, out, tmp, mp_get_bit(exponent, i)); - - if (++i >= exponent->nw * BIGNUM_INT_BITS) - break; - - mp_mul_into(tmp, square, square); - monty_reduce(mc, tmp); - mp_copy_into(square, tmp); - } - - mp_free(square); - mp_free(tmp); - mp_clear(mc->scratch); - return out; -} - -mp_int *mp_modpow(mp_int *base, mp_int *exponent, mp_int *modulus) -{ - assert(modulus->nw > 0); - assert(modulus->w[0] & 1); - - MontyContext *mc = monty_new(modulus); - mp_int *m_base = monty_import(mc, base); - mp_int *m_out = monty_pow(mc, m_base, exponent); - mp_int *out = monty_export(mc, m_out); - mp_free(m_base); - mp_free(m_out); - monty_free(mc); - return out; -} - -/* - * Given two input integers a,b which are not both even, computes d = - * gcd(a,b) and also two integers A,B such that A*a - B*b = d. A,B - * will be the minimal non-negative pair satisfying that criterion, - * which is equivalent to saying that 0 <= A < b/d and 0 <= B < a/d. - * - * This algorithm is an adapted form of Stein's algorithm, which - * computes gcd(a,b) using only addition and bit shifts (i.e. without - * needing general division), using the following rules: - * - * - if both of a,b are even, divide off a common factor of 2 - * - if one of a,b (WLOG a) is even, then gcd(a,b) = gcd(a/2,b), so - * just divide a by 2 - * - if both of a,b are odd, then WLOG a>b, and gcd(a,b) = - * gcd(b,(a-b)/2). - * - * Sometimes this function is used for modular inversion, in which - * case we already know we expect the two inputs to be coprime, so to - * save time the 'both even' initial case is assumed not to arise (or - * to have been handled already by the caller). So this function just - * performs a sequence of reductions in the following form: - * - * - if a,b are both odd, sort them so that a > b, and replace a with - * b-a; otherwise sort them so that a is the even one - * - either way, now a is even and b is odd, so divide a by 2. - * - * The big change to Stein's algorithm is that we need the Bezout - * coefficients as output, not just the gcd. So we need to know how to - * generate those in each case, based on the coefficients from the - * reduced pair of numbers: - * - * - If a is even, and u,v are such that u*(a/2) + v*b = d: - * + if u is also even, then this is just (u/2)*a + v*b = d - * + otherwise, (u+b)*(a/2) + (v-a/2)*b is also equal to d, and - * since u and b are both odd, (u+b)/2 is an integer, so we have - * ((u+b)/2)*a + (v-a/2)*b = d. - * - * - If a,b are both odd, and u,v are such that u*b + v*(a-b) = d, - * then v*a + (u-v)*b = d. - * - * In the case where we passed from (a,b) to (b,(a-b)/2), we regard it - * as having first subtracted b from a and then halved a, so both of - * these transformations must be done in sequence. - * - * The code below transforms this from a recursive to an iterative - * algorithm. We first reduce a,b to 0,1, recording at each stage - * whether we did the initial subtraction, and whether we had to swap - * the two values; then we iterate backwards over that record of what - * we did, applying the above rules for building up the Bezout - * coefficients as we go. Of course, all the case analysis is done by - * the usual bit-twiddling conditionalisation to avoid data-dependent - * control flow. - * - * Also, since these mp_ints are generally treated as unsigned, we - * store the coefficients by absolute value, with the semantics that - * they always have opposite sign, and in the unwinding loop we keep a - * bit indicating whether Aa-Bb is currently expected to be +d or -d, - * so that we can do one final conditional adjustment if it's -d. - * - * Once the reduction rules have managed to reduce the input numbers - * to (0,d), then they are stable (the next reduction will always - * divide the even one by 2, which maps 0 to 0). So it doesn't matter - * if we do more steps of the algorithm than necessary; hence, for - * constant time, we just need to find the maximum number we could - * _possibly_ require, and do that many. - * - * If a,b < 2^n, at most 2n iterations are required. Proof: consider - * the quantity Q = log_2(a) + log_2(b). Every step halves one of the - * numbers (and may also reduce one of them further by doing a - * subtraction beforehand, but in the worst case, not by much or not - * at all). So Q reduces by at least 1 per iteration, and it starts - * off with a value at most 2n. - * - * The worst case inputs (I think) are where x=2^{n-1} and y=2^n-1 - * (i.e. x is a power of 2 and y is all 1s). In that situation, the - * first n-1 steps repeatedly halve x until it's 1, and then there are - * n further steps each of which subtracts 1 from y and halves it. - */ -static void mp_bezout_into(mp_int *a_coeff_out, mp_int *b_coeff_out, - mp_int *gcd_out, mp_int *a_in, mp_int *b_in) -{ - size_t nw = size_t_max(1, size_t_max(a_in->nw, b_in->nw)); - - /* Make mutable copies of the input numbers */ - mp_int *a = mp_make_sized(nw), *b = mp_make_sized(nw); - mp_copy_into(a, a_in); - mp_copy_into(b, b_in); - - /* Space to build up the output coefficients, with an extra word - * so that intermediate values can overflow off the top and still - * right-shift back down to the correct value */ - mp_int *ac = mp_make_sized(nw + 1), *bc = mp_make_sized(nw + 1); - - /* And a general-purpose temp register */ - mp_int *tmp = mp_make_sized(nw); - - /* Space to record the sequence of reduction steps to unwind. We - * make it a BignumInt for no particular reason except that (a) - * mp_make_sized conveniently zeroes the allocation and mp_free - * wipes it, and (b) this way I can use mp_dump() if I have to - * debug this code. */ - size_t steps = 2 * nw * BIGNUM_INT_BITS; - mp_int *record = mp_make_sized( - (steps*2 + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); - - for (size_t step = 0; step < steps; step++) { - /* - * If a and b are both odd, we want to sort them so that a is - * larger. But if one is even, we want to sort them so that a - * is the even one. - */ - unsigned swap_if_both_odd = mp_cmp_hs(b, a); - unsigned swap_if_one_even = a->w[0] & 1; - unsigned both_odd = a->w[0] & b->w[0] & 1; - unsigned swap = swap_if_one_even ^ ( - (swap_if_both_odd ^ swap_if_one_even) & both_odd); - - mp_cond_swap(a, b, swap); - - /* - * If a,b are both odd, then a is the larger number, so - * subtract the smaller one from it. - */ - mp_cond_sub_into(a, a, b, both_odd); - - /* - * Now a is even, so divide it by two. - */ - mp_rshift_fixed_into(a, a, 1); - - /* - * Record the two 1-bit values both_odd and swap. - */ - mp_set_bit(record, step*2, both_odd); - mp_set_bit(record, step*2+1, swap); - } - - /* - * Now we expect to have reduced the two numbers to 0 and d, - * although we don't know which way round. (But we avoid checking - * this by assertion; sometimes we'll need to do this computation - * without giving away that we already know the inputs were bogus. - * So we'd prefer to just press on and return nonsense.) - */ - - if (gcd_out) { - /* - * At this point we can return the actual gcd. Since one of - * a,b is it and the other is zero, the easiest way to get it - * is to add them together. - */ - mp_add_into(gcd_out, a, b); - } - - /* - * If the caller _only_ wanted the gcd, and neither Bezout - * coefficient is even required, we can skip the entire unwind - * stage. - */ - if (a_coeff_out || b_coeff_out) { - - /* - * The Bezout coefficients of a,b at this point are simply 0 - * for whichever of a,b is zero, and 1 for whichever is - * nonzero. The nonzero number equals gcd(a,b), which by - * assumption is odd, so we can do this by just taking the low - * bit of each one. - */ - ac->w[0] = mp_get_bit(a, 0); - bc->w[0] = mp_get_bit(b, 0); - - /* - * Overwrite a,b themselves with those same numbers. This has - * the effect of dividing both of them by d, which will - * arrange that during the unwind stage we generate the - * minimal coefficients instead of a larger pair. - */ - mp_copy_into(a, ac); - mp_copy_into(b, bc); - - /* - * We'll maintain the invariant as we unwind that ac * a - bc - * * b is either +d or -d (or rather, +1/-1 after scaling by - * d), and we'll remember which. (We _could_ keep it at +d the - * whole time, but it would cost more work every time round - * the loop, so it's cheaper to fix that up once at the end.) - * - * Initially, the result is +d if a was the nonzero value after - * reduction, and -d if b was. - */ - unsigned minus_d = b->w[0]; - - for (size_t step = steps; step-- > 0 ;) { - /* - * Recover the data from the step we're unwinding. - */ - unsigned both_odd = mp_get_bit(record, step*2); - unsigned swap = mp_get_bit(record, step*2+1); - - /* - * Unwind the division: if our coefficient of a is odd, we - * adjust the coefficients by +b and +a respectively. - */ - unsigned adjust = ac->w[0] & 1; - mp_cond_add_into(ac, ac, b, adjust); - mp_cond_add_into(bc, bc, a, adjust); - - /* - * Now ac is definitely even, so we divide it by two. - */ - mp_rshift_fixed_into(ac, ac, 1); - - /* - * Now unwind the subtraction, if there