/* * Multiprecision integer arithmetic, implementing mpint.h. */ #include #include #include #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_resize(mp_int *mp, size_t newmaxbits) { mp_int *copy = mp_new(newmaxbits); mp_copy_into(copy, mp); mp_free(mp); return copy; } 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; } #define MODPOW_LOG2_WINDOW_SIZE 5 #define MODPOW_WINDOW_SIZE (1 << MODPOW_LOG2_WINDOW_SIZE) mp_int *monty_pow(MontyContext *mc, mp_int *base, mp_int *exponent) { /* * Modular exponentiation is done from the top down, using a * fixed-window technique. * * We have a table storing every power of the base from base^0 up * to base^{w-1}, where w is a small power of 2, say 2^k. (k is * defined above as MODPOW_LOG2_WINDOW_SIZE, and w = 2^k is * defined as MODPOW_WINDOW_SIZE.) * * We break the exponent up into k-bit chunks, from the bottom up, * that is * * exponent = c_0 + 2^k c_1 + 2^{2k} c_2 + ... + 2^{nk} c_n * * and we compute base^exponent by computing in turn * * base^{c_n} * base^{2^k c_n + c_{n-1}} * base^{2^{2k} c_n + 2^k c_{n-1} + c_{n-2}} * ... * * where each line is obtained by raising the previous line to the * power 2^k (i.e. squaring it k times) and then multiplying in * a value base^{c_i}, which we can look up in our table. * * Side-channel considerations: the exponent is secret, so * actually doing a single table lookup by using a chunk of * exponent bits as an array index would be an obvious leak of * secret information into the cache. So instead, in each * iteration, we read _all_ the table entries, and do a sequence * of mp_select operations to leave just the one we wanted in the * variable that will go into the multiplication. In other * contexts (like software AES) that technique is so prohibitively * slow that it makes you choose a strategy that doesn't use table * lookups at all (we do bitslicing in preference); but here, this * iteration through 2^k table elements is replacing k-1 bignum * _multiplications_ that you'd have to use instead if you did * simple square-and-multiply, and that makes it still a win. */ /* Table that holds base^0, ..., base^{w-1} */ mp_int *table[MODPOW_WINDOW_SIZE]; table[0] = mp_copy(monty_identity(mc)); for (size_t i = 1; i < MODPOW_WINDOW_SIZE; i++) table[i] = monty_mul(mc, table[i-1], base); /* out accumulates the output value */ mp_int *out = mp_make_sized(mc->rw); mp_copy_into(out, monty_identity(mc)); /* table_entry will hold each value we get out of the table */ mp_int *table_entry = mp_make_sized(mc->rw); /* Bit index of the chunk of bits we're working on. Start with the * highest multiple of k strictly less than the size of our * bignum, i.e. the highest-index chunk of bits that might * conceivably contain any nonzero bit. */ size_t i = (exponent->nw * BIGNUM_INT_BITS) - 1; i -= i % MODPOW_LOG2_WINDOW_SIZE; bool first_iteration = true; while (true) { /* Construct the table index */ unsigned table_index = 0; for (size_t j = 0; j < MODPOW_LOG2_WINDOW_SIZE; j++) table_index |= mp_get_bit(exponent, i+j) << j; /* Iterate through the table to do a side-channel-safe lookup, * ending up with table_entry = table[table_index] */ mp_copy_into(table_entry, table[0]); for (size_t j = 1; j < MODPOW_WINDOW_SIZE; j++) { unsigned not_this_one = ((table_index ^ j) + MODPOW_WINDOW_SIZE - 1) >> MODPOW_LOG2_WINDOW_SIZE; mp_select_into(table_entry, table[j], table_entry, not_this_one); } if (!first_iteration) { /* Multiply into the output */ monty_mul_into(mc, out, out, table_entry); } else { /* On the first iteration, we can save one multiplication * by just copying */ mp_copy_into(out, table_entry); first_iteration = false; } /* If that was the bottommost chunk of bits, we're done */ if (i == 0) break; /* Otherwise, square k times and go round again. */ for (size_t j = 0; j < MODPOW_LOG2_WINDOW_SIZE; j++) monty_mul_into(mc, out, out, out); i-= MODPOW_LOG2_WINDOW_SIZE; } for (size_t i = 0; i < MODPOW_WINDOW_SIZE; i++) mp_free(table[i]); mp_free(table_entry); 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); BignumInt top_word = (BignumInt)1 << ((log2_R+1) % BIGNUM_INT_BITS); mp_add_integer_into_shifted_by_words( two_R, two_R, top_word, (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; } uint32_t mp_mod_known_integer(mp_int *x, uint32_t m) { uint64_t reciprocal = ((uint64_t)1 << 48) / m; uint64_t accumulator = 0; for (size_t i = mp_max_bytes(x); i-- > 0 ;) { accumulator = 0x100 * accumulator + mp_get_byte(x, i); /* * Let A be the value in 'accumulator' at this point, and let * R be the value it will have after we subtract quot*m below. * * Lemma 1: if A < 2^48, then R < 2m. * * Proof: * * By construction, we have 2^48/m - 1 < reciprocal <= 2^48/m. * Multiplying that by the accumulator gives * * A/m * 2^48 - A < unshifted_quot <= A/m * 2^48 * i.e. 0 <= (A/m * 2^48) - unshifted_quot < A * i.e. 0 <= A/m - unshifted_quot/2^48 < A/2^48 * * So when we shift this quotient right by 48 bits, i.e. take * the floor of (unshifted_quot/2^48), the value we take the * floor of is at most A/2^48 less than the true rational * value A/m that we _wanted_ to take the floor of. * * Provided A < 2^48, this is less than 1. So the quotient * 'quot' that we've just produced is either the true quotient * floor(A/m), or one less than it. Hence, the output value R * is less than 2m. [] * * Lemma 2: if A < 2^16 m, then the multiplication of * accumulator*reciprocal does not overflow. * * Proof: as above, we have reciprocal <= 2^48/m. Multiplying * by A gives unshifted_quot <= 2^48 * A / m < 2^48 * 2^16 = * 2^64. [] */ uint64_t unshifted_quot = accumulator * reciprocal; uint64_t quot = unshifted_quot >> 48; accumulator -= quot * m; } /* * Theorem 1: accumulator < 2m at the end of every iteration of * this loop. * * Proof: induction on the above loop. * * Base case: at the start of the first loop iteration, the * accumulator is 0, which is certainly < 2m. * * Inductive step: in each loop iteration, we take a value at most * 2m-1, multiply it by 2^8, and add another byte less than 2^8 to * generate the input value A to the reduction process above. So * we have A < 2m * 2^8 - 1. We know m < 2^32 (because it was * passed in as a uint32_t), so A < 2^41, which is enough to allow * us to apply Lemma 1, showing that the value of 'accumulator' at * the end of the loop is still < 2m. [] * * Corollary: we need at most one final subtraction of m to * produce the canonical residue of x mod m, i.e. in the range * [0,m). * * Theorem 2: no multiplication in the inner loop overflows. * * Proof: in Theorem 1 we established A < 2m * 2^8 - 1 in every * iteration. That is less than m * 2^16, so Lemma 2 applies. * * The other multiplication, of quot * m, cannot overflow because * quot is at most A/m, so quot*m <= A < 2^64. [] */ uint32_t result = accumulator; uint32_t reduced = result - m; uint32_t select = -(reduced >> 31); result = reduced ^ ((result ^ reduced) & select); assert(result < m); return result; } 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; }