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Diffstat (limited to 'lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/fpr.c')
| -rw-r--r-- | lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/fpr.c | 1890 |
1 files changed, 1890 insertions, 0 deletions
diff --git a/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/fpr.c b/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/fpr.c new file mode 100644 index 000000000..669c825ee --- /dev/null +++ b/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/fpr.c @@ -0,0 +1,1890 @@ +#include "inner.h" + +/* + * Floating-point operations. + * + * This file implements the non-inline functions declared in + * fpr.h, as well as the constants for FFT / iFFT. + * + * ==========================(LICENSE BEGIN)============================ + * + * Copyright (c) 2017-2019 Falcon Project + * + * Permission is hereby granted, free of charge, to any person obtaining + * a copy of this software and associated documentation files (the + * "Software"), to deal in the Software without restriction, including + * without limitation the rights to use, copy, modify, merge, publish, + * distribute, sublicense, and/or sell copies of the Software, and to + * permit persons to whom the Software is furnished to do so, subject to + * the following conditions: + * + * The above copyright notice and this permission notice shall be + * included in all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, + * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF + * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. + * IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY + * CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, + * TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE + * SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + * + * ===========================(LICENSE END)============================= + * + * @author Thomas Pornin <thomas.pornin@nccgroup.com> + */ + + + +/* + * Normalize a provided unsigned integer to the 2^63..2^64-1 range by + * left-shifting it if necessary. The exponent e is adjusted accordingly + * (i.e. if the value was left-shifted by n bits, then n is subtracted + * from e). If source m is 0, then it remains 0, but e is altered. + * Both m and e must be simple variables (no expressions allowed). + */ +#define FPR_NORM64(m, e) do { \ + uint32_t nt; \ + \ + (e) -= 63; \ + \ + nt = (uint32_t)((m) >> 32); \ + nt = (nt | -nt) >> 31; \ + (m) ^= ((m) ^ ((m) << 32)) & ((uint64_t)nt - 1); \ + (e) += (int)(nt << 5); \ + \ + nt = (uint32_t)((m) >> 48); \ + nt = (nt | -nt) >> 31; \ + (m) ^= ((m) ^ ((m) << 16)) & ((uint64_t)nt - 1); \ + (e) += (int)(nt << 4); \ + \ + nt = (uint32_t)((m) >> 56); \ + nt = (nt | -nt) >> 31; \ + (m) ^= ((m) ^ ((m) << 8)) & ((uint64_t)nt - 1); \ + (e) += (int)(nt << 3); \ + \ + nt = (uint32_t)((m) >> 60); \ + nt = (nt | -nt) >> 31; \ + (m) ^= ((m) ^ ((m) << 4)) & ((uint64_t)nt - 1); \ + (e) += (int)(nt << 2); \ + \ + nt = (uint32_t)((m) >> 62); \ + nt = (nt | -nt) >> 31; \ + (m) ^= ((m) ^ ((m) << 2)) & ((uint64_t)nt - 1); \ + (e) += (int)(nt << 1); \ + \ + nt = (uint32_t)((m) >> 63); \ + (m) ^= ((m) ^ ((m) << 1)) & ((uint64_t)nt - 1); \ + (e) += (int)(nt); \ + } while (0) + +uint64_t +fpr_ursh(uint64_t x, int n) { + x ^= (x ^ (x >> 32)) & -(uint64_t)(n >> 5); + return x >> (n & 31); +} + +int64_t +fpr_irsh(int64_t x, int n) { + x ^= (x ^ (x >> 32)) & -(int64_t)(n >> 5); + return x >> (n & 31); +} + +uint64_t +fpr_ulsh(uint64_t x, int n) { + x ^= (x ^ (x << 32)) & -(uint64_t)(n >> 5); + return x << (n & 31); +} + +fpr +FPR(int s, int e, uint64_t m) { + fpr