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putty/mpint.c

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87 KiB
C
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#include <assert.h>
#include <limits.h>
#include <stdio.h>
#include "defs.h"
#include "misc.h"
#include "puttymem.h"
#include "mpint.h"
#include "mpint_i.h"
#define SIZE_T_BITS (CHAR_BIT * sizeof(size_t))
/*
* Inline helpers to take min and max of size_t values, used
* throughout this code.
*/
static inline size_t size_t_min(size_t a, size_t b)
{
return a < b ? a : b;
}
static inline size_t size_t_max(size_t a, size_t b)
{
return a > b ? a : b;
}
/*
* Helper to fetch a word of data from x with array overflow checking.
* If x is too short to have that word, 0 is returned.
*/
static inline BignumInt mp_word(mp_int *x, size_t i)
{
return i < x->nw ? x->w[i] : 0;
}
/*
* Shift an ordinary C integer by BIGNUM_INT_BITS, in a way that
* avoids writing a shift operator whose RHS is greater or equal to
* the size of the type, because that's undefined behaviour in C.
*
* In fact we must avoid even writing it in a definitely-untaken
* branch of an if, because compilers will sometimes warn about
* that. So you can't just write 'shift too big ? 0 : n >> shift',
* because even if 'shift too big' is a constant-expression
* evaluating to false, you can still get complaints about the
* else clause of the ?:.
*
* So we have to re-check _inside_ that clause, so that the shift
* count is reset to something nonsensical but safe in the case
* where the clause wasn't going to be taken anyway.
*/
static uintmax_t shift_right_by_one_word(uintmax_t n)
{
bool shift_too_big = BIGNUM_INT_BYTES >= sizeof(n);
return shift_too_big ? 0 :
n >> (shift_too_big ? 0 : BIGNUM_INT_BITS);
}
static uintmax_t shift_left_by_one_word(uintmax_t n)
{
bool shift_too_big = BIGNUM_INT_BYTES >= sizeof(n);
return shift_too_big ? 0 :
n << (shift_too_big ? 0 : BIGNUM_INT_BITS);
}
mp_int *mp_make_sized(size_t nw)
{
mp_int *x = snew_plus(mp_int, nw * sizeof(BignumInt));
assert(nw); /* we outlaw the zero-word mp_int */
x->nw = nw;
x->w = snew_plus_get_aux(x);
mp_clear(x);
return x;
}
mp_int *mp_new(size_t maxbits)
{
size_t words = (maxbits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
return mp_make_sized(words);
}
mp_int *mp_from_integer(uintmax_t n)
{
mp_int *x = mp_make_sized(
(sizeof(n) + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES);
for (size_t i = 0; i < x->nw; i++)
x->w[i] = n >> (i * BIGNUM_INT_BITS);
return x;
}
size_t mp_max_bytes(mp_int *x)
{
return x->nw * BIGNUM_INT_BYTES;
}
size_t mp_max_bits(mp_int *x)
{
return x->nw * BIGNUM_INT_BITS;
}
void mp_free(mp_int *x)
{
mp_clear(x);
smemclr(x, sizeof(*x));
sfree(x);
}
void mp_dump(FILE *fp, const char *prefix, mp_int *x, const char *suffix)
{
fprintf(fp, "%s0x", prefix);
for (size_t i = mp_max_bytes(x); i-- > 0 ;)
fprintf(fp, "%02X", mp_get_byte(x, i));
fputs(suffix, fp);
}
void mp_copy_into(mp_int *dest, mp_int *src)
{
size_t copy_nw = size_t_min(dest->nw, src->nw);
memmove(dest->w, src->w, copy_nw * sizeof(BignumInt));
smemclr(dest->w + copy_nw, (dest->nw - copy_nw) * sizeof(BignumInt));
}
void mp_copy_integer_into(mp_int *r, uintmax_t n)
{
for (size_t i = 0; i < r->nw; i++) {
r->w[i] = n;
n = shift_right_by_one_word(n);
}
}
/*
* Conditional selection is done by negating 'which', to give a mask
* word which is all 1s if which==1 and all 0s if which==0. Then you
* can select between two inputs a,b without data-dependent control
* flow by XORing them to get their difference; ANDing with the mask
* word to replace that difference with 0 if which==0; and XORing that
* into a, which will either turn it into b or leave it alone.
*
* This trick will be used throughout this code and taken as read the
* rest of the time (or else I'd be here all week typing comments),
* but I felt I ought to explain it in words _once_.
*/
void mp_select_into(mp_int *dest, mp_int *src0, mp_int *src1,
unsigned which)
{
BignumInt mask = -(BignumInt)(1 & which);
for (size_t i = 0; i < dest->nw; i++) {
BignumInt srcword0 = mp_word(src0, i), srcword1 = mp_word(src1, i);
dest->w[i] = srcword0 ^ ((srcword1 ^ srcword0) & mask);
}
}
void mp_cond_swap(mp_int *x0, mp_int *x1, unsigned swap)
{
assert(x0->nw == x1->nw);
volatile BignumInt mask = -(BignumInt)(1 & swap);
for (size_t i = 0; i < x0->nw; i++) {
BignumInt diff = (x0->w[i] ^ x1->w[i]) & mask;
x0->w[i] ^= diff;
x1->w[i] ^= diff;
}
}
void mp_clear(mp_int *x)
{
smemclr(x->w, x->nw * sizeof(BignumInt));
}
void mp_cond_clear(mp_int *x, unsigned clear)
{
BignumInt mask = ~-(BignumInt)(1 & clear);
for (size_t i = 0; i < x->nw; i++)
x->w[i] &= mask;
}
/*
* Common code between mp_from_bytes_{le,be} which reads bytes in an
* arbitrary arithmetic progression.
*/
static mp_int *mp_from_bytes_int(ptrlen bytes, size_t m, size_t c)
{
size_t nw = (bytes.len + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES;
nw = size_t_max(nw, 1);
mp_int *n = mp_make_sized(nw);
for (size_t i = 0; i < bytes.len; i++)
n->w[i / BIGNUM_INT_BYTES] |=
(BignumInt)(((const unsigned char *)bytes.ptr)[m*i+c]) <<
(8 * (i % BIGNUM_INT_BYTES));
return n;
}
mp_int *mp_from_bytes_le(ptrlen bytes)
{
return mp_from_bytes_int(bytes, 1, 0);
}
mp_int *mp_from_bytes_be(ptrlen bytes)
{
return mp_from_bytes_int(bytes, -1, bytes.len - 1);
}
static mp_int *mp_from_words(size_t nw, const BignumInt *w)
{
mp_int *x = mp_make_sized(nw);
memcpy(x->w, w, x->nw * sizeof(BignumInt));
return x;
}
/*
* Decimal-to-binary conversion: just go through the input string
* adding on the decimal value of each digit, and then multiplying the
* number so far by 10.
*/
mp_int *mp_from_decimal_pl(ptrlen decimal)
{
/* 196/59 is an upper bound (and also a continued-fraction
* convergent) for log2(10), so this conservatively estimates the
* number of bits that will be needed to store any number that can
* be written in this many decimal digits. */
assert(decimal.len < (~(size_t)0) / 196);
size_t bits = 196 * decimal.len / 59;
/* Now round that up to words. */
size_t words = bits / BIGNUM_INT_BITS + 1;
mp_int *x = mp_make_sized(words);
for (size_t i = 0; i < decimal.len; i++) {
mp_add_integer_into(x, x, ((const char *)decimal.ptr)[i] - '0');
if (i+1 == decimal.len)
break;
mp_mul_integer_into(x, x, 10);
}
return x;
}
mp_int *mp_from_decimal(const char *decimal)
{
return mp_from_decimal_pl(ptrlen_from_asciz(decimal));
}
/*
* Hex-to-binary conversion: _algorithmically_ simpler than decimal
* (none of those multiplications by 10), but there's some fiddly
* bit-twiddling needed to process each hex digit without diverging
* control flow depending on whether it's a letter or a number.
*/
mp_int *mp_from_hex_pl(ptrlen hex)
{
assert(hex.len <= (~(size_t)0) / 4);
size_t bits = hex.len * 4;
size_t words = (bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
words = size_t_max(words, 1);
mp_int *x = mp_make_sized(words);
for (size_t nibble = 0; nibble < hex.len; nibble++) {
BignumInt digit = ((const char *)hex.ptr)[hex.len-1 - nibble];
BignumInt lmask = ~-((BignumInt)((digit-'a')|('f'-digit))
>> (BIGNUM_INT_BITS-1));
BignumInt umask = ~-((BignumInt)((digit-'A')|('F'-digit))
>> (BIGNUM_INT_BITS-1));
BignumInt digitval = digit - '0';
digitval ^= (digitval ^ (digit - 'a' + 10)) & lmask;
digitval ^= (digitval ^ (digit - 'A' + 10)) & umask;
digitval &= 0xF; /* at least be _slightly_ nice about weird input */
size_t word_idx = nibble / (BIGNUM_INT_BYTES*2);
size_t nibble_within_word = nibble % (BIGNUM_INT_BYTES*2);
x->w[word_idx] |= digitval << (nibble_within_word * 4);
}
return x;
}
mp_int *mp_from_hex(const char *hex)
{
return mp_from_hex_pl(ptrlen_from_asciz(hex));
}
mp_int *mp_copy(mp_int *x)
{
return mp_from_words(x->nw, x->w);
}
uint8_t mp_get_byte(mp_int *x, size_t byte)
{
return 0xFF & (mp_word(x, byte / BIGNUM_INT_BYTES) >>
(8 * (byte % BIGNUM_INT_BYTES)));
}
unsigned mp_get_bit(mp_int *x, size_t bit)
{
return 1 & (mp_word(x, bit / BIGNUM_INT_BITS) >>
(bit % BIGNUM_INT_BITS));
}
uintmax_t mp_get_integer(mp_int *x)
{
uintmax_t toret = 0;
for (size_t i = x->nw; i-- > 0 ;)
toret = shift_left_by_one_word(toret) | x->w[i];
return toret;
}
void mp_set_bit(mp_int *x, size_t bit, unsigned val)
{
size_t word = bit / BIGNUM_INT_BITS;
assert(word < x->nw);
unsigned shift = (bit % BIGNUM_INT_BITS);
x->w[word] &= ~((BignumInt)1 << shift);
x->w[word] |= (BignumInt)(val & 1) << shift;
}
/*
* Helper function used here and there to normalise any nonzero input
* value to 1.
