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pqcrypto/crypto_sign/falcon-1024/clean/fpr.c

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/*
* 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>
*/
#include "inner.h"
/*
* 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)
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_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_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_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_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);
}
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, 4607009347991985328U,
4603179351334086856U, 4605664432017547683U,
13829036468872323491U, 4603179351334086856U,
4605664432017547683U, 4603179351334086856U,
13826551388188862664U, 4605664432017547683U,
4596196889902818827U, 4607009347991985328U,
13830381384846761136U, 4596196889902818827U,
4607139046673687846U, 4591727299969791020U,
13815099336824566828U, 4607139046673687846U,
4603889326261607894U, 4605137878724712257U,
13828509915579488065U, 4603889326261607894U,
4606118860100255153U, 4602163548591158843U,
13825535585445934651U, 4606118860100255153U,
4598900923775164166U, 4606794571824115162U,
13830166608678890970U, 4598900923775164166U,
4606794571824115162U, 4598900923775164166U,
13822272960629939974U, 4606794571824115162U,
4602163548591158843U, 4606118860100255153U,
13829490896955030961U, 4602163548591158843U,
4605137878724712257U, 4603889326261607894U,
13827261363116383702U, 4605137878724712257U,
4591727299969791020U, 4607139046673687846U,
13830511083528463654U, 4591727299969791020U,
4607171569234046334U, 4587232218149935124U,
13810604255004710932U, 4607171569234046334U,
4604224084862889120U, 4604849113969373103U,
13828221150824148911U, 4604224084862889120U,
4606317631232591731U, 4601373767755717824U,
13824745804610493632U, 4606317631232591731U,
4599740487990714333U, 4606655894547498725U,
13830027931402274533U, 4599740487990714333U,
4606912484326125783U, 4597922303871901467U,
13821294340726677275U, 4606912484326125783U,
4602805845399633902U, 4605900952042040894U,
13829272988896816702U, 4602805845399633902U,
4605409869824231233U, 4603540801876750389U,
13826912838731526197U, 4605409869824231233U,
4594454542771183930U, 4607084929468638487U,
13830456966323414295U, 4594454542771183930U,
4607084929468638487U, 4594454542771183930U,
13817826579625959738U, 4607084929468638487U,
4603540801876750389U, 4605409869824231233U,
13828781906679007041U, 4603540801876750389U,
4605900952042040894U, 4602805845399633902U,
13826177882254409710U, 4605900952042040894U,
4597922303871901467U, 4606912484326125783U,
13830284521180901591U, 4597922303871901467U,
4606655894547498725U, 4599740487990714333U,
13823112524845490141U, 4606655894547498725U,
4601373767755717824U, 4606317631232591731U,
13829689668087367539U, 4601373767755717824U,
4604849113969373103U, 4604224084862889120U,
13827596121717664928U, 4604849113969373103U,
4587232218149935124U, 4607171569234046334U,
13830543606088822142U, 4587232218149935124U,
4607179706000002317U, 4582730748936808062U,
13806102785791583870U, 4607179706000002317U,
4604386048625945823U, 4604698657331085206U,
13828070694185861014U, 4604386048625945823U,
4606409688975526202U, 4600971798440897930U,
13824343835295673738U, 4606409688975526202U,
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4591066993883984169U, 4607146792632922887U,
13830518829487698695U, 4591066993883984169U,
4607167120476811757U, 4588556721781247689U,
13811928758636023497U, 4607167120476811757U,
4604162403772767740U, 4604904503566677638U,
13828276540421453446U, 4604162403772767740U,
4606281842017099424U, 4601523323048804569U,
13824895359903580377U, 4606281842017099424U,
4599584122834874440U, 4606683463531482757U,
13830055500386258565U, 4599584122834874440U,
4606891971185517504U, 4598209407597805010U,
13821581444452580818U, 4606891971185517504U,
4602734543519989142U, 4605943243960030558U,
13829315280814806366U, 4602734543519989142U,
4605360179893335444U, 4603607160562208225U,
13826979197416984033U, 4605360179893335444U,
4594126307716900071U, 4607096716058023245U,
13830468752912799053U, 4594126307716900071U,
4607072388129742377U, 4594782329999411347U,
13818154366854187155U, 4607072388129742377U,
4603473988668005304U, 4605458946901419122U,
13828830983756194930U, 4603473988668005304U,
4605858005670328613U, 4602876755014813164U,
13826248791869588972U, 4605858005670328613U,
4597600270510262682U, 4606932257325205256U,
13830304294179981064U, 4597600270510262682U,
4606627607157935956U, 4599896339047301634U,
13823268375902077442U, 4606627607157935956U,
4601223560006786057U, 4606352730697093817U,
13829724767551869625U, 4601223560006786057U,
4604793159020491611U, 4604285253548209224U,
