[BearSSL] / src / int / i15_core.c

/*
 * Copyright (c) 2017 Thomas Pornin <pornin@bolet.org>
 *
 * 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.
 */

#include "inner.h"

/*
 * This file contains the core "big integer" functions for the i15
 * implementation, that represents integers as sequences of 15-bit
 * words.
 */

/* see inner.h */
uint32_t
br_i15_iszero(const uint16_t *x)
{
        uint32_t z;
        size_t u;

        z = 0;
        for (u = (x[0] + 15) >> 4; u > 0; u --) {
                z |= x[u];
        }
        return ~(z | -z) >> 31;
}

/* see inner.h */
uint16_t
br_i15_ninv15(uint16_t x)
{
        uint32_t y;

        y = 2 - x;
        y = MUL15(y, 2 - MUL15(x, y));
        y = MUL15(y, 2 - MUL15(x, y));
        y = MUL15(y, 2 - MUL15(x, y));
        return MUX(x & 1, -y, 0) & 0x7FFF;
}

/* see inner.h */
uint32_t
br_i15_add(uint16_t *a, const uint16_t *b, uint32_t ctl)
{
        uint32_t cc;
        size_t u, m;

        cc = 0;
        m = (a[0] + 31) >> 4;
        for (u = 1; u < m; u ++) {
                uint32_t aw, bw, naw;

                aw = a[u];
                bw = b[u];
                naw = aw + bw + cc;
                cc = naw >> 15;
                a[u] = MUX(ctl, naw & 0x7FFF, aw);
        }
        return cc;
}

/* see inner.h */
uint32_t
br_i15_sub(uint16_t *a, const uint16_t *b, uint32_t ctl)
{
        uint32_t cc;
        size_t u, m;

        cc = 0;
        m = (a[0] + 31) >> 4;
        for (u = 1; u < m; u ++) {
                uint32_t aw, bw, naw;

                aw = a[u];
                bw = b[u];
                naw = aw - bw - cc;
                cc = naw >> 31;
                a[u] = MUX(ctl, naw & 0x7FFF, aw);
        }
        return cc;
}

/*
 * Constant-time division. The divisor must not be larger than 16 bits,
 * and the quotient must fit on 17 bits.
 */
static uint32_t
divrem16(uint32_t x, uint32_t d, uint32_t *r)
{
        int i;
        uint32_t q;

        q = 0;
        d <<= 16;
        for (i = 16; i >= 0; i --) {
                uint32_t ctl;

                ctl = LE(d, x);
                q |= ctl << i;
                x -= (-ctl) & d;
                d >>= 1;
        }
        if (r != NULL) {
                *r = x;
        }
        return q;
}

/* see inner.h */
void
br_i15_muladd_small(uint16_t *x, uint16_t z, const uint16_t *m)
{
        /*
         * Constant-time: we accept to leak the exact bit length of the
         * modulus m.
         */
        unsigned m_bitlen, mblr;
        size_t u, mlen;
        uint32_t hi, a0, a, b, q;
        uint32_t cc, tb, over, under;

        /*
         * Simple case: the modulus fits on one word.
         */
        m_bitlen = m[0];
        if (m_bitlen == 0) {
                return;
        }
        if (m_bitlen <= 15) {
                uint32_t rem;

                divrem16(((uint32_t)x[1] << 15) | z, m[1], &rem);
                x[1] = rem;
                return;
        }
        mlen = (m_bitlen + 15) >> 4;
        mblr = m_bitlen & 15;

        /*
         * Principle: we estimate the quotient (x*2^15+z)/m by
         * doing a 30/15 division with the high words.
         *
         * Let:
         *   w = 2^15
         *   a = (w*a0 + a1) * w^N + a2
         *   b = b0 * w^N + b2
         * such that:
         *   0 <= a0 < w
         *   0 <= a1 < w
         *   0 <= a2 < w^N
         *   w/2 <= b0 < w
         *   0 <= b2 < w^N
         *   a < w*b
         * I.e. the two top words of a are a0:a1, the top word of b is
         * b0, we ensured that b0 is "full" (high bit set), and a is
         * such that the quotient q = a/b fits on one word (0 <= q < w).
         *
         * If a = b*q + r (with 0 <= r < q), then we can estimate q by
         * using a division on the top words:
         *   a0*w + a1 = b0*u + v (with 0 <= v < b0)
         * Then the following holds:
         *   0 <= u <= w
         *   u-2 <= q <= u
         */
        hi = x[mlen];
        if (mblr == 0) {
                a0 = x[mlen];
                memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
                x[1] = z;
                a = (a0 << 15) + x[mlen];
                b = m[mlen];
        } else {
                a0 = (x[mlen] << (15 - mblr)) | (x[mlen - 1] >> mblr);
                memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
                x[1] = z;
                a = (a0 << 15) | (((x[mlen] << (15 - mblr))
                        | (x[mlen - 1] >> mblr)) & 0x7FFF);
                b = (m[mlen] << (15 - mblr)) | (m[mlen - 1] >> mblr);
        }
        q = divrem16(a, b, NULL);