*/
static inline unsigned normalise_to_1(BignumInt n)
{
n = (n >> 1) | (n & 1); /* ensure top bit is clear */
n = (BignumInt)(-n) >> (BIGNUM_INT_BITS - 1); /* normalise to 0 or 1 */
return n;
}
static inline unsigned normalise_to_1_u64(uint64_t n)
{
n = (n >> 1) | (n & 1); /* ensure top bit is clear */
n = (-n) >> 63; /* normalise to 0 or 1 */
return n;
}
/*
* Find the highest nonzero word in a number. Returns the index of the
* word in x->w, and also a pair of output uint64_t in which that word
* appears in the high one shifted left by 'shift_wanted' bits, the
* words immediately below it occupy the space to the right, and the
* words below _that_ fill up the low one.
*
* If there is no nonzero word at all, the passed-by-reference output
* variables retain their original values.
*/
static inline void mp_find_highest_nonzero_word_pair(
mp_int *x, size_t shift_wanted, size_t *index,
uint64_t *hi, uint64_t *lo)
{
uint64_t curr_hi = 0, curr_lo = 0;
for (size_t curr_index = 0; curr_index < x->nw; curr_index++) {
BignumInt curr_word = x->w[curr_index];
unsigned indicator = normalise_to_1(curr_word);
curr_lo = (BIGNUM_INT_BITS < 64 ? (curr_lo >> BIGNUM_INT_BITS) : 0) |
(curr_hi << (64 - BIGNUM_INT_BITS));
curr_hi = (BIGNUM_INT_BITS < 64 ? (curr_hi >> BIGNUM_INT_BITS) : 0) |
((uint64_t)curr_word << shift_wanted);
if (hi) *hi ^= (curr_hi ^ *hi ) & -(uint64_t)indicator;
if (lo) *lo ^= (curr_lo ^ *lo ) & -(uint64_t)indicator;
if (index) *index ^= (curr_index ^ *index) & -(size_t) indicator;
}
}
size_t mp_get_nbits(mp_int *x)
{
/* Sentinel values in case there are no bits set at all: we
* imagine that there's a word at position -1 (i.e. the topmost
* fraction word) which is all 1s, because that way, we handle a
* zero input by considering its highest set bit to be the top one
* of that word, i.e. just below the units digit, i.e. at bit
* index -1, i.e. so we'll return 0 on output. */
size_t hiword_index = -(size_t)1;
uint64_t hiword64 = ~(BignumInt)0;
/*
* Find the highest nonzero word and its index.
*/
mp_find_highest_nonzero_word_pair(x, 0, &hiword_index, &hiword64, NULL);
BignumInt hiword = hiword64; /* in case BignumInt is a narrower type */
/*
* Find the index of the highest set bit within hiword.
*/
BignumInt hibit_index = 0;
for (size_t i = (1 << (BIGNUM_INT_BITS_BITS-1)); i != 0; i >>= 1) {
BignumInt shifted_word = hiword >> i;
BignumInt indicator =
(BignumInt)(-shifted_word) >> (BIGNUM_INT_BITS-1);
hiword ^= (shifted_word ^ hiword ) & -indicator;
hibit_index += i & -(size_t)indicator;
}
/*
* Put together the result.
*/
return (hiword_index << BIGNUM_INT_BITS_BITS) + hibit_index + 1;
}
/*
* Shared code between the hex and decimal output functions to get rid
* of leading zeroes on the output string. The idea is that we wrote
* out a fixed number of digits and a trailing \0 byte into 'buf', and
* now we want to shift it all left so that the first nonzero digit
* moves to buf[0] (or, if there are no nonzero digits at all, we move
* up by 'maxtrim', so that we return 0 as "0" instead of "").
*/
static void trim_leading_zeroes(char *buf, size_t bufsize, size_t maxtrim)
{
size_t trim = maxtrim;
/*
* Look for the first character not equal to '0', to find the
* shift count.
*/
if (trim > 0) {
for (size_t pos = trim; pos-- > 0 ;) {
uint8_t diff = buf[pos] ^ '0';
size_t mask = -((((size_t)diff) - 1) >> (SIZE_T_BITS - 1));
trim ^= (trim ^ pos) & ~mask;
}
}
/*
* Now do the shift, in log n passes each of which does a
* conditional shift by 2^i bytes if bit i is set in the shift
* count.
*/
uint8_t *ubuf = (uint8_t *)buf;
for (size_t logd = 0; bufsize >> logd; logd++) {
uint8_t mask = -(uint8_t)((trim >> logd) & 1);
size_t d = (size_t)1 << logd;
for (size_t i = 0; i+d < bufsize; i++) {
uint8_t diff = mask & (ubuf[i] ^ ubuf[i+d]);
ubuf[i] ^= diff;
ubuf[i+d] ^= diff;
}
}
}
/*
* Binary to decimal conversion. Our strategy here is to extract each
* decimal digit by finding the input number's residue mod 10, then
* subtract that off to give an exact multiple of 10, which then means
* you can safely divide by 10 by means of shifting right one bit and
* then multiplying by the inverse of 5 mod 2^n.
*/
char *mp_get_decimal(mp_int *x_orig)
{
mp_int *x = mp_copy(x_orig), *y = mp_make_sized(x->nw);
/*
* The inverse of 5 mod 2^lots is 0xccccccccccccccccccccd, for an
* appropriate number of 'c's. Manually construct an integer the
* right size.
*/
mp_int *inv5 = mp_make_sized(x->nw);
assert(BIGNUM_INT_BITS % 8 == 0);
for (size_t i = 0; i < inv5->nw; i++)
inv5->w[i] = BIGNUM_INT_MASK / 5 * 4;
inv5->w[0]++;
/*
* 146/485 is an upper bound (and also a continued-fraction
* convergent) of log10(2), so this is a conservative estimate of
* the number of decimal digits needed to store a value that fits
* in this many binary bits.
*/
assert(x->nw < (~(size_t)1) / (146 * BIGNUM_INT_BITS));
size_t bufsize = size_t_max(x->nw * (146 * BIGNUM_INT_BITS) / 485, 1) + 2;
char *outbuf = snewn(bufsize, char);
outbuf[bufsize - 1] = '\0';
/*
* Loop over the number generating digits from the least
* significant upwards, so that we write to outbuf in reverse
* order.
*/
for (size_t pos = bufsize - 1; pos-- > 0 ;) {
/*
* Find the current residue mod 10. We do this by first
* summing the bytes of the number, with all but the lowest
* one multiplied by 6 (because 256^i == 6 mod 10 for all
* i>0). That gives us a single word congruent mod 10 to the
* input number, and then we reduce it further by manual
* multiplication and shifting, just in case the compiler
* target implements the C division operator in a way that has
* input-dependent timing.
*/
uint32_t low_digit = 0, maxval = 0, mult = 1;
for (size_t i = 0; i < x->nw; i++) {
for (unsigned j = 0; j < BIGNUM_INT_BYTES; j++) {
low_digit += mult * (0xFF & (x->w[i] >> (8*j)));
maxval += mult * 0xFF;
mult = 6;
}
/*
* For _really_ big numbers, prevent overflow of t by
* periodically folding the top half of the accumulator
* into the bottom half, using the same rule 'multiply by
* 6 when shifting down by one or more whole bytes'.
*/
if (maxval > UINT32_MAX - (6 * 0xFF * BIGNUM_INT_BYTES)) {
low_digit = (low_digit & 0xFFFF) + 6 * (low_digit >> 16);
maxval = (maxval & 0xFFFF) + 6 * (maxval >> 16);
}
}
/*
* Final reduction of low_digit. We multiply by 2^32 / 10
* (that's the constant 0x19999999) to get a 64-bit value
* whose top 32 bits are the approximate quotient
* low_digit/10; then we subtract off 10 times that; and
* finally we do one last trial subtraction of 10 by adding 6
* (which sets bit 4 if the number was just over 10) and then
* testing bit 4.
