/*
* Copyright (c) 2018 Thomas Pornin
*
* 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"
/*
* In this file, we handle big integers with a custom format, i.e.
* without the usual one-word header. Value is split into 15-bit words,
* each stored in a 16-bit slot (top bit is zero) in little-endian
* order. The length (in words) is provided explicitly. In some cases,
* the value can be negative (using two's complement representation). In
* some cases, the top word is allowed to have a 16th bit.
*/
/*
* Negate big integer conditionally. The value consists of 'len' words,
* with 15 bits in each word (the top bit of each word should be 0,
* except possibly for the last word). If 'ctl' is 1, the negation is
* computed; otherwise, if 'ctl' is 0, then the value is unchanged.
*/
static void
cond_negate(uint16_t *a, size_t len, uint32_t ctl)
{
size_t k;
uint32_t cc, xm;
cc = ctl;
xm = 0x7FFF & -ctl;
for (k = 0; k < len; k ++) {
uint32_t aw;
aw = a[k];
aw = (aw ^ xm) + cc;
a[k] = aw & 0x7FFF;
cc = (aw >> 15) & 1;
}
}
/*
* Finish modular reduction. Rules on input parameters:
*
* if neg = 1, then -m <= a < 0
* if neg = 0, then 0 <= a < 2*m
*
* If neg = 0, then the top word of a[] may use 16 bits.
*
* Also, modulus m must be odd.
*/
static void
finish_mod(uint16_t *a, size_t len, const uint16_t *m, uint32_t neg)
{
size_t k;
uint32_t cc, xm, ym;
/*
* First pass: compare a (assumed nonnegative) with m.
*/
cc = 0;
for (k = 0; k < len; k ++) {
uint32_t aw, mw;
aw = a[k];
mw = m[k];
cc = (aw - mw - cc) >> 31;
}
/*
* At this point:
* if neg = 1, then we must add m (regardless of cc)
* if neg = 0 and cc = 0, then we must subtract m
* if neg = 0 and cc = 1, then we must do nothing
*/
xm = 0x7FFF & -neg;
ym = -(neg | (1 - cc));
cc = neg;
for (k = 0; k < len; k ++) {
uint32_t aw, mw;
aw = a[k];
mw = (m[k] ^ xm) & ym;
aw = aw - mw - cc;
a[k] = aw & 0x7FFF;
cc = aw >> 31;
}
}
/*
* Compute:
* a <- (a*pa+b*pb)/(2^15)
* b <- (a*qa+b*qb)/(2^15)
* The division is assumed to be exact (i.e. the low word is dropped).
* If the final a is negative, then it is negated. Similarly for b.
* Returned value is the combination of two bits:
* bit 0: 1 if a had to be negated, 0 otherwise
* bit 1: 1 if b had to be negated, 0 otherwise
*
* Factors pa, pb, qa and qb must be at most 2^15 in absolute value.
* Source integers a and b must be nonnegative; top word is not allowed
* to contain an extra 16th bit.
*/
static uint32_t
co_reduce(uint16_t *a, uint16_t *b, size_t len,
int32_t pa, int32_t pb, int32_t qa, int32_t qb)
{
size_t k;
int32_t cca, ccb;
uint32_t nega, negb;
cca = 0;
ccb = 0;
for (k = 0; k < len; k ++) {
uint32_t wa, wb, za, zb;
uint16_t tta, ttb;
/*
* Since:
* |pa| <= 2^15
* |pb| <= 2^15
* 0 <= wa <= 2^15 - 1
* 0 <= wb <= 2^15 - 1
* |cca| <= 2^16 - 1
* Then:
* |za| <= (2^15-1)*(2^16) + (2^16-1) = 2^31 - 1
*
* Thus, the new value of cca is such that |cca| <= 2^16 - 1.
* The same applies to ccb.
*/
wa = a[k];
wb = b[k];
za = wa * (uint32_t)pa + wb * (uint32_t)pb + (uint32_t)cca;
zb = wa * (uint32_t)qa + wb * (uint32_t)qb + (uint32_t)ccb;
if (k > 0) {
a[k - 1] = za & 0x7FFF;
b[k - 1] = zb & 0x7FFF;
}
tta = za >> 15;
ttb = zb >> 15;
cca = *(int16_t *)&tta;
ccb = *(int16_t *)&ttb;
}
a[len - 1] = (uint16_t)cca;
b[len - 1] = (uint16_t)ccb;
nega = (uint32_t)cca >> 31;
negb = (uint32_t)ccb >> 31;
cond_negate(a, len, nega);
cond_negate(b, len, negb);
return nega | (negb << 1);
}
/*
* Compute:
* a <- (a*pa+b*pb)/(2^15) mod m
* b <- (a*qa+b*qb)/(2^15) mod m
*
* m0i is equal to -1/m[0] mod 2^15.
*
* Factors pa, pb, qa and qb must be at most 2^15 in absolute value.