        /*
         * We computed an estimate for q, but the real one may be q,
         * q-1 or q-2; moreover, the division may have returned a value
         * 8000 or even 8001 if the two high words were identical, and
         * we want to avoid values beyond 7FFF. We thus adjust q so
         * that the "true" multiplier will be q+1, q or q-1, and q is
         * in the 0000..7FFF range.
         */
        q = MUX(EQ(b, a0), 0x7FFF, q - 1 + ((q - 1) >> 31));

        /*
         * We subtract q*m from x (x has an extra high word of value 'hi').
         * Since q may be off by 1 (in either direction), we may have to
         * add or subtract m afterwards.
         *
         * The 'tb' flag will be true (1) at the end of the loop if the
         * result is greater than or equal to the modulus (not counting
         * 'hi' or the carry).
         */
        cc = 0;
        tb = 1;
        for (u = 1; u <= mlen; u ++) {
                uint32_t mw, zl, xw, nxw;

                mw = m[u];
                zl = MUL15(mw, q) + cc;
                cc = zl >> 15;
                zl &= 0x7FFF;
                xw = x[u];
                nxw = xw - zl;
                cc += nxw >> 31;
                nxw &= 0x7FFF;
                x[u] = nxw;
                tb = MUX(EQ(nxw, mw), tb, GT(nxw, mw));
        }

        /*
         * If we underestimated q, then either cc < hi (one extra bit
         * beyond the top array word), or cc == hi and tb is true (no
         * extra bit, but the result is not lower than the modulus).
         *
         * If we overestimated q, then cc > hi.
         */
        over = GT(cc, hi);
        under = ~over & (tb | LT(cc, hi));
        br_i15_add(x, m, over);
        br_i15_sub(x, m, under);
}

/* see inner.h */
void
br_i15_montymul(uint16_t *d, const uint16_t *x, const uint16_t *y,
        const uint16_t *m, uint16_t m0i)
{
        size_t len, len4, u, v;
        uint32_t dh;

        len = (m[0] + 15) >> 4;
        len4 = len & ~(size_t)3;
        br_i15_zero(d, m[0]);
        dh = 0;
        for (u = 0; u < len; u ++) {
                uint32_t f, xu, r, zh;

                xu = x[u + 1];
                f = MUL15((d[1] + MUL15(x[u + 1], y[1])) & 0x7FFF, m0i)
                        & 0x7FFF;

                r = 0;
                for (v = 0; v < len4; v += 4) {
                        uint32_t z;

                        z = d[v + 1] + MUL15(xu, y[v + 1])
                                + MUL15(f, m[v + 1]) + r;
                        r = z >> 15;
                        d[v + 0] = z & 0x7FFF;
                        z = d[v + 2] + MUL15(xu, y[v + 2])
                                + MUL15(f, m[v + 2]) + r;
                        r = z >> 15;
                        d[v + 1] = z & 0x7FFF;
                        z = d[v + 3] + MUL15(xu, y[v + 3])
                                + MUL15(f, m[v + 3]) + r;
                        r = z >> 15;
                        d[v + 2] = z & 0x7FFF;
                        z = d[v + 4] + MUL15(xu, y[v + 4])
                                + MUL15(f, m[v + 4]) + r;
                        r = z >> 15;
                        d[v + 3] = z & 0x7FFF;
                }
                for (; v < len; v ++) {
                        uint32_t z;

                        z = d[v + 1] + MUL15(xu, y[v + 1])
                                + MUL15(f, m[v + 1]) + r;
                        r = z >> 15;
                        d[v + 0] = z & 0x7FFF;
                }

                zh = dh + r;
                d[len] = zh & 0x7FFF;
                dh = zh >> 15;
        }

        /*
         * Restore the bit length (it was overwritten in the loop above).
         */
        d[0] = m[0];

        /*
         * d[] may be greater than m[], but it is still lower than twice
         * the modulus.
         */
        br_i15_sub(d, m, NEQ(dh, 0) | NOT(br_i15_sub(d, m, 0)));
}

/* see inner.h */
void
br_i15_to_monty(uint16_t *x, const uint16_t *m)
{
        unsigned k;

        for (k = (m[0] + 15) >> 4; k > 0; k --) {
                br_i15_muladd_small(x, 0, m);
        }
}