*/
low_digit -= 10 * ((0x19999999ULL * low_digit) >> 32);
low_digit -= 10 * ((low_digit + 6) >> 4);
assert(low_digit < 10); /* make sure we did reduce fully */
outbuf[pos] = '0' + low_digit;
/*
* Now subtract off that digit, divide by 2 (using a right
* shift) and by 5 (using the modular inverse), to get the
* next output digit into the units position.
*/
mp_sub_integer_into(x, x, low_digit);
mp_rshift_fixed_into(y, x, 1);
mp_mul_into(x, y, inv5);
}
mp_free(x);
mp_free(y);
mp_free(inv5);
trim_leading_zeroes(outbuf, bufsize, bufsize - 2);
return outbuf;
}
/*
* Binary to hex conversion. Reasonably simple (only a spot of bit
* twiddling to choose whether to output a digit or a letter for each
* nibble).
*/
static char *mp_get_hex_internal(mp_int *x, uint8_t letter_offset)
{
size_t nibbles = x->nw * BIGNUM_INT_BYTES * 2;
size_t bufsize = nibbles + 1;
char *outbuf = snewn(bufsize, char);
outbuf[nibbles] = '\0';
for (size_t nibble = 0; nibble < nibbles; nibble++) {
size_t word_idx = nibble / (BIGNUM_INT_BYTES*2);
size_t nibble_within_word = nibble % (BIGNUM_INT_BYTES*2);
uint8_t digitval = 0xF & (x->w[word_idx] >> (nibble_within_word * 4));
uint8_t mask = -((digitval + 6) >> 4);
char digit = digitval + '0' + (letter_offset & mask);
outbuf[nibbles-1 - nibble] = digit;
}
trim_leading_zeroes(outbuf, bufsize, nibbles - 1);
return outbuf;
}
char *mp_get_hex(mp_int *x)
{
return mp_get_hex_internal(x, 'a' - ('0'+10));
}
char *mp_get_hex_uppercase(mp_int *x)
{
return mp_get_hex_internal(x, 'A' - ('0'+10));
}
/*
* Routines for reading and writing the SSH-1 and SSH-2 wire formats
* for multiprecision integers, declared in marshal.h.
*
* These can't avoid having control flow dependent on the true bit
* size of the number, because the wire format requires the number of
* output bytes to depend on that.
*/
void BinarySink_put_mp_ssh1(BinarySink *bs, mp_int *x)
{
size_t bits = mp_get_nbits(x);
size_t bytes = (bits + 7) / 8;
assert(bits < 0x10000);
put_uint16(bs, bits);
for (size_t i = bytes; i-- > 0 ;)
put_byte(bs, mp_get_byte(x, i));
}
void BinarySink_put_mp_ssh2(BinarySink *bs, mp_int *x)
{
size_t bytes = (mp_get_nbits(x) + 8) / 8;
put_uint32(bs, bytes);
for (size_t i = bytes; i-- > 0 ;)
put_byte(bs, mp_get_byte(x, i));
}
mp_int *BinarySource_get_mp_ssh1(BinarySource *src)
{
unsigned bitc = get_uint16(src);
ptrlen bytes = get_data(src, (bitc + 7) / 8);
if (get_err(src)) {
return mp_from_integer(0);
} else {
mp_int *toret = mp_from_bytes_be(bytes);
/* SSH-1.5 spec says that it's OK for the prefix uint16 to be
* _greater_ than the actual number of bits */
if (mp_get_nbits(toret) > bitc) {
src->err = BSE_INVALID;
mp_free(toret);
toret = mp_from_integer(0);
}
return toret;
}
}
mp_int *BinarySource_get_mp_ssh2(BinarySource *src)
{
ptrlen bytes = get_string(src);
if (get_err(src)) {
return mp_from_integer(0);
} else {
const unsigned char *p = bytes.ptr;
if ((bytes.len > 0 &&
((p[0] & 0x80) ||
(p[0] == 0 && (bytes.len <= 1 || !(p[1] & 0x80)))))) {
src->err = BSE_INVALID;
return mp_from_integer(0);
}
return mp_from_bytes_be(bytes);
}
}
/*
* Make an mp_int structure whose words array aliases a subinterval of
* some other mp_int. This makes it easy to read or write just the low
* or high words of a number, e.g. to add a number starting from a
* high bit position, or to reduce mod 2^{n*BIGNUM_INT_BITS}.
*
* The convention throughout this code is that when we store an mp_int
* directly by value, we always expect it to be an alias of some kind,
* so its words array won't ever need freeing. Whereas an 'mp_int *'
* has an owner, who knows whether it needs freeing or whether it was
* created by address-taking an alias.
*/
static mp_int mp_make_alias(mp_int *in, size_t offset, size_t len)
{
/*
* Bounds-check the offset and length so that we always return
* something valid, even if it's not necessarily the length the
* caller asked for.
*/
if (offset > in->nw)
offset = in->nw;
if (len > in->nw - offset)
len = in->nw - offset;
mp_int toret;
toret.nw = len;
toret.w = in->w + offset;
return toret;
}
/*
* A special case of mp_make_alias: in some cases we preallocate a
* large mp_int to use as scratch space (to avoid pointless
* malloc/free churn in recursive or iterative work).
*
* mp_alloc_from_scratch creates an alias of size 'len' to part of
* 'pool', and adjusts 'pool' itself so that further allocations won't
* overwrite that space.
*
* There's no free function to go with this. Typically you just copy
* the pool mp_int by value, allocate from the copy, and when you're
* done with those allocations, throw the copy away and go back to the
* original value of pool. (A mark/release system.)
*/
static mp_int mp_alloc_from_scratch(mp_int *pool, size_t len)
{
assert(len <= pool->nw);
mp_int toret = mp_make_alias(pool, 0, len);
*pool = mp_make_alias(pool, len, pool->nw);
return toret;
}
/*
* Internal component common to lots of assorted add/subtract code.
* Reads words from a,b; writes into w_out (which might be NULL if the
* output isn't even needed). Takes an input carry flag in 'carry',
* and returns the output carry. Each word read from b is ANDed with
* b_and and then XORed with b_xor.
*
* So you can implement addition by setting b_and to all 1s and b_xor
* to 0; you can subtract by making b_xor all 1s too (effectively
* bit-flipping b) and also passing 1 as the input carry (to turn
* one's complement into two's complement). And you can do conditional
* add/subtract by choosing b_and to be all 1s or all 0s based on a
* condition, because the value of b will be totally ignored if b_and
* == 0.
*/
static BignumCarry mp_add_masked_into(
BignumInt *w_out, size_t rw, mp_int *a, mp_int *b,
BignumInt b_and, BignumInt b_xor, BignumCarry carry)
{
for (size_t i = 0; i < rw; i++) {
BignumInt aword = mp_word(a, i), bword = mp_word(b, i), out;
bword = (bword & b_and) ^ b_xor;
BignumADC(out, carry, aword, bword, carry);
if (w_out)
w_out[i] = out;
}
return carry;
}
/*
* Like the public mp_add_into except that it returns the output carry.
*/
static inline BignumCarry mp_add_into_internal(mp_int *r, mp_int *a, mp_int *b)
{
return mp_add_masked_into(r->w, r->nw, a, b, ~(BignumInt)0, 0, 0);
}
void mp_add_into(mp_int *r, mp_int *a, mp_int *b)
{
mp_add_into_internal(r, a, b);
}
void mp_sub_into(mp_int *r, mp_int *a, mp_int *b)
{
mp_add_masked_into(r->w, r->nw, a, b, ~(BignumInt)0, ~(BignumInt)0, 1);
}
void mp_and_into(mp_int *r, mp_int *a, mp_int *b)
{
for (size_t i = 0; i < r->nw; i++) {
BignumInt aword = mp_word(a, i), bword = mp_word(b, i);
r->w[i] = aword & bword;
}
}
void mp_or_into(mp_int *r, mp_int *a, mp_int *b)
{
for (size_t i = 0; i < r->nw; i++) {
BignumInt aword = mp_word(a, i), bword = mp_word(b, i);
r->w[i] = aword | bword;
}
}
void mp_xor_into(mp_int *r, mp_int *a, mp_int *b)
{
for (size_t i = 0; i < r->nw; i++) {
BignumInt aword = mp_word(a, i), bword = mp_word(b, i);
r->w[i] = aword ^ bword;
}
}
void mp_bic_into(mp_int *r, mp_int *a, mp_int *b)
{
for (size_t i = 0; i < r->nw; i++) {
BignumInt aword = mp_word(a, i), bword = mp_word(b, i);
r->w[i] = aword & ~bword;
}
}
static void mp_cond_negate(mp_int *r, mp_int *x, unsigned yes)
{
BignumCarry carry = yes;
BignumInt flip = -(BignumInt)yes;
for (size_t i = 0; i < r->nw; i++) {
BignumInt xword = mp_word(x, i);
xword ^= flip;
BignumADC(r->w[i], carry, 0, xword, carry);
}
}
/*
* Similar to mp_add_masked_into, but takes a C integer instead of an
* mp_int as the masked operand.