was one, by adding - * ac to bc. - */ - mp_cond_add_into(bc, bc, ac, both_odd); - - /* - * Undo the transformation of the input numbers, by - * multiplying a by 2 and then adding b to a (the latter - * only if both_odd). - */ - mp_lshift_fixed_into(a, a, 1); - mp_cond_add_into(a, a, b, both_odd); - - /* - * Finally, undo the swap. If we do swap, this also - * reverses the sign of the current result ac*a+bc*b. - */ - mp_cond_swap(a, b, swap); - mp_cond_swap(ac, bc, swap); - minus_d ^= swap; - } - - /* - * Now we expect to have recovered the input a,b (or rather, - * the versions of them divided by d). But we might find that - * our current result is -d instead of +d, that is, we have - * A',B' such that A'a - B'b = -d. - * - * In that situation, we set A = b-A' and B = a-B', giving us - * Aa-Bb = ab - A'a - ab + B'b = +1. - */ - mp_sub_into(tmp, b, ac); - mp_select_into(ac, ac, tmp, minus_d); - mp_sub_into(tmp, a, bc); - mp_select_into(bc, bc, tmp, minus_d); - - /* - * Now we really are done. Return the outputs. - */ - if (a_coeff_out) - mp_copy_into(a_coeff_out, ac); - if (b_coeff_out) - mp_copy_into(b_coeff_out, bc); - - } - - mp_free(a); - mp_free(b); - mp_free(ac); - mp_free(bc); - mp_free(tmp); - mp_free(record); -} - -mp_int *mp_invert(mp_int *x, mp_int *m) -{ - mp_int *result = mp_make_sized(m->nw); - mp_bezout_into(result, NULL, NULL, x, m); - return result; -} - -void mp_gcd_into(mp_int *a, mp_int *b, mp_int *gcd, mp_int *A, mp_int *B) -{ - /* - * Identify shared factors of 2. To do this we OR the two numbers - * to get something whose lowest set bit is in the right place, - * remove all higher bits by ANDing it with its own negation, and - * use mp_get_nbits to find the location of the single remaining - * set bit. - */ - mp_int *tmp = mp_make_sized(size_t_max(a->nw, b->nw)); - for (size_t i = 0; i < tmp->nw; i++) - tmp->w[i] = mp_word(a, i) | mp_word(b, i); - BignumCarry carry = 1; - for (size_t i = 0; i < tmp->nw; i++) { - BignumInt negw; - BignumADC(negw, carry, 0, ~tmp->w[i], carry); - tmp->w[i] &= negw; - } - size_t shift = mp_get_nbits(tmp) - 1; - mp_free(tmp); - - /* - * Make copies of a,b with those shared factors of 2 divided off, - * so that at least one is odd (which is the precondition for - * mp_bezout_into). Compute the gcd of those. - */ - mp_int *as = mp_rshift_safe(a, shift); - mp_int *bs = mp_rshift_safe(b, shift); - mp_bezout_into(A, B, gcd, as, bs); - mp_free(as); - mp_free(bs); - - /* - * And finally shift the gcd back up (unless the caller didn't - * even ask for it), to put the shared factors of 2 back in. - */ - if (gcd) - mp_lshift_safe_in_place(gcd, shift); -} - -mp_int *mp_gcd(mp_int *a, mp_int *b) -{ - mp_int *gcd = mp_make_sized(size_t_min(a->nw, b->nw)); - mp_gcd_into(a, b, gcd, NULL, NULL); - return gcd; -} - -unsigned mp_coprime(mp_int *a, mp_int *b) -{ - mp_int *gcd = mp_gcd(a, b); - unsigned toret = mp_eq_integer(gcd, 1); - mp_free(gcd); - return toret; -} - -static uint32_t recip_approx_32(uint32_t x) -{ - /* - * Given an input x in [2^31,2^32), i.e. a uint32_t with its high - * bit set, this function returns an approximation to 2^63/x, - * computed using only multiplications and bit shifts just in case - * the C divide operator has non-constant time (either because the - * underlying machine instruction does, or because the operator - * expands to a library function on a CPU without hardware - * division). - * - * The coefficients are derived from those of the degree-9 - * polynomial which is the minimax-optimal approximation to that - * function on the given interval (generated using the Remez - * algorithm), converted into integer arithmetic with shifts used - * to maximise the number of significant bits at every state. (A - * sort of 'static floating point' - the exponent is statically - * known at every point in the code, so it never needs to be - * stored at run time or to influence runtime decisions.) - * - * Exhaustive iteration over the whole input space shows the - * largest possible error to be 1686.54. (The input value - * attaining that bound is 4226800006 == 0xfbefd986, whose true - * reciprocal is 2182116973.540... == 0x8210766d.8a6..., whereas - * this function returns 2182115287 == 0x82106fd7.) - */ - uint64_t r = 0x92db03d6ULL; - r = 0xf63e71eaULL - ((r*x) >> 34); - r = 0xb63721e8ULL - ((r*x) >> 34); - r = 0x9c2da00eULL - ((r*x) >> 33); - r = 0xaada0bb8ULL - ((r*x) >> 32); - r = 0xf75cd403ULL - ((r*x) >> 31); - r = 0xecf97a41ULL - ((r*x) >> 31); - r = 