x; + uint32_t t; + unsigned f; + + /* + * If e >= -1076, then the value is "normal"; otherwise, it + * should be a subnormal, which we clamp down to zero. + */ + e += 1076; + t = (uint32_t)e >> 31; + m &= (uint64_t)t - 1; + + /* + * If m = 0 then we want a zero; make e = 0 too, but conserve + * the sign. + */ + t = (uint32_t)(m >> 54); + e &= -(int)t; + + /* + * The 52 mantissa bits come from m. Value m has its top bit set + * (unless it is a zero); we leave it "as is": the top bit will + * increment the exponent by 1, except when m = 0, which is + * exactly what we want. + */ + x = (((uint64_t)s << 63) | (m >> 2)) + ((uint64_t)(uint32_t)e << 52); + + /* + * Rounding: if the low three bits of m are 011, 110 or 111, + * then the value should be incremented to get the next + * representable value. This implements the usual + * round-to-nearest rule (with preference to even values in case + * of a tie). Note that the increment may make a carry spill + * into the exponent field, which is again exactly what we want + * in that case. + */ + f = (unsigned)m & 7U; + x += (0xC8U >> f) & 1; + return x; +} + +fpr +fpr_scaled(int64_t i, int sc) { + /* + * To convert from int to float, we have to do the following: + * 1. Get the absolute value of the input, and its sign + * 2. Shift right or left the value as appropriate + * 3. Pack the result + * + * We can assume that the source integer is not -2^63. + */ + int s, e; + uint32_t t; + uint64_t m; + + /* + * Extract sign bit. + * We have: -i = 1 + ~i + */ + s = (int)((uint64_t)i >> 63); + i ^= -(int64_t)s; + i += s; + + /* + * For now we suppose that i != 0. + * Otherwise, we set m to i and left-shift it as much as needed + * to get a 1 in the top bit. We can do that in a logarithmic + * number of conditional shifts. + */ + m = (uint64_t)i; + e = 9 + sc; + FPR_NORM64(m, e); + + /* + * Now m is in the 2^63..2^64-1 range. We must divide it by 512; + * if one of the dropped bits is a 1, this should go into the + * "sticky bit". + */ + m |= ((uint32_t)m & 0x1FF) + 0x1FF; + m >>= 9; + + /* + * Corrective action: if i = 0 then all of the above was + * incorrect, and we clamp e and m down to zero. + */ + t = (uint32_t)((uint64_t)(i | -i) >> 63); + m &= -(uint64_t)t; + e &= -(int)t; + + /* + * Assemble back everything. The FPR() function will handle cases + * where e is too low. + */ + return FPR(s, e, m); +} + +fpr +fpr_of(int64_t i) { + return fpr_scaled(i, 0); +} + +int64_t +fpr_rint(fpr x) { + uint64_t m, d; + int e; + uint32_t s, dd, f; + + /* + * We assume that the value fits in -(2^63-1)..+(2^63-1). We can + * thus extract the mantissa as a 63-bit integer, then right-shift + * it as needed. + */ + m = ((x << 10) | ((uint64_t)1 << 62)) & (((uint64_t)1 << 63) - 1); + e = 1085 - ((int)(x >> 52) & 0x7FF); + + /* + * If a shift of more than 63 bits is needed, then simply set m + * to zero. This also covers the case of an input operand equal + * to zero. + */ + m &= -(uint64_t)((uint32_t)(e - 64) >> 31); + e &= 63; + + /* + * Right-shift m as needed. Shift count is e. Proper rounding + * mandates that: + * - If the highest dropped bit is zero, then round low. + * - If the highest dropped bit is one, and at least one of the + * other dropped bits is one, then round up. + * - If the highest dropped bit is one, and all other dropped + * bits are zero, then round up if the lowest kept bit is 1, + * or low otherwise (i.e. ties are broken by "rounding to even"). + * + * We thus first extract a word consisting of all the dropped bit + * AND the lowest kept bit; then we shrink it down to three bits, + * the lowest being "sticky". + */ + d = fpr_ulsh(m, 63 - e); + dd = (uint32_t)d | ((uint32_t)(d >> 32) & 0x1FFFFFFF); + f = (uint32_t)(d >> 61) | ((dd | -dd) >> 31); + m = fpr_ursh(m, e) + (uint64_t)((0xC8U >> f) & 1U); + + /* + * Apply the sign bit. + */ + s = (uint32_t)(x >> 63); + return ((int64_t)m ^ -(int64_t)s) + (int64_t)s; +} + +int64_t +fpr_floor(fpr x) { + uint64_t t; + int64_t xi; + int e, cc; + + /* + * We extract the integer as a _signed_ 64-bit integer with + * a scaling factor. Since we assume that the value fits + * in the -(2^63-1)..