*/
static BignumCarry mp_add_masked_integer_into(
BignumInt *w_out, size_t rw, mp_int *a, uintmax_t b,
BignumInt b_and, BignumInt b_xor, BignumCarry carry)
{
for (size_t i = 0; i < rw; i++) {
BignumInt aword = mp_word(a, i);
BignumInt bword = b;
b = shift_right_by_one_word(b);
BignumInt out;
bword = (bword ^ b_xor) & b_and;
BignumADC(out, carry, aword, bword, carry);
if (w_out)
w_out[i] = out;
}
return carry;
}
void mp_add_integer_into(mp_int *r, mp_int *a, uintmax_t n)
{
mp_add_masked_integer_into(r->w, r->nw, a, n, ~(BignumInt)0, 0, 0);
}
void mp_sub_integer_into(mp_int *r, mp_int *a, uintmax_t n)
{
mp_add_masked_integer_into(r->w, r->nw, a, n,
~(BignumInt)0, ~(BignumInt)0, 1);
}
/*
* Sets r to a + n << (word_index * BIGNUM_INT_BITS), treating
* word_index as secret data.
*/
static void mp_add_integer_into_shifted_by_words(
mp_int *r, mp_int *a, uintmax_t n, size_t word_index)
{
unsigned indicator = 0;
BignumCarry carry = 0;
for (size_t i = 0; i < r->nw; i++) {
/* indicator becomes 1 when we reach the index that the least
* significant bits of n want to be placed at, and it stays 1
* thereafter. */
indicator |= 1 ^ normalise_to_1(i ^ word_index);
/* If indicator is 1, we add the low bits of n into r, and
* shift n down. If it's 0, we add zero bits into r, and
* leave n alone. */
BignumInt bword = n & -(BignumInt)indicator;
uintmax_t new_n = shift_right_by_one_word(n);
n ^= (n ^ new_n) & -(uintmax_t)indicator;
BignumInt aword = mp_word(a, i);
BignumInt out;
BignumADC(out, carry, aword, bword, carry);
r->w[i] = out;
}
}
void mp_mul_integer_into(mp_int *r, mp_int *a, uint16_t n)
{
BignumInt carry = 0, mult = n;
for (size_t i = 0; i < r->nw; i++) {
BignumInt aword = mp_word(a, i);
BignumMULADD(carry, r->w[i], aword, mult, carry);
}
assert(!carry);
}
void mp_cond_add_into(mp_int *r, mp_int *a, mp_int *b, unsigned yes)
{
BignumInt mask = -(BignumInt)(yes & 1);
mp_add_masked_into(r->w, r->nw, a, b, mask, 0, 0);
}
void mp_cond_sub_into(mp_int *r, mp_int *a, mp_int *b, unsigned yes)
{
BignumInt mask = -(BignumInt)(yes & 1);
mp_add_masked_into(r->w, r->nw, a, b, mask, mask, 1 & mask);
}
/*
* Ordered comparison between unsigned numbers is done by subtracting
* one from the other and looking at the output carry.
*/
unsigned mp_cmp_hs(mp_int *a, mp_int *b)
{
size_t rw = size_t_max(a->nw, b->nw);
return mp_add_masked_into(NULL, rw, a, b, ~(BignumInt)0, ~(BignumInt)0, 1);
}
unsigned mp_hs_integer(mp_int *x, uintmax_t n)
{
BignumInt carry = 1;
size_t nwords = sizeof(n)/BIGNUM_INT_BYTES;
for (size_t i = 0, e = size_t_max(x->nw, nwords); i < e; i++) {
BignumInt nword = n;
n = shift_right_by_one_word(n);
BignumInt dummy_out;
BignumADC(dummy_out, carry, mp_word(x, i), ~nword, carry);
(void)dummy_out;
}
return carry;
}
/*
* Equality comparison is done by bitwise XOR of the input numbers,
* ORing together all the output words, and normalising the result
* using our careful normalise_to_1 helper function.
*/
unsigned mp_cmp_eq(mp_int *a, mp_int *b)
{
BignumInt diff = 0;
for (size_t i = 0, limit = size_t_max(a->nw, b->nw); i < limit; i++)
diff |= mp_word(a, i) ^ mp_word(b, i);
return 1 ^ normalise_to_1(diff); /* return 1 if diff _is_ zero */
}
unsigned mp_eq_integer(mp_int *x, uintmax_t n)
{
BignumInt diff = 0;
size_t nwords = sizeof(n)/BIGNUM_INT_BYTES;
for (size_t i = 0, e = size_t_max(x->nw, nwords); i < e; i++) {
BignumInt nword = n;
n = shift_right_by_one_word(n);
diff |= mp_word(x, i) ^ nword;
}
return 1 ^ normalise_to_1(diff); /* return 1 if diff _is_ zero */
}
static void mp_neg_into(mp_int *r, mp_int *a)
{
mp_int zero;
zero.nw = 0;
mp_sub_into(r, &zero, a);
}
mp_int *mp_add(mp_int *x, mp_int *y)
{
mp_int *r = mp_make_sized(size_t_max(x->nw, y->nw) + 1);
mp_add_into(r, x, y);
return r;
}
mp_int *mp_sub(mp_int *x, mp_int *y)
{
mp_int *r = mp_make_sized(size_t_max(x->nw, y->nw));
mp_sub_into(r, x, y);
return r;
}
/*
* Internal routine: multiply and accumulate in the trivial O(N^2)
* way. Sets r <- r + a*b.
*/
static void mp_mul_add_simple(mp_int *r, mp_int *a, mp_int *b)
{
BignumInt *aend = a->w + a->nw, *bend = b->w + b->nw, *rend = r->w + r->nw;
for (BignumInt *ap = a->w, *rp = r->w;
ap < aend && rp < rend; ap++, rp++) {
BignumInt adata = *ap, carry = 0, *rq = rp;
for (BignumInt *bp = b->w; bp < bend && rq < rend; bp++, rq++) {
BignumInt bdata = bp < bend ? *bp : 0;
BignumMULADD2(carry, *rq, adata, bdata, *rq, carry);
}
for (; rq < rend; rq++)
BignumADC(*rq, carry, carry, *rq, 0);
}
}
#ifndef KARATSUBA_THRESHOLD /* allow redefinition via -D for testing */
#define KARATSUBA_THRESHOLD 24
#endif
static inline size_t mp_mul_scratchspace_unary(size_t n)
{
/*
* Simplistic and overcautious bound on the amount of scratch
* space that the recursive multiply function will need.
*
* The rationale is: on the main Karatsuba branch of
* mp_mul_internal, which is the most space-intensive one, we
* allocate space for (a0+a1) and (b0+b1) (each just over half the
* input length n) and their product (the sum of those sizes, i.e.
* just over n itself). Then in order to actually compute the
* product, we do a recursive multiplication of size just over n.
*
* If all those 'just over' weren't there, and everything was
* _exactly_ half the length, you'd get the amount of space for a
* size-n multiply defined by the recurrence M(n) = 2n + M(n/2),
* which is satisfied by M(n) = 4n. But instead it's (2n plus a
* word or two) and M(n/2 plus a word or two). On the assumption
* that there's still some constant k such that M(n) <= kn, this
* gives us kn = 2n + w + k(n/2 + w), where w is a small constant
* (one or two words). That simplifies to kn/2 = 2n + (k+1)w, and
* since we don't even _start_ needing scratch space until n is at
* least 50, we can bound 2n + (k+1)w above by 3n, giving k=6.
*
* So I claim that 6n words of scratch space will suffice, and I
* check that by assertion at every stage of the recursion.
*/
return n * 6;
}
static size_t mp_mul_scratchspace(size_t rw, size_t aw, size_t bw)
{
size_t inlen = size_t_min(rw, size_t_max(aw, bw));
return mp_mul_scratchspace_unary(inlen);
}
static void mp_mul_internal(mp_int *r, mp_int *a, mp_int *b, mp_int scratch)
{
size_t inlen = size_t_min(r->nw, size_t_max(a->nw, b->nw));
assert(scratch.nw >= mp_mul_scratchspace_unary(inlen));
mp_clear(r);
if (inlen < KARATSUBA_THRESHOLD || a->nw == 0 || b->nw == 0) {
/*
* The input numbers are too small to bother optimising. Go
* straight to the simple primitive approach.
*/
mp_mul_add_simple(r, a, b);
return;
}
/*
* Karatsuba divide-and-conquer algorithm. We cut each input in
* half, so that it's expressed as two big 'digits' in a giant
* base D:
*
* a = a_1 D + a_0
* b = b_1 D + b_0
*
* Then the product is of course
*
* ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
*
* and we compute the three coefficients by recursively calling
* ourself to do half-length multiplications.
*
* The clever bit that makes this worth doing is that we only need
* _one_ half-length multiplication for the central coefficient
* rather than the two that it obviouly looks like, because we can
* use a single multiplication to compute
*
* (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0
*
* and then we subtract the other two coefficients (a_1 b_1 and
* a_0 b_0) which we were computing anyway.
*
* Hence we get to multiply two numbers of length N in about three
* times as much work as it takes to multiply numbers of length
* N/2, which is obviously better than the four times as much work
* it would take if we just did a long conventional multiply.
*/
/* Break up the input as botlen + toplen, with botlen >= toplen.