* Source integers a and b must be nonnegative; top word is not allowed
* to contain an extra 16th bit.
*/
static void
co_reduce_mod(uint16_t *a, uint16_t *b, size_t len,
int32_t pa, int32_t pb, int32_t qa, int32_t qb,
const uint16_t *m, uint16_t m0i)
{
size_t k;
int32_t cca, ccb, fa, fb;
cca = 0;
ccb = 0;
fa = ((a[0] * (uint32_t)pa + b[0] * (uint32_t)pb) * m0i) & 0x7FFF;
fb = ((a[0] * (uint32_t)qa + b[0] * (uint32_t)qb) * m0i) & 0x7FFF;
for (k = 0; k < len; k ++) {
uint32_t wa, wb, za, zb;
uint32_t tta, ttb;
/*
* In this loop, carries 'cca' and 'ccb' always fit on
* 17 bits (in absolute value).
*/
wa = a[k];
wb = b[k];
za = wa * (uint32_t)pa + wb * (uint32_t)pb
+ m[k] * (uint32_t)fa + (uint32_t)cca;
zb = wa * (uint32_t)qa + wb * (uint32_t)qb
+ m[k] * (uint32_t)fb + (uint32_t)ccb;
if (k > 0) {
a[k - 1] = za & 0x7FFF;
b[k - 1] = zb & 0x7FFF;
}
/*
* The XOR-and-sub construction below does an arithmetic
* right shift in a portable way (technically, right-shifting
* a negative signed value is implementation-defined in C).
*/
#define M ((uint32_t)1 << 16)
tta = za >> 15;
ttb = zb >> 15;
tta = (tta ^ M) - M;
ttb = (ttb ^ M) - M;
cca = *(int32_t *)&tta;
ccb = *(int32_t *)&ttb;
#undef M
}
a[len - 1] = (uint32_t)cca;
b[len - 1] = (uint32_t)ccb;
/*
* At this point:
* -m <= a < 2*m
* -m <= b < 2*m
* (this is a case of Montgomery reduction)
* The top word of 'a' and 'b' may have a 16-th bit set.
* We may have to add or subtract the modulus.
*/
finish_mod(a, len, m, (uint32_t)cca >> 31);
finish_mod(b, len, m, (uint32_t)ccb >> 31);
}
/* see inner.h */
uint32_t
br_i15_moddiv(uint16_t *x, const uint16_t *y, const uint16_t *m, uint16_t m0i,
uint16_t *t)
{
/*
* Algorithm is an extended binary GCD. We maintain four values
* a, b, u and v, with the following invariants:
*
* a * x = y * u mod m
* b * x = y * v mod m
*
* Starting values are:
*
* a = y
* b = m
* u = x
* v = 0
*
* The formal definition of the algorithm is a sequence of steps:
*
* - If a is even, then a <- a/2 and u <- u/2 mod m.
* - Otherwise, if b is even, then b <- b/2 and v <- v/2 mod m.
* - Otherwise, if a > b, then a <- (a-b)/2 and u <- (u-v)/2 mod m.
* - Otherwise, b <- (b-a)/2 and v <- (v-u)/2 mod m.
*
* Algorithm stops when a = b. At that point, they both are equal
* to GCD(y,m); the modular division succeeds if that value is 1.
* The result of the modular division is then u (or v: both are
* equal at that point).
*
* Each step makes either a or b shrink by at least one bit; hence,
* if m has bit length k bits, then 2k-2 steps are sufficient.
*
*
* Though complexity is quadratic in the size of m, the bit-by-bit
* processing is not very efficient. We can speed up processing by
* remarking that the decisions are taken based only on observation
* of the top and low bits of a and b.
*
* In the loop below, at each iteration, we use the two top words
* of a and b, and the low words of a and b, to compute reduction
* parameters pa, pb, qa and qb such that the new values for a
* and b are:
*
* a' = (a*pa + b*pb) / (2^15)
* b' = (a*qa + b*qb) / (2^15)
*
* the division being exact.
*
* Since the choices are based on the top words, they may be slightly
* off, requiring an optional correction: if a' < 0, then we replace
* pa with -pa, and pb with -pb. The total length of a and b is
* thus reduced by at least 14 bits at each iteration.
*
* The stopping conditions are still the same, though: when a
* and b become equal, they must be both odd (since m is odd,
* the GCD cannot be even), therefore the next operation is a
* subtraction, and one of the values becomes 0. At that point,
* nothing else happens, i.e. one value is stuck at 0, and the
* other one is the GCD.
*/
size_t len, k;
uint16_t *a, *b, *u, *v;
uint32_t num, r;
len = (m[0] + 15) >> 4;
a = t;
b = a + len;
u = x + 1;
v = b + len;
memcpy(a, y + 1, len * sizeof *y);
memcpy(b, m + 1, len * sizeof *m);
memset(v, 0, len * sizeof *v);
/*
* Loop below ensures that a and b are reduced by some bits each,
* for a total of at least 14 bits.