+(2^63-1) range, we can left-shift the + * absolute value to make it in the 2^62..2^63-1 range: we + * will only need a right-shift afterwards. + */ + e = (int)(x >> 52) & 0x7FF; + t = x >> 63; + xi = (int64_t)(((x << 10) | ((uint64_t)1 << 62)) + & (((uint64_t)1 << 63) - 1)); + xi = (xi ^ -(int64_t)t) + (int64_t)t; + cc = 1085 - e; + + /* + * We perform an arithmetic right-shift on the value. This + * applies floor() semantics on both positive and negative values + * (rounding toward minus infinity). + */ + xi = fpr_irsh(xi, cc & 63); + + /* + * If the true shift count was 64 or more, then we should instead + * replace xi with 0 (if nonnegative) or -1 (if negative). Edge + * case: -0 will be floored to -1, not 0 (whether this is correct + * is debatable; in any case, the other functions normalize zero + * to +0). + * + * For an input of zero, the non-shifted xi was incorrect (we used + * a top implicit bit of value 1, not 0), but this does not matter + * since this operation will clamp it down. + */ + xi ^= (xi ^ -(int64_t)t) & -(int64_t)((uint32_t)(63 - cc) >> 31); + return xi; +} + +int64_t +fpr_trunc(fpr x) { + uint64_t t, xu; + int e, cc; + + /* + * Extract the absolute value. Since we assume that the value + * fits in the -(2^63-1)..+(2^63-1) range, we can left-shift + * the absolute value into the 2^62..2^63-1 range, and then + * do a right shift afterwards. + */ + e = (int)(x >> 52) & 0x7FF; + xu = ((x << 10) | ((uint64_t)1 << 62)) & (((uint64_t)1 << 63) - 1); + cc = 1085 - e; + xu = fpr_ursh(xu, cc & 63); + + /* + * If the exponent is too low (cc > 63), then the shift was wrong + * and we must clamp the value to 0. This also covers the case + * of an input equal to zero. + */ + xu &= -(uint64_t)((uint32_t)(cc - 64) >> 31); + + /* + * Apply back the sign, if the source value is negative. + */ + t = x >> 63; + xu = (xu ^ -t) + t; + return *(int64_t *)&xu; +} + +fpr +fpr_add(fpr x, fpr y) { + uint64_t m, xu, yu, za; + uint32_t cs; + int ex, ey, sx, sy, cc; + + /* + * Make sure that the first operand (x) has the larger absolute + * value. This guarantees that the exponent of y is less than + * or equal to the exponent of x, and, if they are equal, then + * the mantissa of y will not be greater than the mantissa of x. + * + * After this swap, the result will have the sign x, except in + * the following edge case: abs(x) = abs(y), and x and y have + * opposite sign bits; in that case, the result shall be +0 + * even if the sign bit of x is 1. To handle this case properly, + * we do the swap is abs(x) = abs(y) AND the sign of x is 1. + */ + m = ((uint64_t)1 << 63) - 1; + za = (x & m) - (y & m); + cs = (uint32_t)(za >> 63) + | ((1U - (uint32_t)(-za >> 63)) & (uint32_t)(x >> 63)); + m = (x ^ y) & -(uint64_t)cs; + x ^= m; + y ^= m; + + /* + * Extract sign bits, exponents and mantissas. The mantissas are + * scaled up to 2^55..2^56-1, and the exponent is unbiased. If + * an operand is zero, its mantissa is set to 0 at this step, and + * its exponent will be -1078. + */ + ex = (int)(x >> 52); + sx = ex >> 11; + ex &= 0x7FF; + m = (uint64_t)(uint32_t)((ex + 0x7FF) >> 11) << 52; + xu = ((x & (((uint64_t)1 << 52) - 1)) | m) << 3; + ex -= 1078; + ey = (int)(y >> 52); + sy = ey >> 11; + ey &= 0x7FF; + m = (uint64_t)(uint32_t)((ey + 0x7FF) >> 11) << 52; + yu = ((y & (((uint64_t)1 << 52) - 1)) | m) << 3; + ey -= 1078; + + /* + * x has the larger exponent; hence, we only need to right-shift y. + * If the shift count is larger than 59 bits then we clamp the + * value to zero. + */ + cc = ex - ey; + yu &= -(uint64_t)((uint32_t)(cc - 60) >> 31); + cc &= 63; + + /* + * The lowest bit of yu is "sticky". + */ + m = fpr_ulsh(1, cc) - 1; + yu |= (yu & m) + m; + yu = fpr_ursh(yu, cc); + + /* + * If the operands have the same sign, then we add the mantissas; + * otherwise, we subtract the mantissas. + */ + xu += yu - ((yu << 1) & -(uint64_t)(sx ^ sy)); + + /* + * The result may be smaller, or slightly larger. We normalize + * it to the 2^63..2^64-1 range (if xu is zero, then it stays + * at zero). + */ + FPR_NORM64(xu, ex); + + /* + * Scale down the value to 2^54..s^55-1, handling the last bit + * as sticky. + */ + xu |= ((uint32_t)xu & 0x1FF) + 0x1FF; + xu >>= 9; + ex += 9; + + /* + * In general, the result has the sign of x. However, if the + * result is exactly zero, then the following situations may + * be encountered: + * x > 0, y = -x -> result should be +0 + * x < 0, y = -x -> result should be +0 + * x = +0, y = +0 -> result should be +0 + * x = -0, y = +0 -> result should be +0 + * x = +0, y = -0 -> result should be +0 + * x = -0, y = -0 -> result should be -0 + * + * But at the conditional swap step at the start of the + * function, we ensured that if abs(x) = abs(y) and the + * sign of x was 1, then x and y were swapped. Thus, the + * two following cases cannot actually happen: + * x < 0, y = -x + * x = -0, y = +0 + * In all other cases, the sign bit of x is conserved, which + * is what the FPR() function does. The FPR() function also + * properly clamps values to zero when the exponent is too + * low, but does not alter the sign in that case. + */ + return FPR(sx, ex, xu); +} + +fpr +fpr_sub(fpr x, fpr y) { + y ^= (uint64_t)1 << 63; + return fpr_add(x, y); +} + +fpr +fpr_neg(fpr x) { + x ^= (uint64_t)1 << 63; + return x; +} + +fpr +fpr_half(fpr x) { + /* + * To divide a value by 2, we just have to subtract 1 from its + * exponent, but we have to take care of zero. + */ + uint32_t t; + + x -= (uint64_t)1 << 52; + t = (((uint32_t)(x >> 52) & 0x7FF) + 1) >> 11; + x &= (uint64_t)t - 1; + return x; +} + +fpr +fpr_double(fpr x) { + /* + * To double a value, we just increment by one the exponent. We + * don't care about infinites or NaNs; however, 0 is a + * special case. + */ + x += (uint64_t)((((unsigned)(x >> 52) & 0x7FFU) + 0x7FFU) >> 11) << 52; + return x; +} + +fpr +fpr_mul(fpr x, fpr y) { + uint64_t xu, yu, w, zu, zv; + uint32_t x0, x1, y0, y1, z0, z1, z2; + int ex, ey, d, e, s; + + /* + * Extract absolute values as scaled unsigned integers. We + * don't extract exponents yet. + */ + xu = (x & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); + yu = (y & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); + + /* + * We have two 53-bit integers to multiply; we need to split + * each into a lower half and a upper half. Moreover, we + * prefer to have lower halves to be of 25 bits each, for + * reasons explained later on. + */ + x0 = (uint32_t)xu & 0x01FFFFFF; + x1 = (uint32_t)(xu >> 25); + y0 = (uint32_t)yu & 0x01FFFFFF; + y1 = (uint32_t)(yu >> 25); + w = (uint64_t)x0 * (uint64_t)y0; + z0 = (uint32_t)w & 0x01FFFFFF; + z1 = (uint32_t)(w >> 25); + w = (uint64_t)x0 * (uint64_t)y1; + z1 += (uint32_t)w & 0x01FFFFFF; + z2 = (uint32_t)(w >> 25); + w = (uint64_t)x1 * (uint64_t)y0; + z1 += (uint32_t)w & 0x01FFFFFF; + z2 += (uint32_t)(w >> 25); + zu = (uint64_t)x1 * (uint64_t)y1; + z2 += (z1 >> 25); + z1 &= 0x01FFFFFF; + zu += z2; + + /* + * Since xu and yu are both in the 2^52..2^53-1 range, the + * product is in the 2^104..2^106-1 range. We first reassemble + * it and round it into the 2^54..2^56-1 range; the bottom bit + * is made "sticky". Since the low limbs z0 and z1 are 25 bits + * each, we just take the upper part (zu), and consider z0 and + * z1 only for purposes of stickiness. + * (This is the reason why we chose 25-bit limbs above.) + */ + zu |= ((z0 | z1) + 0x01FFFFFF) >> 25; + + /* + * We normalize zu to the 2^54..s^55-1 range: it could be one + * bit too large at this point. This is done with a conditional + * right-shift that takes into account the sticky bit. + */ + zv = (zu >> 1) | (zu & 1); + w = zu >> 55; + zu ^= (zu ^ zv) & -w; + + /* + * Get the aggregate scaling factor: + * + * - Each exponent is biased by 1023. + * + * - Integral mantissas are scaled by 2^52, hence an + * extra 52 bias for each exponent. + * + * - However, we right-shifted z by 50 bits, and then + * by 0 or 1 extra bit (depending on the value of w). + * + * In total, we must add the exponents, then subtract + * 2 * (1023 + 52), then add 50 + w. + */ + ex = (int)((x >> 52) & 0x7FF); + ey = (int)((y >> 52) & 0x7FF); + e = ex + ey - 2100 + (int)w; + + /* + * Sign bit is the XOR of the operand sign bits. + */ + s = (int)((x ^ y) >> 63); + + /* + * Corrective actions for zeros: if either of the operands is + * zero, then the computations above were wrong. Test for zero + * is whether ex or ey is zero. We just have to set the mantissa + * (zu) to zero, the FPR() function will normalize e. + */ + d = ((ex + 0x7FF) & (ey + 0x7FF)) >> 11; + zu &= -(uint64_t)d; + + /* + * FPR() packs the result and applies proper rounding. + */ + return FPR(s, e, zu); +} + +fpr +fpr_sqr(fpr x) { + return fpr_mul(x, x); +} + +fpr +fpr_div(fpr x, fpr y) { + uint64_t xu, yu, q, q2, w; + int i, ex, ey, e, d, s; + + /* + * Extract mantissas of x and y (unsigned). + */ + xu = (x & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); + yu = (y & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); + + /* + * Perform bit-by-bit division of xu by yu. We run it for 55 bits. + */ + q = 0; + for (i = 0; i < 55; i ++) { + /* + * If yu is less than or equal xu, then subtract it and + * push a 1 in the quotient; otherwise, leave xu unchanged + * and push a 0. + */ + uint64_t b; + + b = ((xu - yu) >> 63) - 1; + xu -= b & yu; + q |= b & 1; + xu <<= 1; + q <<= 1; + } + + /* + * We got 55 bits in the quotient, followed by an extra zero. We + * want that 56th bit to be "sticky": it should be a 1 if and + * only if the remainder (xu) is non-zero. + */ + q |= (xu | -xu) >> 63; + + /* + * Quotient is at most 2^56-1. Its top bit may be zero, but in + * that case the next-to-top bit will be a one, since the + * initial xu and yu were both in the 2^52..2^53-1 range. + * We perform a conditional shift to normalize q to the + * 2^54..2^55-1 range (with the bottom bit being sticky). + */ + q2 = (q >> 1) | (q & 1); + w = q >> 55; + q ^= (q ^ q2) & -w; + + /* + * Extract exponents to compute the scaling factor: + * + * - Each exponent is biased and we scaled them up by + * 52 bits; but these biases will cancel out. + * + * - The division loop produced a 55-bit shifted result, + * so we must scale it down by 55 bits. + * + * - If w = 1, we right-shifted the integer by 1 bit, + * hence we must add 1 to the scaling. + */ + ex = (int)((x >> 52) & 0x7FF); + ey = (int)((y >> 52) & 0x7FF); + e = ex - ey - 55 + (int)w; + + /* + * Sign is the XOR of the signs of the operands. + */ + s = (int)((x ^ y) >> 63); + + /* + * Corrective actions for zeros: if x = 0, then the computation + * is wrong, and we must clamp e and q to 0. We do not care + * about the case y = 0 (as per assumptions in this module, + * the caller does not perform divisions by zero). + */ + d = (ex + 0x7FF) >> 11; + s &= d; + e &= -d; + q &= -(uint64_t)d; + + /* + * FPR() packs the result and applies proper rounding. + */ + return FPR(s, e, q); +} + +fpr +fpr_inv(fpr x) { + return fpr_div(4607182418800017408u, x); +} + +fpr +fpr_sqrt(fpr x) { + uint64_t xu, q, s, r; + int ex, e; + + /* + * Extract the mantissa and the exponent. We don't care about + * the sign: by assumption, the operand is nonnegative. + * We want the "true" exponent corresponding to a mantissa + * in the 1..2 range. + */ + xu = (x & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52); + ex = (int)((x >> 52) & 0x7FF); + e = ex - 1023; + + /* + * If the exponent is odd, double the mantissa and decrement + * the exponent. The exponent is then halved to account for + * the square root. + */ + xu += xu & -(uint64_t)(e & 1); + e >>= 1; + + /* + * Double the mantissa. + */ + xu <<= 1; + + /* + * We now have a mantissa in the 2^53..2^55-1 range. It + * represents a value between 1 (inclusive) and 4 (exclusive) + * in fixed point notation (with 53 fractional bits). We + * compute the square root bit by bit. + */ + q = 0; + s = 0; + r = (uint64_t)1 << 53; + for (int i = 0; i < 54; i ++) { + uint64_t t, b; + + t = s + r; + b = ((xu - t) >> 63) - 1; + s += (r << 1) & b; + xu -= t & b; + q += r & b; + xu <<= 1; + r >>= 1; + } + + /* + * Now, q is a rounded-low 54-bit value, with a leading 1, + * 52 fractional digits, and an additional guard bit. We add + * an extra sticky bit to account for what remains of the operand. + */ + q <<= 1; + q |= (xu | -xu) >> 63; + + /* + * Result q is in the 2^54..2^55-1 range; we bias the exponent + * by 54 bits (the value e at that point contains the "true" + * exponent, but q is now considered an integer, i.e. scaled + * up. + */ + e -= 54; + + /* + * Corrective action for an operand of value zero. + */ + q &= -(uint64_t)((ex + 0x7FF) >> 11); + + /* + * Apply rounding and back result. + */ + return FPR(0, e, q); +} + +int +fpr_lt(fpr x, fpr y) { + /* + * If both x and y are positive, then a signed comparison yields + * the proper result: + * - For positive values, the order is preserved. + * - The sign bit is at the same place as in integers, so + * sign is preserved. + * Moreover, we can compute [x < y] as sgn(x-y) and the computation + * of x-y will not overflow. + * + * If the signs differ, then sgn(x) gives the proper result. + * + * If both x and y are negative, then the order is reversed. + * Hence [x < y] = sgn(y-x). We must compute this separately from + * sgn(x-y); simply inverting sgn(x-y) would not handle the edge + * case x = y properly. + */ + int cc0, cc1; + int64_t sx; + int64_t sy; + + sx = *(int64_t *)&x; + sy = *(int64_t *)&y; + sy &= ~((sx ^ sy) >> 63); /* set sy=0 if signs differ */ + + cc0 = (int)((sx - sy) >> 63) & 1; /* Neither subtraction overflows when */ + cc1 = (int)((sy - sx) >> 63) & 1; /* the signs are the same. */ + + return cc0 ^ ((cc0 ^ cc1) & (int)((x & y) >> 63)); +} + +uint64_t +fpr_expm_p63(fpr x, fpr ccs) { + /* + * Polynomial approximation of exp(-x) is taken from FACCT: + * https://eprint.iacr.org/2018/1234 + * Specifically, values are extracted from the implementation + * referenced