* The 'base' D is equal to 2^{botlen * BIGNUM_INT_BITS}. */
size_t toplen = inlen / 2;
size_t botlen = inlen - toplen;
/* Alias bignums that address the two halves of a,b, and useful
* pieces of r. */
mp_int a0 = mp_make_alias(a, 0, botlen);
mp_int b0 = mp_make_alias(b, 0, botlen);
mp_int a1 = mp_make_alias(a, botlen, toplen);
mp_int b1 = mp_make_alias(b, botlen, toplen);
mp_int r0 = mp_make_alias(r, 0, botlen*2);
mp_int r1 = mp_make_alias(r, botlen, r->nw);
mp_int r2 = mp_make_alias(r, botlen*2, r->nw);
/* Recurse to compute a0*b0 and a1*b1, in their correct positions
* in the output bignum. They can't overlap. */
mp_mul_internal(&r0, &a0, &b0, scratch);
mp_mul_internal(&r2, &a1, &b1, scratch);
if (r->nw < inlen*2) {
/*
* The output buffer isn't large enough to require the whole
* product, so some of a1*b1 won't have been stored. In that
* case we won't try to do the full Karatsuba optimisation;
* we'll just recurse again to compute a0*b1 and a1*b0 - or at
* least as much of them as the output buffer size requires -
* and add each one in.
*/
mp_int s = mp_alloc_from_scratch(
&scratch, size_t_min(botlen+toplen, r1.nw));
mp_mul_internal(&s, &a0, &b1, scratch);
mp_add_into(&r1, &r1, &s);
mp_mul_internal(&s, &a1, &b0, scratch);
mp_add_into(&r1, &r1, &s);
return;
}
/* a0+a1 and b0+b1 */
mp_int asum = mp_alloc_from_scratch(&scratch, botlen+1);
mp_int bsum = mp_alloc_from_scratch(&scratch, botlen+1);
mp_add_into(&asum, &a0, &a1);
mp_add_into(&bsum, &b0, &b1);
/* Their product */
mp_int product = mp_alloc_from_scratch(&scratch, botlen*2+1);
mp_mul_internal(&product, &asum, &bsum, scratch);
/* Subtract off the outer terms we already have */
mp_sub_into(&product, &product, &r0);
mp_sub_into(&product, &product, &r2);
/* And add it in with the right offset. */
mp_add_into(&r1, &r1, &product);
}
void mp_mul_into(mp_int *r, mp_int *a, mp_int *b)
{
mp_int *scratch = mp_make_sized(mp_mul_scratchspace(r->nw, a->nw, b->nw));
mp_mul_internal(r, a, b, *scratch);
mp_free(scratch);
}
mp_int *mp_mul(mp_int *x, mp_int *y)
{
mp_int *r = mp_make_sized(x->nw + y->nw);
mp_mul_into(r, x, y);
return r;
}
void mp_lshift_fixed_into(mp_int *r, mp_int *a, size_t bits)
{
size_t words = bits / BIGNUM_INT_BITS;
size_t bitoff = bits % BIGNUM_INT_BITS;
for (size_t i = r->nw; i-- > 0 ;) {
if (i < words) {
r->w[i] = 0;
} else {
r->w[i] = mp_word(a, i - words);
if (bitoff != 0) {
r->w[i] <<= bitoff;
if (i > words)
r->w[i] |= mp_word(a, i - words - 1) >>
(BIGNUM_INT_BITS - bitoff);
}
}
}
}
void mp_rshift_fixed_into(mp_int *r, mp_int *a, size_t bits)
{
size_t words = bits / BIGNUM_INT_BITS;
size_t bitoff = bits % BIGNUM_INT_BITS;
for (size_t i = 0; i < r->nw; i++) {
r->w[i] = mp_word(a, i + words);
if (bitoff != 0) {
r->w[i] >>= bitoff;
r->w[i] |= mp_word(a, i + words + 1) << (BIGNUM_INT_BITS - bitoff);
}
}
}
mp_int *mp_lshift_fixed(mp_int *x, size_t bits)
{
size_t words = (bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
mp_int *r = mp_make_sized(x->nw + words);
mp_lshift_fixed_into(r, x, bits);
return r;
}
mp_int *mp_rshift_fixed(mp_int *x, size_t bits)
{
size_t words = bits / BIGNUM_INT_BITS;
size_t nw = x->nw - size_t_min(x->nw, words);
mp_int *r = mp_make_sized(size_t_max(nw, 1));
mp_rshift_fixed_into(r, x, bits);
return r;
}
/*
* Safe right shift is done using the same technique as
* trim_leading_zeroes above: you make an n-word left shift by
* composing an appropriate subset of power-of-2-sized shifts, so it
* takes log_2(n) loop iterations each of which does a different shift
* by a power of 2 words, using the usual bit twiddling to make the
* whole shift conditional on the appropriate bit of n.
*/
static void mp_rshift_safe_in_place(mp_int *r, size_t bits)
{
size_t wordshift = bits / BIGNUM_INT_BITS;
size_t bitshift = bits % BIGNUM_INT_BITS;
unsigned clear = (r->nw - wordshift) >> (CHAR_BIT * sizeof(size_t) - 1);
mp_cond_clear(r, clear);
for (unsigned bit = 0; r->nw >> bit; bit++) {
size_t word_offset = 1 << bit;
BignumInt mask = -(BignumInt)((wordshift >> bit) & 1);
for (size_t i = 0; i < r->nw; i++) {
BignumInt w = mp_word(r, i + word_offset);
r->w[i] ^= (r->w[i] ^ w) & mask;
}
}
/*
* That's done the shifting by words; now we do the shifting by
* bits.
*/
for (unsigned bit = 0; bit < BIGNUM_INT_BITS_BITS; bit++) {
unsigned shift = 1 << bit, upshift = BIGNUM_INT_BITS - shift;
BignumInt mask = -(BignumInt)((bitshift >> bit) & 1);
for (size_t i = 0; i < r->nw; i++) {
BignumInt w = ((r->w[i] >> shift) | (mp_word(r, i+1) << upshift));
r->w[i] ^= (r->w[i] ^ w) & mask;
}
}
}
mp_int *mp_rshift_safe(mp_int *x, size_t bits)
{
mp_int *r = mp_copy(x);
mp_rshift_safe_in_place(r, bits);
return r;
}
void mp_rshift_safe_into(mp_int *r, mp_int *x, size_t bits)
{
mp_copy_into(r, x);
mp_rshift_safe_in_place(r, bits);
}
static void mp_lshift_safe_in_place(mp_int *r, size_t bits)
{
size_t wordshift = bits / BIGNUM_INT_BITS;
size_t bitshift = bits % BIGNUM_INT_BITS;
/*
* Same strategy as mp_rshift_safe_in_place, but of course the
* other way up.
*/
unsigned clear = (r->nw - wordshift) >> (CHAR_BIT * sizeof(size_t) - 1);
mp_cond_clear(r, clear);
for (unsigned bit = 0; r->nw >> bit; bit++) {
size_t word_offset = 1 << bit;
BignumInt mask = -(BignumInt)((wordshift >> bit) & 1);
for (size_t i = r->nw; i-- > 0 ;) {
BignumInt w = mp_word(r, i - word_offset);
r->w[i] ^= (r->w[i] ^ w) & mask;
}
}
size_t downshift = BIGNUM_INT_BITS - bitshift;
size_t no_shift = (downshift >> BIGNUM_INT_BITS_BITS);
downshift &= ~-(size_t)no_shift;
BignumInt downshifted_mask = ~-(BignumInt)no_shift;
for (size_t i = r->nw; i-- > 0 ;) {
r->w[i] = (r->w[i] << bitshift) |
((mp_word(r, i-1) >> downshift) & downshifted_mask);
}
}
void mp_lshift_safe_into(mp_int *r, mp_int *x, size_t bits)
{
mp_copy_into(r, x);
mp_lshift_safe_in_place(r, bits);
}
void mp_reduce_mod_2to(mp_int *x, size_t p)
{
size_t word = p / BIGNUM_INT_BITS;
size_t mask = ((size_t)1 << (p % BIGNUM_INT_BITS)) - 1;
for (; word < x->nw; word++) {
x->w[word] &= mask;
mask = 0;
}
}
/*
* Inverse mod 2^n is computed by an iterative technique which doubles
* the number of bits at each step.
*/
mp_int *mp_invert_mod_2to(mp_int *x, size_t p)
{
/* Input checks: x must be coprime to the modulus, i.e. odd, and p
* can't be zero */
assert(x->nw > 0);
assert(x->w[0] & 1);
assert(p > 0);
size_t rw = (p + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
rw = size_t_max(rw, 1);
mp_int *r = mp_make_sized(rw);
size_t mul_scratchsize = mp_mul_scratchspace(2*rw, rw, rw);
mp_int *scratch_orig = mp_make_sized(6 * rw + mul_scratchsize);
mp_int scratch_per_iter = *scratch_orig;
mp_int mul_scratch = mp_alloc_from_scratch(
&scratch_per_iter, mul_scratchsize);
r->w[0] = 1;
for (size_t b = 1; b < p; b <<= 1) {
/*
* In each step of this iteration, we have the inverse of x
* mod 2^b, and we want the inverse of x mod 2^{2b}.
*
* Write B = 2^b for convenience, so we want x^{-1} mod B^2.
* Let x = x_0 + B x_1 + k B^2, with 0 <= x_0,x_1 < B.
*
* We want to find r_0 and r_1 such that
* (r_1 B + r_0) (x_1 B + x_0) == 1 (mod B^2)
*
* To begin with, we know r_0 must be the inverse mod B of
* x_0, i.e. of x, i.e. it is the inverse we computed in the
* previous iteration. So now all we need is r_1.