*/
for (num = ((m[0] - (m[0] >> 4)) << 1) + 14; num >= 14; num -= 14) {
size_t j;
uint32_t c0, c1;
uint32_t a0, a1, b0, b1;
uint32_t a_hi, b_hi, a_lo, b_lo;
int32_t pa, pb, qa, qb;
int i;
/*
* Extract top words of a and b. If j is the highest
* index >= 1 such that a[j] != 0 or b[j] != 0, then we want
* (a[j] << 15) + a[j - 1], and (b[j] << 15) + b[j - 1].
* If a and b are down to one word each, then we use a[0]
* and b[0].
*/
c0 = (uint32_t)-1;
c1 = (uint32_t)-1;
a0 = 0;
a1 = 0;
b0 = 0;
b1 = 0;
j = len;
while (j -- > 0) {
uint32_t aw, bw;
aw = a[j];
bw = b[j];
a0 ^= (a0 ^ aw) & c0;
a1 ^= (a1 ^ aw) & c1;
b0 ^= (b0 ^ bw) & c0;
b1 ^= (b1 ^ bw) & c1;
c1 = c0;
c0 &= (((aw | bw) + 0xFFFF) >> 16) - (uint32_t)1;
}
/*
* If c1 = 0, then we grabbed two words for a and b.
* If c1 != 0 but c0 = 0, then we grabbed one word. It
* is not possible that c1 != 0 and c0 != 0, because that
* would mean that both integers are zero.
*/
a1 |= a0 & c1;
a0 &= ~c1;
b1 |= b0 & c1;
b0 &= ~c1;
a_hi = (a0 << 15) + a1;
b_hi = (b0 << 15) + b1;
a_lo = a[0];
b_lo = b[0];
/*
* Compute reduction factors:
*
* a' = a*pa + b*pb
* b' = a*qa + b*qb
*
* such that a' and b' are both multiple of 2^15, but are
* only marginally larger than a and b.
*/
pa = 1;
pb = 0;
qa = 0;
qb = 1;
for (i = 0; i < 15; i ++) {
/*
* At each iteration:
*
* a <- (a-b)/2 if: a is odd, b is odd, a_hi > b_hi
* b <- (b-a)/2 if: a is odd, b is odd, a_hi <= b_hi
* a <- a/2 if: a is even
* b <- b/2 if: a is odd, b is even
*
* We multiply a_lo and b_lo by 2 at each
* iteration, thus a division by 2 really is a
* non-multiplication by 2.
*/
uint32_t r, oa, ob, cAB, cBA, cA;
/*
* cAB = 1 if b must be subtracted from a
* cBA = 1 if a must be subtracted from b
* cA = 1 if a is divided by 2, 0 otherwise
*
* Rules:
*
* cAB and cBA cannot be both 1.
* if a is not divided by 2, b is.
*/
r = GT(a_hi, b_hi);
oa = (a_lo >> i) & 1;
ob = (b_lo >> i) & 1;
cAB = oa & ob & r;
cBA = oa & ob & NOT(r);
cA = cAB | NOT(oa);
/*
* Conditional subtractions.
*/
a_lo -= b_lo & -cAB;
a_hi -= b_hi & -cAB;
pa -= qa & -(int32_t)cAB;
pb -= qb & -(int32_t)cAB;
b_lo -= a_lo & -cBA;
b_hi -= a_hi & -cBA;
qa -= pa & -(int32_t)cBA;
qb -= pb & -(int32_t)cBA;
/*
* Shifting.
*/
a_lo += a_lo & (cA - 1);
pa += pa & ((int32_t)cA - 1);
pb += pb & ((int32_t)cA - 1);
a_hi ^= (a_hi ^ (a_hi >> 1)) & -cA;
b_lo += b_lo & -cA;
qa += qa & -(int32_t)cA;
qb += qb & -(int32_t)cA;
b_hi ^= (b_hi ^ (b_hi >> 1)) & (cA - 1);
}
/*
* Replace a and b with new values a' and b'.
*/
r = co_reduce(a, b, len, pa, pb, qa, qb);
pa -= pa * ((r & 1) << 1);
pb -= pb * ((r & 1) << 1);
qa -= qa * (r & 2);
qb -= qb * (r & 2);
co_reduce_mod(u, v, len, pa, pb, qa, qb, m + 1, m0i);
}
/*
* Now one of the arrays should be 0, and the other contains
* the GCD. If a is 0, then u is 0 as well, and v contains
* the division result.
* Result is correct if and only if GCD is 1.
*/
r = (a[0] | b[0]) ^ 1;
u[0] |= v[0];
for (k = 1; k < len; k ++) {
r |= a[k] | b[k];
u[k] |= v[k];
}
return EQ0(r);
}