13827657290402985032U, 4604793159020491611U,
4585907115494236537U, 4607175255902437396U,
13830547292757213204U, 4585907115494236537U,
4607177290141793710U, 4585023436363055487U,
13808395473217831295U, 4607177290141793710U,
4604325745441780828U, 4604755543975806820U,
13828127580830582628U, 4604325745441780828U,
4606375745674388705U, 4601123065313358619U,
13824495102168134427U, 4606375745674388705U,
4599999947619525579U, 4606608350964852124U,
13829980387819627932U, 4599999947619525579U,
4606945027305114062U, 4597385183080791534U,
13820757219935567342U, 4606945027305114062U,
4602923807199184054U, 4605829012964735987U,
13829201049819511795U, 4602923807199184054U,
4605491322423429598U, 4603429196809300824U,
13826801233664076632U, 4605491322423429598U,
4595000592312171144U, 4607063608453868552U,
13830435645308644360U, 4595000592312171144U,
4607104153983298999U, 4593907249284540294U,
13817279286139316102U, 4607104153983298999U,
4603651144395358093U, 4605326714874986465U,
13828698751729762273U, 4603651144395358093U,
4605971073215153165U, 4602686793990243041U,
13826058830845018849U, 4605971073215153165U,
4598316292140394014U, 4606877885424248132U,
13830249922279023940U, 4598316292140394014U,
4606701442584137310U, 4599479600326345459U,
13822851637181121267U, 4606701442584137310U,
4601622657843474729U, 4606257600839867033U,
13829629637694642841U, 4601622657843474729U,
4604941113561600762U, 4604121000955189926U,
13827493037809965734U, 4604941113561600762U,
4589303678145802340U, 4607163731439411601U,
13830535768294187409U, 4589303678145802340U,
4607151534426937478U, 4590626485056654602U,
13813998521911430410U, 4607151534426937478U,
4603995455647851249U, 4605049409688478101U,
13828421446543253909U, 4603995455647851249U,
4606183055233559255U, 4601918851211878557U,
13825290888066654365U, 4606183055233559255U,
4599164736579548843U, 4606753451050079834U,
13830125487904855642U, 4599164736579548843U,
4606833664420673202U, 4598635880488956483U,
13822007917343732291U, 4606833664420673202U,
4602406247776385022U, 4606052795787882823U,
13829424832642658631U, 4602406247776385022U,
4605224709411790590U, 4603781852316960384U,
13827153889171736192U, 4605224709411790590U,
4592826452951465409U, 4607124449686274900U,
13830496486541050708U, 4592826452951465409U,
4607035262954517034U, 4595654028864046335U,
13819026065718822143U, 4607035262954517034U,
4603293641160266722U, 4605586791482848547U,
13828958828337624355U, 4603293641160266722U,
4605740310302420207U, 4603063884010218172U,
13826435920864993980U, 4605740310302420207U,
4596738097012783531U, 4606981354314050484U,
13830353391168826292U, 4596738097012783531U,
4606548680329491866U, 4600309328230211502U,
13823681365084987310U, 4606548680329491866U,
4600819913163773071U, 4606442934727379583U,
13829814971582155391U, 4600819913163773071U,
4604641218080103285U, 4604445825685214043U,
13827817862539989851U, 4604641218080103285U,
4579996072175835083U, 4607181359080094673U,
13830553395934870481U, 4579996072175835083U,
4607180341788068727U, 4581846703643734566U,
13805218740498510374U, 4607180341788068727U,
4604406033021674239U, 4604679572075463103U,
13828051608930238911U, 4604406033021674239U,
4606420848538580260U, 4600921238092511730U,
13824293274947287538U, 4606420848538580260U,
4600206446098256018U, 4606568886807728474U,
13829940923662504282U, 4600206446098256018U,
4606969576261663845U, 4596954088216812973U,
13820326125071588781U, 4606969576261663845U,
4603017373458244943U, 4605770164172969910U,
13829142201027745718U, 4603017373458244943U,
4605555245917486022U, 4603339021357904144U,
13826711058212679952U, 4605555245917486022U,
4595436449949385485U, 4607045045516813836U,
13830417082371589644U, 4595436449949385485U,
4607118021058468598U, 4593265590854265407U,
13816637627709041215U, 4607118021058468598U,
4603738491917026584U, 4605258978359093269U,
13828631015213869077U, 4603738491917026584U,
4606025850160239809U, 4602502755147763107U,
13825874792002538915U, 4606025850160239809U,
4598529532600161144U, 4606848731493011465U,
13830220768347787273U, 4598529532600161144U,
4606736437002195879U, 4599269903251194481U,
13822641940105970289U, 4606736437002195879U,
4601820425647934753U, 4606208206518262803U,
13829580243373038611U, 4601820425647934753U,
4605013567986435066U, 4604037525321326463U,
13827409562176102271U, 4605013567986435066U,
4590185751760970393U, 4607155938267770208U,
13830527975122546016U, 4590185751760970393U,
4607160003989618959U, 4589744810590291021U,
13813116847445066829U, 4607160003989618959U,
4604079374282302598U, 4604977468824438271U,
13828349505679214079U, 4604079374282302598U,
4606233055365547081U, 4601721693286060937U,
13825093730140836745U, 4606233055365547081U,
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
};
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