*
* Multiplying out, neglecting multiples of B^2, and writing
* x_0 r_0 = K B + 1, we have
*
* r_1 x_0 B + r_0 x_1 B + K B == 0 (mod B^2)
* => r_1 x_0 B == - r_0 x_1 B - K B (mod B^2)
* => r_1 x_0 == - r_0 x_1 - K (mod B)
* => r_1 == r_0 (- r_0 x_1 - K) (mod B)
*
* (the last step because we multiply through by the inverse
* of x_0, which we already know is r_0).
*/
mp_int scratch_this_iter = scratch_per_iter;
size_t Bw = (b + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
size_t B2w = (2*b + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
/* Start by finding K: multiply x_0 by r_0, and shift down. */
mp_int x0 = mp_alloc_from_scratch(&scratch_this_iter, Bw);
mp_copy_into(&x0, x);
mp_reduce_mod_2to(&x0, b);
mp_int r0 = mp_make_alias(r, 0, Bw);
mp_int Kshift = mp_alloc_from_scratch(&scratch_this_iter, B2w);
mp_mul_internal(&Kshift, &x0, &r0, mul_scratch);
mp_int K = mp_alloc_from_scratch(&scratch_this_iter, Bw);
mp_rshift_fixed_into(&K, &Kshift, b);
/* Now compute the product r_0 x_1, reusing the space of Kshift. */
mp_int x1 = mp_alloc_from_scratch(&scratch_this_iter, Bw);
mp_rshift_fixed_into(&x1, x, b);
mp_reduce_mod_2to(&x1, b);
mp_int r0x1 = mp_make_alias(&Kshift, 0, Bw);
mp_mul_internal(&r0x1, &r0, &x1, mul_scratch);
/* Add K to that. */
mp_add_into(&r0x1, &r0x1, &K);
/* Negate it. */
mp_neg_into(&r0x1, &r0x1);
/* Multiply by r_0. */
mp_int r1 = mp_alloc_from_scratch(&scratch_this_iter, Bw);
mp_mul_internal(&r1, &r0, &r0x1, mul_scratch);
mp_reduce_mod_2to(&r1, b);
/* That's our r_1, so add it on to r_0 to get the full inverse
* output from this iteration. */
mp_lshift_fixed_into(&K, &r1, (b % BIGNUM_INT_BITS));
size_t Bpos = b / BIGNUM_INT_BITS;
mp_int r1_position = mp_make_alias(r, Bpos, B2w-Bpos);
mp_add_into(&r1_position, &r1_position, &K);
}
/* Finally, reduce mod the precise desired number of bits. */
mp_reduce_mod_2to(r, p);
mp_free(scratch_orig);
return r;
}
static size_t monty_scratch_size(MontyContext *mc)
{
return 3*mc->rw + mc->pw + mp_mul_scratchspace(mc->pw, mc->rw, mc->rw);
}
MontyContext *monty_new(mp_int *modulus)
{
MontyContext *mc = snew(MontyContext);
mc->rw = modulus->nw;
mc->rbits = mc->rw * BIGNUM_INT_BITS;
mc->pw = mc->rw * 2 + 1;
mc->m = mp_copy(modulus);
mc->minus_minv_mod_r = mp_invert_mod_2to(mc->m, mc->rbits);
mp_neg_into(mc->minus_minv_mod_r, mc->minus_minv_mod_r);
mp_int *r = mp_make_sized(mc->rw + 1);
r->w[mc->rw] = 1;
mc->powers_of_r_mod_m[0] = mp_mod(r, mc->m);
mp_free(r);
for (size_t j = 1; j < lenof(mc->powers_of_r_mod_m); j++)
mc->powers_of_r_mod_m[j] = mp_modmul(
mc->powers_of_r_mod_m[0], mc->powers_of_r_mod_m[j-1], mc->m);
mc->scratch = mp_make_sized(monty_scratch_size(mc));
return mc;
}
void monty_free(MontyContext *mc)
{
mp_free(mc->m);
for (size_t j = 0; j < 3; j++)
mp_free(mc->powers_of_r_mod_m[j]);
mp_free(mc->minus_minv_mod_r);
mp_free(mc->scratch);
smemclr(mc, sizeof(*mc));
sfree(mc);
}
/*
* The main Montgomery reduction step.
*/
static mp_int monty_reduce_internal(MontyContext *mc, mp_int *x, mp_int scratch)
{
/*
* The trick with Montgomery reduction is that on the one hand we
* want to reduce the size of the input by a factor of about r,
* and on the other hand, the two numbers we just multiplied were
* both stored with an extra factor of r multiplied in. So we
* computed ar*br = ab r^2, but we want to return abr, so we need
* to divide by r - and if we can do that by _actually dividing_
* by r then this also reduces the size of the number.
*
* But we can only do that if the number we're dividing by r is a
* multiple of r. So first we must add an adjustment to it which
* clears its bottom 'rbits' bits. That adjustment must be a
* multiple of m in order to leave the residue mod n unchanged, so
* the question is, what multiple of m can we add to x to make it
* congruent to 0 mod r? And the answer is, x * (-m)^{-1} mod r.
*/
/* x mod r */
mp_int x_lo = mp_make_alias(x, 0, mc->rbits);
/* x * (-m)^{-1}, i.e. the number we want to multiply by m */
mp_int k = mp_alloc_from_scratch(&scratch, mc->rw);
mp_mul_internal(&k, &x_lo, mc->minus_minv_mod_r, scratch);
/* m times that, i.e. the number we want to add to x */
mp_int mk = mp_alloc_from_scratch(&scratch, mc->pw);
mp_mul_internal(&mk, mc->m, &k, scratch);
/* Add it to x */
mp_add_into(&mk, x, &mk);
/* Reduce mod r, by simply making an alias to the upper words of x */
mp_int toret = mp_make_alias(&mk, mc->rw, mk.nw - mc->rw);
/*
* We'll generally be doing this after a multiplication of two
* fully reduced values. So our input could be anything up to m^2,
* and then we added up to rm to it. Hence, the maximum value is
* rm+m^2, and after dividing by r, that becomes r + m(m/r) < 2r.
* So a single trial-subtraction will finish reducing to the
* interval [0,m).
*/
mp_cond_sub_into(&toret, &toret, mc->m, mp_cmp_hs(&toret, mc->m));
return toret;
}
void monty_mul_into(MontyContext *mc, mp_int *r, mp_int *x, mp_int *y)
{
assert(x->nw <= mc->rw);
assert(y->nw <= mc->rw);
mp_int scratch = *mc->scratch;
mp_int tmp = mp_alloc_from_scratch(&scratch, 2*mc->rw);
mp_mul_into(&tmp, x, y);
mp_int reduced = monty_reduce_internal(mc, &tmp, scratch);
mp_copy_into(r, &reduced);
mp_clear(mc->scratch);
}
mp_int *monty_mul(MontyContext *mc, mp_int *x, mp_int *y)
{
mp_int *toret = mp_make_sized(mc->rw);
monty_mul_into(mc, toret, x, y);
return toret;
}
mp_int *monty_modulus(MontyContext *mc)
{
return mc->m;
}
mp_int *monty_identity(MontyContext *mc)
{
return mc->powers_of_r_mod_m[0];
}
mp_int *monty_invert(MontyContext *mc, mp_int *x)
{
/* Given xr, we want to return x^{-1}r = (xr)^{-1} r^2 =
* monty_reduce((xr)^{-1} r^3) */
mp_int *tmp = mp_invert(x, mc->m);
mp_int *toret = monty_mul(mc, tmp, mc->powers_of_r_mod_m[2]);
mp_free(tmp);
return toret;
}
/*
* Importing a number into Montgomery representation involves
* multiplying it by r and reducing mod m. We use the general-purpose
* mp_modmul for this, in case the input number is out of range.
*/
mp_int *monty_import(MontyContext *mc, mp_int *x)
{
return mp_modmul(x, mc->powers_of_r_mod_m[0], mc->m);
}
void monty_import_into(MontyContext *mc, mp_int *r, mp_int *x)
{
mp_int *imported = monty_import(mc, x);
mp_copy_into(r, imported);
mp_free(imported);
}
/*
* Exporting a number means multiplying it by r^{-1}, which is exactly
* what monty_reduce does anyway, so we just do that.