from the FACCT article, and available at: + * https://github.com/raykzhao/gaussian + * Here, the coefficients have been scaled up by 2^63 and + * converted to integers. + * + * Tests over more than 24 billions of random inputs in the + * 0..log(2) range have never shown a deviation larger than + * 2^(-50) from the true mathematical value. + */ + static const uint64_t C[] = { + 0x00000004741183A3u, + 0x00000036548CFC06u, + 0x0000024FDCBF140Au, + 0x0000171D939DE045u, + 0x0000D00CF58F6F84u, + 0x000680681CF796E3u, + 0x002D82D8305B0FEAu, + 0x011111110E066FD0u, + 0x0555555555070F00u, + 0x155555555581FF00u, + 0x400000000002B400u, + 0x7FFFFFFFFFFF4800u, + 0x8000000000000000u + }; + + uint64_t z, y; + size_t u; + uint32_t z0, z1, y0, y1; + uint64_t a, b; + + y = C[0]; + z = (uint64_t)fpr_trunc(fpr_mul(x, fpr_ptwo63)) << 1; + for (u = 1; u < (sizeof C) / sizeof(C[0]); u ++) { + /* + * Compute product z * y over 128 bits, but keep only + * the top 64 bits. + * + * TODO: On some architectures/compilers we could use + * some intrinsics (__umulh() on MSVC) or other compiler + * extensions (unsigned __int128 on GCC / Clang) for + * improved speed; however, most 64-bit architectures + * also have appropriate IEEE754 floating-point support, + * which is better. + */ + uint64_t c; + + z0 = (uint32_t)z; + z1 = (uint32_t)(z >> 32); + y0 = (uint32_t)y; + y1 = (uint32_t)(y >> 32); + a = ((uint64_t)z0 * (uint64_t)y1) + + (((uint64_t)z0 * (uint64_t)y0) >> 32); + b = ((uint64_t)z1 * (uint64_t)y0); + c = (a >> 32) + (b >> 32); + c += (((uint64_t)(uint32_t)a + (uint64_t)(uint32_t)b) >> 32); + c += (uint64_t)z1 * (uint64_t)y1; + y = C[u] - c; + } + + /* + * The scaling factor must be applied at the end. Since y is now + * in fixed-point notation, we have to convert the factor to the + * same format, and do an extra integer multiplication. + */ + z = (uint64_t)fpr_trunc(fpr_mul(ccs, fpr_ptwo63)) << 1; + z0 = (uint32_t)z; + z1 = (uint32_t)(z >> 32); + y0 = (uint32_t)y; + y1 = (uint32_t)(y >> 32); + a = ((uint64_t)z0 * (uint64_t)y1) + + (((uint64_t)z0 * (uint64_t)y0) >> 32); + b = ((uint64_t)z1 * (uint64_t)y0); + y = (a >> 32) + (b >> 32); + y += (((uint64_t)(uint32_t)a + (uint64_t)(uint32_t)b) >> 32); + y += (uint64_t)z1 * (uint64_t)y1; + + return y; +} + +const fpr fpr_gm_tab[] = { + 0, 0, + 9223372036854775808U, 4607182418800017408U, + 4604544271217802189U, 4604544271217802189U, + 13827916308072577997U, 4604544271217802189U, + 4606496786581982534U, 4600565431771507043U, + 13823937468626282851U, 4606496786581982534U, + 4600565431771507043U, 4606496786581982534U, + 13829868823436758342U, 4600565431771507043U, + 4607009347991985328U, 4596196889902818827U, + 13819568926757594635U, 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4599374859150636784U, 4606719100629313491U, + 13830091137484089299U, 4599374859150636784U, + 4606863472012527185U, 4598423001813699022U, + 13821795038668474830U, 4606863472012527185U, + 4602598930031891166U, 4605998608960791335U, + 13829370645815567143U, 4602598930031891166U, + 4605292980606880364U, 4603694922063032361U, + 13827066958917808169U, 4605292980606880364U, + 4593688012422887515U, 4607111255739239816U, + 13830483292594015624U, 4593688012422887515U, + 4607054494135176056U, 4595218635031890910U, + 13818590671886666718U, 4607054494135176056U, + 4603384207141321914U, 4605523422498301790U, + 13828895459353077598U, 4603384207141321914U, + 4605799732098147061U, 4602970680601913687U, + 13826342717456689495U, 4605799732098147061U, + 4597169786279785693U, 4606957467106717424U, + 13830329503961493232U, 4597169786279785693U, + 4606588777269136769U, 