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using System;
namespace Org.BouncyCastle.Pqc.Crypto.Falcon
{
class SamplerZ
{
FalconRNG p;
FalconFPR sigma_min;
FPREngine fpre;
internal SamplerZ(FalconRNG p, FalconFPR sigma_min, FPREngine fpre) {
this.p = p;
this.sigma_min = sigma_min;
this.fpre = fpre;
}
internal int Sample(FalconFPR mu, FalconFPR isigma) {
return this.sampler(mu, isigma);
}
/*
* Sample an integer value along a half-gaussian distribution centered
* on zero and standard deviation 1.8205, with a precision of 72 bits.
*/
int gaussian0_sampler(FalconRNG p)
{
uint[] dist = {
10745844u, 3068844u, 3741698u,
5559083u, 1580863u, 8248194u,
2260429u, 13669192u, 2736639u,
708981u, 4421575u, 10046180u,
169348u, 7122675u, 4136815u,
30538u, 13063405u, 7650655u,
4132u, 14505003u, 7826148u,
417u, 16768101u, 11363290u,
31u, 8444042u, 8086568u,
1u, 12844466u, 265321u,
0u, 1232676u, 13644283u,
0u, 38047u, 9111839u,
0u, 870u, 6138264u,
0u, 14u, 12545723u,
0u, 0u, 3104126u,
0u, 0u, 28824u,
0u, 0u, 198u,
0u, 0u, 1u
};
uint v0, v1, v2, hi;
ulong lo;
int u;
int z;
/*
* Get a random 72-bit value, into three 24-bit limbs v0..v2.
*/
lo = p.prng_get_u64();
hi = p.prng_get_u8();
v0 = (uint)lo & 0xFFFFFF;
v1 = (uint)(lo >> 24) & 0xFFFFFF;
v2 = (uint)(lo >> 48) | (hi << 16);
/*
* Sampled value is z, such that v0..v2 is lower than the first
* z elements of the table.
*/
z = 0;
for (u = 0; u < dist.Length; u += 3) {
uint w0, w1, w2, cc;
w0 = dist[u + 2];
w1 = dist[u + 1];
w2 = dist[u + 0];
cc = (v0 - w0) >> 31;
cc = (v1 - w1 - cc) >> 31;
cc = (v2 - w2 - cc) >> 31;
z += (int)cc;
}
return z;
}
/*
* Sample a bit with probability exp(-x) for some x >= 0.
*/
int BerExp(FalconRNG p, FalconFPR x, FalconFPR ccs)
{
int s, i;
FalconFPR r;
uint sw, w;
ulong z;
/*
* Reduce x modulo log(2): x = s*log(2) + r, with s an integer,
* and 0 <= r < log(2). Since x >= 0, we can use this.fpre.fpr_trunc().
*/
s = (int)this.fpre.fpr_trunc(this.fpre.fpr_mul(x, this.fpre.fpr_inv_log2));
r = this.fpre.fpr_sub(x, this.fpre.fpr_mul(this.fpre.fpr_of(s), this.fpre.fpr_log2));
/*
* It may happen (quite rarely) that s >= 64; if sigma = 1.2
* (the minimum value for sigma), r = 0 and b = 1, then we get
* s >= 64 if the half-Gaussian produced a z >= 13, which happens
* with probability about 0.000000000230383991, which is
* approximatively equal to 2^(-32). In any case, if s >= 64,
* then BerExp will be non-zero with probability less than
* 2^(-64), so we can simply saturate s at 63.
*/
sw = (uint)s;
sw ^= (uint)((sw ^ 63) & -((63 - sw) >> 31));
s = (int)sw;
/*
* Compute exp(-r); we know that 0 <= r < log(2) at this point, so
* we can use this.fpre.fpr_expm_p63(), which yields a result scaled to 2^63.
* We scale it up to 2^64, then right-shift it by s bits because
* we really want exp(-x) = 2^(-s)*exp(-r).