0x90d876cdULL - ((r*x) >> 31); - r = 0x6682799a0ULL - ((r*x) >> 26); - return r; -} - -void mp_divmod_into(mp_int *n, mp_int *d, mp_int *q_out, mp_int *r_out) -{ - assert(!mp_eq_integer(d, 0)); - - /* - * We do division by using Newton-Raphson iteration to converge to - * the reciprocal of d (or rather, R/d for R a sufficiently large - * power of 2); then we multiply that reciprocal by n; and we - * finish up with conditional subtraction. - * - * But we have to do it in a fixed number of N-R iterations, so we - * need some error analysis to know how many we might need. - * - * The iteration is derived by defining f(r) = d - R/r. - * Differentiating gives f'(r) = R/r^2, and the Newton-Raphson - * formula applied to those functions gives - * - * r_{i+1} = r_i - f(r_i) / f'(r_i) - * = r_i - (d - R/r_i) r_i^2 / R - * = r_i (2 R - d r_i) / R - * - * Now let e_i be the error in a given iteration, in the sense - * that - * - * d r_i = R + e_i - * i.e. e_i/R = (r_i - r_true) / r_true - * - * so e_i is the _relative_ error in r_i. - * - * We must also introduce a rounding-error term, because the - * division by R always gives an integer. This might make the - * output off by up to 1 (in the negative direction, because - * right-shifting gives floor of the true quotient). So when we - * divide by R, we must imagine adding some f in [0,1). Then we - * have - * - * d r_{i+1} = d r_i (2 R - d r_i) / R - d f - * = (R + e_i) (R - e_i) / R - d f - * = (R^2 - e_i^2) / R - d f - * = R - (e_i^2 / R + d f) - * => e_{i+1} = - (e_i^2 / R + d f) - * - * The sum of two positive quantities is bounded above by twice - * their max, and max |f| = 1, so we can bound this as follows: - * - * |e_{i+1}| <= 2 max (e_i^2/R, d) - * |e_{i+1}/R| <= 2 max ((e_i/R)^2, d/R) - * log2 |R/e_{i+1}| <= min (2 log2 |R/e_i|, log2 |R/d|) - 1 - * - * which tells us that the number of 'good' bits - i.e. - * log2(R/e_i) - very nearly doubles at every iteration (apart - * from that subtraction of 1), until it gets to the same size as - * log2(R/d). In other words, the size of R in bits has to be the - * size of denominator we're putting in, _plus_ the amount of - * precision we want to get back out. - * - * So when we multiply n (the input numerator) by our final - * reciprocal approximation r, but actually r differs from R/d by - * up to 2, then it follows that - * - * n/d - nr/R = n/d - [ n (R/d + e) ] / R - * = n/d - [ (n/d) R + n e ] / R - * = -ne/R - * => 0 <= n/d - nr/R < 2n/R - * - * so our computed quotient can differ from the true n/d by up to - * 2n/R. Hence, as long as we also choose R large enough that 2n/R - * is bounded above by a constant, we can guarantee a bounded - * number of final conditional-subtraction steps. - */ - - /* - * Get at least 32 of the most significant bits of the input - * number. - */ - size_t hiword_index = 0; - uint64_t hibits = 0, lobits = 0; - mp_find_highest_nonzero_word_pair(d, 64 - BIGNUM_INT_BITS, - &hiword_index, &hibits, &lobits); - - /* - * Make a shifted combination of those two words which puts the - * topmost bit of the number at bit 63. - */ - size_t shift_up = 0; - for (size_t i = BIGNUM_INT_BITS_BITS; i-- > 0;) { - size_t sl = (size_t)1 << i; /* left shift count */ - size_t sr = 64 - sl; /* complementary right-shift count */ - - /* Should we shift up? */ - unsigned indicator = 1 ^ normalise_to_1_u64(hibits >> sr); - - /* If we do, what will we get? */ - uint64_t new_hibits = (hibits << sl) | (lobits >> sr); - uint64_t new_lobits = lobits << sl; - size_t new_shift_up = shift_up + sl; - - /* Conditionally swap those values in. */ - hibits ^= (hibits ^ new_hibits ) & -(uint64_t)indicator; - lobits ^= (lobits ^ new_lobits ) & -(uint64_t)indicator; - shift_up ^= (shift_up ^ new_shift_up ) & -(size_t) indicator; - } - - /* - * So now we know the most significant 32 bits of d are at the top - * of hibits. Approximate the reciprocal of those bits. - */ - lobits = (uint64_t)recip_approx_32(hibits >> 32) << 32; - hibits = 0; - - /* - * And shift that up by as many bits as the input was shifted up - * just now, so that the product of this approximation and the - * actual input will be close to a fixed power of two regardless - * of where the MSB was. - * - * I do this in another log n individual passes, partly in case - * the CPU's register-controlled shift operation isn't - * time-constant, and also in case the compiler code-generates - * uint64_t shifts out of a variable number of smaller-word shift - * instructions, e.g. by splitting