4600103317933788342U, + 13823475354788564150U, 4606588777269136769U, + 4601022290077223616U, 4606398451906509788U, + 13829770488761285596U, 4601022290077223616U, + 4604717681185626434U, 4604366005771528720U, + 13827738042626304528U, 4604717681185626434U, + 4583614727651146525U, 4607178985458280057U, + 13830551022313055865U, 4583614727651146525U, + 4607172882816799076U, 4586790578280679046U, + 13810162615135454854U, 4607172882816799076U, + 4604244531615310815U, 4604830524903495634U, + 13828202561758271442U, 4604244531615310815U, + 4606329407841126011U, 4601323770373937522U, + 13824695807228713330U, 4606329407841126011U, + 4599792496117920694U, 4606646545123403481U, + 13830018581978179289U, 4599792496117920694U, + 4606919157647773535U, 4597815040470278984U, + 13821187077325054792U, 4606919157647773535U, + 4602829525820289164U, 4605886709123365959U, + 13829258745978141767U, 4602829525820289164U, + 4605426297151190466U, 4603518581031047189U, + 13826890617885822997U, 4605426297151190466U, + 4594563856311064231U, 4607080832832247697U, + 13830452869687023505U, 4594563856311064231U, + 4607088942243446236U, 4594345179472540681U, + 13817717216327316489U, 4607088942243446236U, + 4603562972219549215U, 4605393374401988274U, + 13828765411256764082U, 4603562972219549215U, + 4605915122243179241U, 4602782121393764535U, + 13826154158248540343U, 4605915122243179241U, + 4598029484874872834U, 4606905728766014348U, + 13830277765620790156U, 4598029484874872834U, + 4606665164148251002U, 4599688422741010356U, + 13823060459595786164U, 4606665164148251002U, + 4601423692641949331U, 4606305777984577632U, + 13829677814839353440U, 4601423692641949331U, + 4604867640218014515U, 4604203581176243359U, + 13827575618031019167U, 4604867640218014515U, + 4587673791460508439U, 4607170170974224083U, + 13830542207828999891U, 4587673791460508439U, + 4607141713064252300U, 4591507261658050721U, + 13814879298512826529U, 4607141713064252300U, + 4603910660507251362U, 4605120315324767624U, + 13828492352179543432U, 4603910660507251362U, + 4606131849150971908U, 4602114767134999006U, + 13825486803989774814U, 4606131849150971908U, + 4598953786765296928U, 4606786509620734768U, + 13830158546475510576U, 4598953786765296928U, + 4606802552898869248U, 4598848011564831930U, + 13822220048419607738U, 4606802552898869248U, + 4602212250118051877U, 4606105796280968177U, + 13829477833135743985U, 4602212250118051877U, + 4605155376589456981U, 4603867938232615808U, + 13827239975087391616U, 4605155376589456981U, + 4591947271803021404U, 4607136295912168606U, + 13830508332766944414U, 4591947271803021404U, + 4607014697483910382U, 4596088445927168004U, + 13819460482781943812U, 4607014697483910382U, + 4603202304363743346U, 4605649044311923410U, + 13829021081166699218U, 4603202304363743346U, + 4605679749231851918U, 4603156351203636159U, + 13826528388058411967U, 4605679749231851918U, + 4596305267720071930U, 4607003915349878877U, + 13830375952204654685U, 4596305267720071930U, + 4606507322377452870U, 4600514338912178239U, + 13823886375766954047U, 4606507322377452870U, + 4600616459743653188U, 4606486172460753999U, + 13829858209315529807U, 4600616459743653188U, + 4604563781218984604U, 4604524701268679793U, + 13827896738123455601U, 4604563781218984604U, + 4569220649180767418U, 4607182376410422530U, + 13830554413265198338U, 4569220649180767418U +}; + +const fpr fpr_p2_tab[] = { + 4611686018427387904U, + 4607182418800017408U, + 4602678819172646912U, + 4598175219545276416U, + 4593671619917905920U, + 4589168020290535424U, + 4584664420663164928U, + 4580160821035794432U, + 4575657221408423936U, + 4571153621781053440U, + 4566650022153682944U +}; |