*/
void monty_export_into(MontyContext *mc, mp_int *r, mp_int *x)
{
assert(x->nw <= 2*mc->rw);
mp_int reduced = monty_reduce_internal(mc, x, *mc->scratch);
mp_copy_into(r, &reduced);
mp_clear(mc->scratch);
}
mp_int *monty_export(MontyContext *mc, mp_int *x)
{
mp_int *toret = mp_make_sized(mc->rw);
monty_export_into(mc, toret, x);
return toret;
}
static void monty_reduce(MontyContext *mc, mp_int *x)
{
mp_int reduced = monty_reduce_internal(mc, x, *mc->scratch);
mp_copy_into(x, &reduced);
mp_clear(mc->scratch);
}
mp_int *monty_pow(MontyContext *mc, mp_int *base, mp_int *exponent)
{
/* square builds up powers of the form base^{2^i}. */
mp_int *square = mp_copy(base);
size_t i = 0;
/* out accumulates the output value. Starts at 1 (in Montgomery
* representation) and we multiply in each base^{2^i}. */
mp_int *out = mp_copy(mc->powers_of_r_mod_m[0]);
/* tmp holds each product we compute and reduce. */
mp_int *tmp = mp_make_sized(mc->rw * 2);
while (true) {
mp_mul_into(tmp, out, square);
monty_reduce(mc, tmp);
mp_select_into(out, out, tmp, mp_get_bit(exponent, i));
if (++i >= exponent->nw * BIGNUM_INT_BITS)
break;
mp_mul_into(tmp, square, square);
monty_reduce(mc, tmp);
mp_copy_into(square, tmp);
}
mp_free(square);
mp_free(tmp);
mp_clear(mc->scratch);
return out;
}
mp_int *mp_modpow(mp_int *base, mp_int *exponent, mp_int *modulus)
{
assert(modulus->nw > 0);
assert(modulus->w[0] & 1);
MontyContext *mc = monty_new(modulus);
mp_int *m_base = monty_import(mc, base);
mp_int *m_out = monty_pow(mc, m_base, exponent);
mp_int *out = monty_export(mc, m_out);
mp_free(m_base);
mp_free(m_out);
monty_free(mc);
return out;
}
/*
* Given two input integers a,b which are not both even, computes d =
* gcd(a,b) and also two integers A,B such that A*a - B*b = d. A,B
* will be the minimal non-negative pair satisfying that criterion,
* which is equivalent to saying that 0 <= A < b/d and 0 <= B < a/d.
*
* This algorithm is an adapted form of Stein's algorithm, which
* computes gcd(a,b) using only addition and bit shifts (i.e. without
* needing general division), using the following rules:
*
* - if both of a,b are even, divide off a common factor of 2
* - if one of a,b (WLOG a) is even, then gcd(a,b) = gcd(a/2,b), so
* just divide a by 2
* - if both of a,b are odd, then WLOG a>b, and gcd(a,b) =
* gcd(b,(a-b)/2).
*
* Sometimes this function is used for modular inversion, in which
* case we already know we expect the two inputs to be coprime, so to
* save time the 'both even' initial case is assumed not to arise (or
* to have been handled already by the caller). So this function just
* performs a sequence of reductions in the following form:
*
* - if a,b are both odd, sort them so that a > b, and replace a with
* b-a; otherwise sort them so that a is the even one
* - either way, now a is even and b is odd, so divide a by 2.
*
* The big change to Stein's algorithm is that we need the Bezout
* coefficients as output, not just the gcd. So we need to know how to
* generate those in each case, based on the coefficients from the
* reduced pair of numbers:
*
* - If a is even, and u,v are such that u*(a/2) + v*b = d:
* + if u is also even, then this is just (u/2)*a + v*b = d
* + otherwise, (u+b)*(a/2) + (v-a/2)*b is also equal to d, and
* since u and b are both odd, (u+b)/2 is an integer, so we have
* ((u+b)/2)*a + (v-a/2)*b = d.
*
* - If a,b are both odd, and u,v are such that u*b + v*(a-b) = d,
* then v*a + (u-v)*b = d.
*
* In the case where we passed from (a,b) to (b,(a-b)/2), we regard it
* as having first subtracted b from a and then halved a, so both of
* these transformations must be done in sequence.
*
* The code below transforms this from a recursive to an iterative
* algorithm. We first reduce a,b to 0,1, recording at each stage
* whether we did the initial subtraction, and whether we had to swap
* the two values; then we iterate backwards over that record of what
* we did, applying the above rules for building up the Bezout
* coefficients as we go. Of course, all the case analysis is done by
* the usual bit-twiddling conditionalisation to avoid data-dependent
* control flow.
*
* Also, since these mp_ints are generally treated as unsigned, we
* store the coefficients by absolute value, with the semantics that
* they always have opposite sign, and in the unwinding loop we keep a
* bit indicating whether Aa-Bb is currently expected to be +d or -d,
* so that we can do one final conditional adjustment if it's -d.
*
* Once the reduction rules have managed to reduce the input numbers
* to (0,d), then they are stable (the next reduction will always
* divide the even one by 2, which maps 0 to 0). So it doesn't matter
* if we do more steps of the algorithm than necessary; hence, for
* constant time, we just need to find the maximum number we could
* _possibly_ require, and do that many.
*
* If a,b < 2^n, at most 2n iterations are required. Proof: consider
* the quantity Q = log_2(a) + log_2(b). Every step halves one of the
* numbers (and may also reduce one of them further by doing a
* subtraction beforehand, but in the worst case, not by much or not
* at all). So Q reduces by at least 1 per iteration, and it starts
* off with a value at most 2n.
*
* The worst case inputs (I think) are where x=2^{n-1} and y=2^n-1
* (i.e. x is a power of 2 and y is all 1s). In that situation, the
* first n-1 steps repeatedly halve x until it's 1, and then there are
* n further steps each of which subtracts 1 from y and halves it.
*/
static void mp_bezout_into(mp_int *a_coeff_out, mp_int *b_coeff_out,
mp_int *gcd_out, mp_int *a_in, mp_int *b_in)
{
size_t nw = size_t_max(1, size_t_max(a_in->nw, b_in->nw));
/* Make mutable copies of the input numbers */
mp_int *a = mp_make_sized(nw), *b = mp_make_sized(nw);
mp_copy_into(a, a_in);
mp_copy_into(b, b_in);
/* Space to build up the output coefficients, with an extra word
* so that intermediate values can overflow off the top and still
* right-shift back down to the correct value */
mp_int *ac = mp_make_sized(nw + 1), *bc = mp_make_sized(nw + 1);
/* And a general-purpose temp register */
mp_int *tmp = mp_make_sized(nw);
/* Space to record the sequence of reduction steps to unwind. We
* make it a BignumInt for no particular reason except that (a)
* mp_make_sized conveniently zeroes the allocation and mp_free
* wipes it, and (b) this way I can use mp_dump() if I have to
* debug this code. */
size_t steps = 2 * nw * BIGNUM_INT_BITS;
mp_int *record = mp_make_sized(
(steps*2 + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS);
for (size_t step = 0; step < steps; step++) {
/*
* If a and b are both odd, we want to sort them so that a is
* larger. But if one is even, we want to sort them so that a
* is the even one.
*/
unsigned swap_if_both_odd = mp_cmp_hs(b, a);
unsigned swap_if_one_even = a->w[0] & 1;
unsigned both_odd = a->w[0] & b->w[0] & 1;
unsigned swap = swap_if_one_even ^ (
(swap_if_both_odd ^ swap_if_one_even) & both_odd);
mp_cond_swap(a, b, swap);
/*
* If a,b are both odd, then a is the larger number, so
* subtract the smaller one from it.
*/
mp_cond_sub_into(a, a, b, both_odd);
/*
* Now a is even, so divide it by two.
*/
mp_rshift_fixed_into(a, a, 1);
/*
* Record the two 1-bit values both_odd and swap.
*/
mp_set_bit(record, step*2, both_odd);
mp_set_bit(record, step*2+1, swap);
}
/*
* Now we expect to have reduced the two numbers to 0 and d,
* although we don't know which way round. (But we avoid checking
* this by assertion; sometimes we'll need to do this computation
* without giving away that we already know the inputs were bogus.
* So we'd prefer to just press on and return nonsense.)
*/
if (gcd_out) {
/*
* At this point we can return the actual gcd. Since one of
* a,b is it and the other is zero, the easiest way to get it
* is to add them together.
*/
mp_add_into(gcd_out, a, b);
}
/*
* If the caller _only_ wanted the gcd, and neither Bezout
* coefficient is even required, we can skip the entire unwind
* stage.
*/
if (a_coeff_out || b_coeff_out) {
/*
* The Bezout coefficients of a,b at this point are simply 0
* for whichever of a,b is zero, and 1 for whichever is
* nonzero. The nonzero number equals gcd(a,b), which by
* assumption is odd, so we can do this by just taking the low
* bit of each one.
*/
ac->w[0] = mp_get_bit(a, 0);
bc->w[0] = mp_get_bit(b, 0);
/*
* Overwrite a,b themselves with those same numbers. This has
* the effect of dividing both of them by d, which will
* arrange that during the unwind stage we generate the
* minimal coefficients instead of a larger pair.
*/
mp_copy_into(a, ac);
mp_copy_into(b, bc);
/*
* We'll maintain the invariant as we unwind that ac * a - bc
* * b is either +d or -d (or rather, +1/-1 after scaling by
* d), and we'll remember which. (We _could_ keep it at +d the
* whole time, but it would cost more work every time round
* the loop, so it's cheaper to fix that up once at the end.)
*
* Initially, the result is +d if a was the nonzero value after
* reduction, and -d if b was.
*/
unsigned minus_d = b->w[0];
for (size_t step = steps; step-- > 0 ;) {
/*
* Recover the data from the step we're unwinding.
*/
unsigned both_odd = mp_get_bit(record, step*2);
unsigned swap = mp_get_bit(record, step*2+1);
/*
* Unwind the division: if our coefficient of a is odd, we
* adjust the coefficients by +b and +a respectively.
*/
unsigned adjust = ac->w[0] & 1;
mp_cond_add_into(ac, ac, b, adjust);
mp_cond_add_into(bc, bc, a, adjust);
/*
* Now ac is definitely even, so we divide it by two.