/* see inner.h */
void
br_i15_modpow(uint16_t *x,
        const unsigned char *e, size_t elen,
        const uint16_t *m, uint16_t m0i, uint16_t *t1, uint16_t *t2)
{
        size_t mlen;
        unsigned k;

        mlen = ((m[0] + 31) >> 4) * sizeof m[0];
        memcpy(t1, x, mlen);
        br_i15_to_monty(t1, m);
        br_i15_zero(x, m[0]);
        x[1] = 1;
        for (k = 0; k < ((unsigned)elen << 3); k ++) {
                uint32_t ctl;

                ctl = (e[elen - 1 - (k >> 3)] >> (k & 7)) & 1;
                br_i15_montymul(t2, x, t1, m, m0i);
                CCOPY(ctl, x, t2, mlen);
                br_i15_montymul(t2, t1, t1, m, m0i);
                memcpy(t1, t2, mlen);
        }
}

/* see inner.h */
void
br_i15_encode(void *dst, size_t len, const uint16_t *x)
{
        unsigned char *buf;
        size_t u, xlen;
        uint32_t acc;
        int acc_len;

        xlen = (x[0] + 15) >> 4;
        if (xlen == 0) {
                memset(dst, 0, len);
                return;
        }
        u = 1;
        acc = 0;
        acc_len = 0;
        buf = dst;
        while (len -- > 0) {
                if (acc_len < 8) {
                        if (u <= xlen) {
                                acc += (uint32_t)x[u ++] << acc_len;
                        }
                        acc_len += 15;
                }
                buf[len] = (unsigned char)acc;
                acc >>= 8;
                acc_len -= 8;
        }
}

/* see inner.h */
uint32_t
br_i15_decode_mod(uint16_t *x, const void *src, size_t len, const uint16_t *m)
{
        /*
         * Two-pass algorithm: in the first pass, we determine whether the
         * value fits; in the second pass, we do the actual write.
         *
         * During the first pass, 'r' contains the comparison result so
         * far:
         *  0x00000000   value is equal to the modulus
         *  0x00000001   value is greater than the modulus
         *  0xFFFFFFFF   value is lower than the modulus
         *
         * Since we iterate starting with the least significant bytes (at
         * the end of src[]), each new comparison overrides the previous
         * except when the comparison yields 0 (equal).
         *
         * During the second pass, 'r' is either 0xFFFFFFFF (value fits)
         * or 0x00000000 (value does not fit).
         *
         * We must iterate over all bytes of the source, _and_ possibly
         * some extra virutal bytes (with value 0) so as to cover the
         * complete modulus as well. We also add 4 such extra bytes beyond
         * the modulus length because it then guarantees that no accumulated
         * partial word remains to be processed.
         */
        const unsigned char *buf;
        size_t mlen, tlen;
        int pass;
        uint32_t r;

        buf = src;
        mlen = (m[0] + 15) >> 4;
        tlen = (mlen << 1);
        if (tlen < len) {
                tlen = len;
        }
        tlen += 4;
        r = 0;
        for (pass = 0; pass < 2; pass ++) {
                size_t u, v;
                uint32_t acc;
                int acc_len;

                v = 1;
                acc = 0;
                acc_len = 0;
                for (u = 0; u < tlen; u ++) {
                        uint32_t b;

                        if (u < len) {
                                b = buf[len - 1 - u];
                        } else {
                                b = 0;
                        }
                        acc |= (b << acc_len);
                        acc_len += 8;
                        if (acc_len >= 15) {
                                uint32_t xw;

                                xw = acc & (uint32_t)0x7FFF;
                                acc_len -= 15;
                                acc = b >> (8 - acc_len);
                                if (v <= mlen) {
                                        if (pass) {
                                                x[v] = r & xw;
                                        } else {
                                                uint32_t cc;

                                                cc = (uint32_t)CMP(xw, m[v]);
                                                r = MUX(EQ(cc, 0), r, cc);
                                        }
                                } else {
                                        if (!pass) {
                                                r = MUX(EQ(xw, 0), r, 1);
                                        }
                                }
                                v ++;
                        }
                }

                /*
                 * When we reach this point at the end of the first pass:
                 * r is either 0, 1 or -1; we want to set r to 0 if it
                 * is equal to 0 or 1, and leave it to -1 otherwise.
                 *
                 * When we reach this point at the end of the second pass:
                 * r is either 0 or -1; we want to leave that value
                 * untouched. This is a subcase of the previous.
                 */
                r >>= 1;
                r |= (r << 1);
        }

        x[0] = m[0];
        return r & (uint32_t)1;
}