*
* The "-1" operation makes sure that the value fits on 64 bits
* (i.e. if r = 0, we may get 2^64, and we prefer 2^64-1 in that
* case). The bias is negligible since this.fpre.fpr_expm_p63() only computes
* with 51 bits of precision or so.
*/
z = ((this.fpre.fpr_expm_p63(r, ccs) << 1) - 1) >> s;
/*
* Sample a bit with probability exp(-x). Since x = s*log(2) + r,
* exp(-x) = 2^-s * exp(-r), we compare lazily exp(-x) with the
* PRNG output to limit its consumption, the sign of the difference
* yields the expected result.
*/
i = 64;
do {
i -= 8;
w = p.prng_get_u8() - ((uint)(z >> i) & 0xFF);
} while (w == 0 && i > 0);
return (int)(w >> 31);
}
/*
* The sampler produces a random integer that follows a discrete Gaussian
* distribution, centered on mu, and with standard deviation sigma. The
* provided parameter isigma is equal to 1/sigma.
*
* The value of sigma MUST lie between 1 and 2 (i.e. isigma lies between
* 0.5 and 1); in Falcon, sigma should always be between 1.2 and 1.9.
*/
int sampler(FalconFPR mu, FalconFPR isigma)
{
int s;
FalconFPR r, dss, ccs;
/*
* Center is mu. We compute mu = s + r where s is an integer
* and 0 <= r < 1.
*/
s = (int)this.fpre.fpr_floor(mu);
r = this.fpre.fpr_sub(mu, this.fpre.fpr_of(s));
/*
* dss = 1/(2*sigma^2) = 0.5*(isigma^2).
*/
dss = this.fpre.fpr_half(this.fpre.fpr_sqr(isigma));
/*
* ccs = sigma_min / sigma = sigma_min * isigma.
*/
ccs = this.fpre.fpr_mul(isigma, this.sigma_min);
/*
* We now need to sample on center r.
*/
for (;;) {
int z0, z, b;
FalconFPR x;
/*
* Sample z for a Gaussian distribution. Then get a
* random bit b to turn the sampling into a bimodal
* distribution: if b = 1, we use z+1, otherwise we
* use -z. We thus have two situations:
*
* - b = 1: z >= 1 and sampled against a Gaussian
* centered on 1.
* - b = 0: z <= 0 and sampled against a Gaussian
* centered on 0.
*/
z0 = gaussian0_sampler(this.p);
b = (int)this.p.prng_get_u8() & 1;
z = b + ((b << 1) - 1) * z0;
/*
* Rejection sampling. We want a Gaussian centered on r;
* but we sampled against a Gaussian centered on b (0 or
* 1). But we know that z is always in the range where
* our sampling distribution is greater than the Gaussian
* distribution, so rejection works.
*
* We got z with distribution:
* G(z) = exp(-((z-b)^2)/(2*sigma0^2))
* We target distribution:
* S(z) = exp(-((z-r)^2)/(2*sigma^2))
* Rejection sampling works by keeping the value z with
* probability S(z)/G(z), and starting again otherwise.
* This requires S(z) <= G(z), which is the case here.
* Thus, we simply need to keep our z with probability:
* P = exp(-x)
* where:
* x = ((z-r)^2)/(2*sigma^2) - ((z-b)^2)/(2*sigma0^2)
*
* Here, we scale up the Bernouilli distribution, which
* makes rejection more probable, but makes rejection
* rate sufficiently decorrelated from the Gaussian
* center and standard deviation that the whole sampler
* can be said to be constant-time.
*/
x = this.fpre.fpr_mul(this.fpre.fpr_sqr(this.fpre.fpr_sub(this.fpre.fpr_of(z), r)), dss);
x = this.fpre.fpr_sub(x, this.fpre.fpr_mul(this.fpre.fpr_of(z0 * z0), this.fpre.fpr_inv_2sqrsigma0));
if (BerExp(this.p, x, ccs) != 0) {
/*
* Rejection sampling was centered on r, but the
* actual center is mu = s + r.
*/
return s + z;
}
}
}
}
}
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