up into cases. - */ - for (size_t i = BIGNUM_INT_BITS_BITS; i-- > 0;) { - size_t sl = (size_t)1 << i; /* left shift count */ - size_t sr = 64 - sl; /* complementary right-shift count */ - - /* Should we shift up? */ - unsigned indicator = 1 & (shift_up >> i); - - /* If we do, what will we get? */ - uint64_t new_hibits = (hibits << sl) | (lobits >> sr); - uint64_t new_lobits = lobits << sl; - - /* Conditionally swap those values in. */ - hibits ^= (hibits ^ new_hibits ) & -(uint64_t)indicator; - lobits ^= (lobits ^ new_lobits ) & -(uint64_t)indicator; - } - - /* - * The product of the 128-bit value now in hibits:lobits with the - * 128-bit value we originally retrieved in the same variables - * will be in the vicinity of 2^191. So we'll take log2(R) to be - * 191, plus a multiple of BIGNUM_INT_BITS large enough to allow R - * to hold the combined sizes of n and d. - */ - size_t log2_R; - { - size_t max_log2_n = (n->nw + d->nw) * BIGNUM_INT_BITS; - log2_R = max_log2_n + 3; - log2_R -= size_t_min(191, log2_R); - log2_R = (log2_R + BIGNUM_INT_BITS - 1) & ~(BIGNUM_INT_BITS - 1); - log2_R += 191; - } - - /* Number of words in a bignum capable of holding numbers the size - * of twice R. */ - size_t rw = ((log2_R+2) + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS; - - /* - * Now construct our full-sized starting reciprocal approximation. - */ - mp_int *r_approx = mp_make_sized(rw); - size_t output_bit_index; - { - /* Where in the input number did the input 128-bit value come from? */ - size_t input_bit_index = - (hiword_index * BIGNUM_INT_BITS) - (128 - BIGNUM_INT_BITS); - - /* So how far do we need to shift our 64-bit output, if the - * product of those two fixed-size values is 2^191 and we want - * to make it 2^log2_R instead? */ - output_bit_index = log2_R - 191 - input_bit_index; - - /* If we've done all that right, it should be a whole number - * of words. */ - assert(output_bit_index % BIGNUM_INT_BITS == 0); - size_t output_word_index = output_bit_index / BIGNUM_INT_BITS; - - mp_add_integer_into_shifted_by_words( - r_approx, r_approx, lobits, output_word_index); - mp_add_integer_into_shifted_by_words( - r_approx, r_approx, hibits, - output_word_index + 64 / BIGNUM_INT_BITS); - } - - /* - * Make the constant 2*R, which we'll need in the iteration. - */ - mp_int *two_R = mp_make_sized(rw); - mp_add_integer_into_shifted_by_words( - two_R, two_R, (BignumInt)1 << ((log2_R+1) % BIGNUM_INT_BITS), - (log2_R+1) / BIGNUM_INT_BITS); - - /* - * Scratch space. - */ - mp_int *dr = mp_make_sized(rw + d->nw); - mp_int *diff = mp_make_sized(size_t_max(rw, dr->nw)); - mp_int *product = mp_make_sized(rw + diff->nw); - size_t scratchsize = size_t_max( - mp_mul_scratchspace(dr->nw, r_approx->nw, d->nw), - mp_mul_scratchspace(product->nw, r_approx->nw, diff->nw)); - mp_int *scratch = mp_make_sized(scratchsize); - mp_int product_shifted = mp_make_alias( - product, log2_R / BIGNUM_INT_BITS, product->nw); - - /* - * Initial error estimate: the 32-bit output of recip_approx_32 - * differs by less than 2048 (== 2^11) from the true top 32 bits - * of the reciprocal, so the relative error is at most 2^11 - * divided by the 32-bit reciprocal, which at worst is 2^11/2^31 = - * 2^-20. So even in the worst case, we have 20 good bits of - * reciprocal to start with. - */ - size_t good_bits = 31 - 11; - size_t good_bits_needed = BIGNUM_INT_BITS * n->nw + 4; /* add a few */ - - /* - * Now do Newton-Raphson iterations until we have reason to think - * they're not converging any more. - */ - while (good_bits < good_bits_needed) { - /* - * Compute the next iterate. - */ - mp_mul_internal(dr, r_approx, d, *scratch); - mp_sub_into(diff, two_R, dr); - mp_mul_internal(product, r_approx, diff, *scratch); - mp_rshift_fixed_into(r_approx, &product_shifted, - log2_R % BIGNUM_INT_BITS); - - /* - * Adjust the error estimate. - */ - good_bits = good_bits * 2 - 1; - } - - mp_free(dr); - mp_free(diff); - mp_free(product); - mp_free(scratch); - - /* - * Now we've got our reciprocal, we can compute the quotient, by - * multiplying in n and then shifting down by log2_R bits. - */ - mp_int *quotient_full = mp_mul(r_approx, n); - mp_int quotient_alias = mp_make_alias( - quotient_full, log2_R / BIGNUM_INT_BITS, quotient_full->nw); - mp_int *quotient = mp_make_sized(n->nw); - mp_rshift_fixed_into(quotient, "ient_alias, log2_R % BIGNUM_INT_BITS); - - /* - * Next, compute the remainder. - */ - mp_int *remainder = mp_make_sized(d->nw); - mp_mul_into(remainder, quotient, d); - mp_sub_into(remainder, n, remainder); - - /* - * Finally, two conditional subtractions to fix up any remaining - * rounding error. (I _think_ one should be enough, but this - * routine isn't time-critical enough to take chances.) - */ - unsigned q_correction = 0; - for (unsigned iter = 0; iter < 2; iter++) { - unsigned need_correction = mp_cmp_hs(remainder, d); - mp_cond_sub_into(remainder, remainder, d, need_correction); - q_correction += need_correction; - } - mp_add_integer_into(quotient, quotient, q_correction); - - /* - * Now we should have a perfect answer, i.e. 0 <= r < d. - */ - assert(!mp_cmp_hs(remainder, d)); - - if (q_out) - mp_copy_into(q_out, quotient); - if (r_out) - mp_copy_into(r_out, remainder); - - mp_free(r_approx); - mp_free(two_R); - mp_free(quotient_full); - mp_free(quotient); - mp_free(remainder); -} - -mp_int *mp_div(mp_int *n, mp_int *d) -{ - mp_int *q = mp_make_sized(n->nw); - mp_divmod_into(n, d, q, NULL); - return q; -} - -mp_int *mp_mod(mp_int *n, mp_int *d) -{ - mp_int *r = mp_make_sized(d->nw); - mp_divmod_into(n, d, NULL, r); - return r; -} - -mp_int *mp_nthroot(mp_int *y, unsigned n, mp_int *remainder_out) -{ - /* - * Allocate scratch space. - */ - mp_int **alloc, **powers, **newpowers, *scratch; - size_t nalloc = 2*(n+1)+1; - alloc = snewn(nalloc, mp_int *); - for (size_t i = 0; i < nalloc; i++) - alloc[i] = mp_make_sized(y->nw + 1); - powers = alloc; - newpowers = alloc + (n+1); - scratch = alloc[2*n+2]; - - /* - * We're computing the rounded-down nth root of y, i.e. the - * maximal x such that x^n <= y. We try to add 2^i to it for each - * possible value of i, starting from the largest one that might - * fit (i.e. such that 2^{n*i} fits in the size of y) downwards to - * i=0. - * - * We track all the smaller powers of x in the array 'powers'. In - * each iteration, if we update x, we update all of those values - * to match. - */ - mp_copy_integer_into(powers[0], 1); - for (size_t s = mp_max_bits(y) / n + 1; s-- > 0 ;) { - /* - * Let b = 2^s. We need to compute the powers (x+b)^i for each - * i, starting from our recorded values of x^i. - */ - for (size_t i = 0; i < n+1; i++) { - /* - * (x+b)^i = x^i - * + (i choose 1) x^{i-1} b - * + (i choose 2) x^{i-2} b^2 - * + ... - * + b^i - */ - uint16_t binom = 1; /* coefficient of b^i */ - mp_copy_into(newpowers[i], powers[i]); - for (size_t j = 0; j < i; j++) { - /* newpowers[i] += binom * powers[j] * 2^{(i-j)*s} */ - mp_mul_integer_into(scratch, powers[j], binom); - mp_lshift_fixed_into(scratch, scratch, (i-j) * s); - mp_add_into(newpowers[i], newpowers[i], scratch); - - uint32_t binom_mul = binom; - binom_mul *= (i-j); - binom_mul /= (j+1); - assert(binom_mul < 0x10000); - binom = binom_mul; - } - } - - /* - * Now, is the new value of x^n still <= y? If so, update. - */ - unsigned newbit = mp_cmp_hs(y, newpowers[n]); - for (size_t i = 0; i < n+1; i++) - mp_select_into(powers[i], powers[i], newpowers[i], newbit); - } - - if (remainder_out) - mp_sub_into(remainder_out, y, powers[n]); - - mp_int *root = mp_new(mp_max_bits(y) / n); - mp_copy_into(root, powers[1]); - - for (size_t i = 0; i < nalloc; i++) - mp_free(alloc[i]); - sfree(alloc); - - return root; -} - -mp_int *mp_modmul(mp_int *x, mp_int *y, mp_int *modulus) -{ - mp_int *product = mp_mul(x, y); - mp_int *reduced = mp_mod(product, modulus); - mp_free(product); - return reduced; -} - -mp_int *mp_modadd(mp_int *x, mp_int *y, mp_int *modulus) -{ - mp_int *sum = mp_add(x, y); - mp_int *reduced = mp_mod(sum, modulus); - mp_free(sum); - return reduced; -} - -mp_int *mp_modsub(mp_int *x, mp_int *y, mp_int *modulus) -{ - mp_int *diff = mp_make_sized(size_t_max(x->nw, y->nw)); - mp_sub_into(diff, x, y); - unsigned negate = mp_cmp_hs(y, x); - mp_cond_negate(diff, diff, negate); - mp_int *residue = mp_mod(diff, modulus); - mp_cond_negate(residue, residue, negate); - /* If we've just negated the residue, then it will be < 0 and need - * the modulus adding to it to make it positive - *except* if the - * residue was zero when we negated it. */ - unsigned make_positive = negate & ~mp_eq_integer(residue, 0); - mp_cond_add_into(residue, residue, modulus, make_positive); - mp_free(diff); - return residue; -} - -static mp_int *mp_modadd_in_range(mp_int *x, mp_int *y, mp_int *modulus) -{ - mp_int *sum = mp_make_sized(modulus->nw); - unsigned carry = mp_add_into_internal(sum, x, y); - mp_cond_sub_into(sum, sum, modulus, carry | mp_cmp_hs(sum, modulus)); - return sum; -} - -static mp_int *mp_modsub_in_range(mp_int *x, mp_int *y, mp_int *modulus) -{ - mp_int *diff = mp_make_sized(modulus->nw); - mp_sub_into(diff, x, y); - mp_cond_add_into(diff, diff, modulus, 1 ^ mp_cmp_hs(x, y)); - return diff; -} - -mp_int *monty_add(MontyContext *mc, mp_int *x, mp_int *y) -{ - return mp_modadd_in_range(x, y, mc->m); -} - -mp_int *monty_sub(MontyContext *mc, mp_int *x, mp_int *y) -{ - return mp_modsub_in_range(x, y, mc->m); -} - -void mp_min_into(mp_int *r, mp_int *x, mp_int *y) -{ - mp_select_into(r, x, y, mp_cmp_hs(x, y)); -} - -void mp_max_into(mp_int *r, mp_int *x, mp_int *y) -{ - mp_select_into(r, y, x, mp_cmp_hs(x, y)); -} - -mp_int *mp_min(mp_int *x, mp_int *y) -{ - mp_int *r = mp_make_sized(size_t_min(x->nw, y->nw)); - mp_min_into(r, x, y); - return r; -} - -mp_int *mp_max(mp_int *x, mp_int *y) -{ - mp_int *r = mp_make_sized(size_t_max(x->nw, y->nw)); - mp_max_into(r, x, y); - return r; -} - -mp_int *mp_power_2(size_t power) -{ - mp_int *x = mp_new(power + 1); - mp_set_bit(x, power, 1); - return x; -} - -struct ModsqrtContext { - mp_int *p; /* the prime */ - MontyContext *mc; /* for doing arithmetic mod p */ - - /* Decompose p-1 as 2^e k, for positive integer e and odd k */ - size_t e; - mp_int *k; - mp_int *km1o2; /* (k-1)/2 */ - - /* The user-provided value z which is not a quadratic residue mod - * p, and its kth power. Both in Montgomery form. */ - mp_int *z, *zk; -}; - -ModsqrtContext *modsqrt_new(mp_int *p, mp_int *any_nonsquare_mod_p) -{ - ModsqrtContext *sc = snew(ModsqrtContext); - memset(sc, 0, sizeof(ModsqrtContext)); - - sc->p = mp_copy(p); - sc->mc = monty_new(sc->p); - sc->z = monty_import(sc->mc, any_nonsquare_mod_p); - - /* Find the lowest set bit in p-1. Since this routine expects p to - * be non-secret (typically a well-known standard elliptic curve - * parameter), for once we don't need clever bit tricks. */ - for (sc->e = 1; sc->e < BIGNUM_INT_BITS * p->nw; sc->e++) - if (mp_get_bit(p, sc->e)) - break; - - sc->k = mp_rshift_fixed(p, sc->e); - sc->km1o2 = mp_rshift_fixed(sc->k, 1); - - /* Leave zk to be filled in lazily, since it's more expensive to - * compute. If this context turns out never to be needed, we can - * save the bulk of the setup time this way. */ - - return sc; -} - -static void modsqrt_lazy_setup(ModsqrtContext *sc) -{ - if (!sc->zk) - sc->zk = monty_pow(sc->mc, sc->z, sc->k); -} - -void modsqrt_free(ModsqrtContext *sc) -{ - monty_free(sc->mc); - mp_free(sc->p); - mp_free(sc->z); - mp_free(sc->k); - mp_free(sc->km1o2); - - if (sc->zk) - mp_free(sc->zk); - - sfree(sc); -} - -mp_int *mp_modsqrt(ModsqrtContext *sc, mp_int *x, unsigned *success) -{ - mp_int *mx = monty_import(sc->mc, x); - mp_int *mroot = monty_modsqrt(sc, mx, success); - mp_free(mx); - mp_int *root = monty_export(sc->mc, mroot); - mp_free(mroot); - return root; -} - -/* - * Modular square root, using an algorithm more or less similar to - * Tonelli-Shanks but adapted for constant time. - * - * The basic idea is to write p-1 = k 2^e, where k is odd and e > 0. - * Then the multiplicative group mod p (call it G) has a sequence of - * e+1 nested subgroups G = G_0 > G_1 > G_2 > ... > G_e, where each - * G_i is exactly half the size of G_{i-1} and consists of all the - * squares of elements in G_{i-1}. So the innermost group G_e has - * order k, which is odd, and hence within that group you can take a - * square root by raising to the power (k+1)/2. - * - * Our strategy is to iterate over these groups one by one and make - * sure the number x we're trying to take the square root of is inside - * each one, by adjusting it if it isn't. - * - * Suppose g is a primitive root of p, i.e. a generator of G_0. (We - * don't actually need to know what g _is_; we just imagine it for the - * sake of understanding.) Then G_i consists of precisely the (2^i)th - * powers of g, and hence, you can tell if a number is in G_i if - * raising it to the power k 2^{e-i} gives 1. So the conceptual - * algorithm goes: for each i, test whether x is in G_i by that - * method. If it isn't, then the previous iteration ensured it's in - * G_{i-1}, so it will be an odd power of g^{2^{i-1}}, and hence - * multiplying by any other odd power of g^{2^{i-1}} will give x' in - * G_i. And we have one of those, because our non-square z is an odd - * power of g, so z^{2^{i-1}} is an odd power of g^{2^{i-1}}. - * - * (There's a special case in the very first iteration, where we don't - * have a G_{i-1}. If it turns out that x is not even in G_1, that - * means it's not a square, so we set *success to 0. We still run the - * rest of the algorithm anyway, for the sake of constant time, but we - * don't give a hoot what it returns.) - * - * When we get to the end and have x in G_e, then we can take its - * square root by raising to (k+1)/2. But of course that's not the - * square root of the original input - it's only the square root of - * the adjusted version we produced during the algorithm. To get the - * true output answer we also have to multiply by a power of z, - * namely, z to the power of _half_ whatever we've been multiplying in - * as we go along. (The power of z we multiplied in must have been - * even, because the case in which we would have multiplied in an odd - * power of z is the i=0 case, in which we instead set the failure - * flag.) - * - * The code below is an optimised version of that basic idea, in which - * we _start_ by computing x^k so as to be able to test membership in - * G_i by only a few squarings rather than a full from-scratch modpow - * every time; we also start by computing our candidate output value - * x^{(k+1)/2}. So when the above description says 'adjust x by z^i' - * for some i, we have to adjust our running values of x^k and - * x^{(k+1)/2} by z^{ik} and z^{ik/2} respectively (the latter is safe - * because, as above, i is always even). And it turns out that we - * don't actually have to store the adjusted version of x itself at - * all - we _only_ keep those two powers of it. - */ -mp_int *monty_modsqrt(ModsqrtContext *sc, mp_int *x, unsigned *success) -{ - modsqrt_lazy_setup(sc); - - mp_int *scratch_to_free = mp_make_sized(3 * sc->mc->rw); - mp_int scratch = *scratch_to_free; - - /* - * Compute toret = x^{(k+1)/2}, our starting point for the output - * square root, and also xk = x^k which we'll use as we go along - * for knowing when to apply correction factors. We do this by - * first computing x^{(k-1)/2}, then multiplying it by x, then - * multiplying the two together. - */ - mp_int *toret = monty_pow(sc->mc, x, sc->km1o2); - mp_int xk = mp_alloc_from_scratch(&scratch, sc->mc->rw); - mp_copy_into(&xk, toret); - monty_mul_into(sc->mc, toret, toret, x); - monty_mul_into(sc->mc, &xk, toret, &xk); - - mp_int tmp = mp_alloc_from_scratch(&scratch, sc->mc->rw); - - mp_int power_of_zk = mp_alloc_from_scratch(&scratch, sc->mc->rw); - mp_copy_into(&power_of_zk, sc->zk); - - for (size_t i = 0; i < sc->e; i++) { - mp_copy_into(&tmp, &xk); - for (size_t j = i+1; j < sc->e; j++) - monty_mul_into(sc->mc, &tmp, &tmp, &tmp); - unsigned eq1 = mp_cmp_eq(&tmp, monty_identity(sc->mc)); - - if (i == 0) { - /* One special case: if x=0, then no power of x will ever - * equal 1, but we should still report success on the - * grounds that 0 does have a square root mod p. */ - *success = eq1 | mp_eq_integer(x, 0); - } else { - monty_mul_into(sc->mc, &tmp, toret, &power_of_zk); - mp_select_into(toret, &tmp, toret, eq1); - - monty_mul_into(sc->mc, &power_of_zk, - &power_of_zk, &power_of_zk); - - monty_mul_into(sc->mc, &tmp, &xk, &power_of_zk); - mp_select_into(&xk, &tmp, &xk, eq1); - } - } - - mp_free(scratch_to_free); - - return toret; -} - -mp_int *mp_random_bits_fn(size_t bits, random_read_fn_t random_read) -{ - size_t bytes = (bits + 7) / 8; - uint8_t *randbuf = snewn(bytes, uint8_t); - random_read(randbuf, bytes); - if (bytes) - randbuf[0] &= (2 << ((bits-1) & 7)) - 1; - mp_int *toret = mp_from_bytes_be(make_ptrlen(randbuf, bytes)); - smemclr(randbuf, bytes); - sfree(randbuf); - return toret; -} - -mp_int *mp_random_upto_fn(mp_int *limit, random_read_fn_t rf) -{ - /* - * It would be nice to generate our random numbers in such a way - * as to make every possible outcome literally equiprobable. But - * we can't do that in constant time, so we have to go for a very - * close approximation instead. I'm going to take the view that a - * factor of (1+2^-128) between the probabilities of two outcomes - * is acceptable on the grounds that you'd have to examine so many - * outputs to even detect it. - */ - mp_int *unreduced = mp_random_bits_fn(mp_max_bits(limit) + 128, rf); - mp_int *reduced = mp_mod(unreduced, limit); - mp_free(unreduced); - return reduced; -} - -mp_int *mp_random_in_range_fn(mp_int *lo, mp_int *hi, random_read_fn_t rf) -{ - mp_int *n_outcomes = mp_sub(hi, lo); - mp_int *addend = mp_random_upto_fn(n_outcomes, rf); - mp_int *result = mp_make_sized(hi->nw); - mp_add_into(result, addend, lo); - mp_free(addend); - mp_free(n_outcomes); - return result; -} |