*/
mp_rshift_fixed_into(ac, ac, 1);
/*
* Now unwind the subtraction, if there was one, by adding
* ac to bc.
*/
mp_cond_add_into(bc, bc, ac, both_odd);
/*
* Undo the transformation of the input numbers, by
* multiplying a by 2 and then adding b to a (the latter
* only if both_odd).
*/
mp_lshift_fixed_into(a, a, 1);
mp_cond_add_into(a, a, b, both_odd);
/*
* Finally, undo the swap. If we do swap, this also
* reverses the sign of the current result ac*a+bc*b.
*/
mp_cond_swap(a, b, swap);
mp_cond_swap(ac, bc, swap);
minus_d ^= swap;
}
/*
* Now we expect to have recovered the input a,b (or rather,
* the versions of them divided by d). But we might find that
* our current result is -d instead of +d, that is, we have
* A',B' such that A'a - B'b = -d.
*
* In that situation, we set A = b-A' and B = a-B', giving us
* Aa-Bb = ab - A'a - ab + B'b = +1.
*/
mp_sub_into(tmp, b, ac);
mp_select_into(ac, ac, tmp, minus_d);
mp_sub_into(tmp, a, bc);
mp_select_into(bc, bc, tmp, minus_d);
/*
* Now we really are done. Return the outputs.
*/
if (a_coeff_out)
mp_copy_into(a_coeff_out, ac);
if (b_coeff_out)
mp_copy_into(b_coeff_out, bc);
}
mp_free(a);
mp_free(b);
mp_free(ac);
mp_free(bc);
mp_free(tmp);
mp_free(record);
}
mp_int *mp_invert(mp_int *x, mp_int *m)
{
mp_int *result = mp_make_sized(m->nw);
mp_bezout_into(result, NULL, NULL, x, m);
return result;
}
void mp_gcd_into(mp_int *a, mp_int *b, mp_int *gcd, mp_int *A, mp_int *B)
{
/*
* Identify shared factors of 2. To do this we OR the two numbers
* to get something whose lowest set bit is in the right place,
* remove all higher bits by ANDing it with its own negation, and
* use mp_get_nbits to find the location of the single remaining
* set bit.
*/
mp_int *tmp = mp_make_sized(size_t_max(a->nw, b->nw));
for (size_t i = 0; i < tmp->nw; i++)
tmp->w[i] = mp_word(a, i) | mp_word(b, i);
BignumCarry carry = 1;
for (size_t i = 0; i < tmp->nw; i++) {
BignumInt negw;
BignumADC(negw, carry, 0, ~tmp->w[i], carry);
tmp->w[i] &= negw;
}
size_t shift = mp_get_nbits(tmp) - 1;
mp_free(tmp);
/*
* Make copies of a,b with those shared factors of 2 divided off,
* so that at least one is odd (which is the precondition for
* mp_bezout_into). Compute the gcd of those.
*/
mp_int *as = mp_rshift_safe(a, shift);
mp_int *bs = mp_rshift_safe(b, shift);
mp_bezout_into(A, B, gcd, as, bs);
mp_free(as);
mp_free(bs);
/*
* And finally shift the gcd back up (unless the caller didn't
* even ask for it), to put the shared factors of 2 back in.
*/
if (gcd)
mp_lshift_safe_in_place(gcd, shift);
}
mp_int *mp_gcd(mp_int *a, mp_int *b)
{
mp_int *gcd = mp_make_sized(size_t_min(a->nw, b->nw));
mp_gcd_into(a, b, gcd, NULL, NULL);
return gcd;
}
unsigned mp_coprime(mp_int *a, mp_int *b)
{
mp_int *gcd = mp_gcd(a, b);
unsigned toret = mp_eq_integer(gcd, 1);
mp_free(gcd);
return toret;
}
static uint32_t recip_approx_32(uint32_t x)
{
/*
* Given an input x in [2^31,2^32), i.e. a uint32_t with its high
* bit set, this function returns an approximation to 2^63/x,
* computed using only multiplications and bit shifts just in case
* the C divide operator has non-constant time (either because the
* underlying machine instruction does, or because the operator
* expands to a library function on a CPU without hardware
* division).
*
* The coefficients are derived from those of the degree-9
* polynomial which is the minimax-optimal approximation to that
* function on the given interval (generated using the Remez
* algorithm), converted into integer arithmetic with shifts used
* to maximise the number of significant bits at every state. (A
* sort of 'static floating point' - the exponent is statically
* known at every point in the code, so it never needs to be
* stored at run time or to influence runtime decisions.)
*
* Exhaustive iteration over the whole input space shows the
* largest possible error to be 1686.54. (The input value
* attaining that bound is 4226800006 == 0xfbefd986, whose true
* reciprocal is 2182116973.540... == 0x8210766d.8a6..., whereas
* this function returns 2182115287 == 0x82106fd7.)
*/
uint64_t r = 0x92db03d6ULL;
r = 0xf63e71eaULL - ((r*x) >> 34);
r = 0xb63721e8ULL - ((r*x) >> 34);
r = 0x9c2da00eULL - ((r*x) >> 33);
r = 0xaada0bb8ULL - ((r*x) >> 32);
r = 0xf75cd403ULL - ((r*x) >> 31);
r = 0xecf97a41ULL - ((r*x) >> 31);
r = 0x90d876cdULL - ((r*x) >> 31);
r = 0x6682799a0ULL - ((r*x) >> 26);
return r;
}
void mp_divmod_into(mp_int *n, mp_int *d, mp_int *q_out, mp_int *r_out)
{
assert(!mp_eq_integer(d, 0));
/*
* We do division by using Newton-Raphson iteration to converge to
* the reciprocal of d (or rather, R/d for R a sufficiently large
* power of 2); then we multiply that reciprocal by n; and we
* finish up with conditional subtraction.
*
* But we have to do it in a fixed number of N-R iterations, so we
* need some error analysis to know how many we might need.
*
* The iteration is derived by defining f(r) = d - R/r.
* Differentiating gives f'(r) = R/r^2, and the Newton-Raphson
* formula applied to those functions gives
*
* r_{i+1} = r_i - f(r_i) / f'(r_i)
* = r_i - (d - R/r_i) r_i^2 / R
* = r_i (2 R - d r_i) / R
*
* Now let e_i be the error in a given iteration, in the sense
* that
*
* d r_i = R + e_i
* i.e. e_i/R = (r_i - r_true) / r_true
*
* so e_i is the _relative_ error in r_i.
*
* We must also introduce a rounding-error term, because the
* division by R always gives an integer. This might make the
* output off by up to 1 (in the negative direction, because
* right-shifting gives floor of the true quotient). So when we
* divide by R, we must imagine adding some f in [0,1). Then we
* have
*
* d r_{i+1} = d r_i (2 R - d r_i) / R - d f
* = (R + e_i) (R - e_i) / R - d f
* = (R^2 - e_i^2) / R - d f
* = R - (e_i^2 / R + d f)
* => e_{i+1} = - (e_i^2 / R + d f)
*
* The sum of two positive quantities is bounded above by twice
* their max, and max |f| = 1, so we can bound this as follows:
*
* |e_{i+1}| <= 2 max (e_i^2/R, d)
* |e_{i+1}/R| <= 2 max ((e_i/R)^2, d/R)
* log2 |R/e_{i+1}| <= min (2 log2 |R/e_i|, log2 |R/d|) - 1
*
* which tells us that the number of 'good' bits - i.e.
* log2(R/e_i) - very nearly doubles at every iteration (apart
* from that subtraction of 1), until it gets to the same size as
* log2(R/d). In other words, the size of R in bits has to be the
* size of denominator we're putting in, _plus_ the amount of
* precision we want to get back out.
*
* So when we multiply n (the input numerator) by our final
* reciprocal approximation r, but actually r differs from R/d by
* up to 2, then it follows that
*
* n/d - nr/R = n/d - [ n (R/d + e) ] / R
* = n/d - [ (n/d) R + n e ] / R
* = -ne/R
* => 0 <= n/d - nr/R < 2n/R
*
* so our computed quotient can differ from the true n/d by up to
* 2n/R. Hence, as long as we also choose R large enough that 2n/R
* is bounded above by a constant, we can guarantee a bounded
* number of final conditional-subtraction steps.
*/
/*
* Get at least 32 of the most significant bits of the input
* number.
*/
size_t hiword_index = 0;
uint64_t hibits = 0, lobits = 0;
mp_find_highest_nonzero_word_pair(d, 64 - BIGNUM_INT_BITS,
&hiword_index, &hibits, &lobits);
/*
* Make a shifted combination of those two words which puts the
* topmost bit of the number at bit 63.
*/
size_t shift_up = 0;
for (size_t i = BIGNUM_INT_BITS_BITS; i-- > 0;) {
size_t sl = 1 << i; /* left shift count */
size_t sr = 64 - sl; /* complementary right-shift count */
/* Should we shift up? */
unsigned indicator = 1 ^ normalise_to_1_u64(hibits >> sr);
/* If we do, what will we get? */
uint64_t new_hibits = (hibits << sl) | (lobits >> sr);
uint64_t new_lobits = lobits << sl;
size_t new_shift_up = shift_up + sl;
/* Conditionally swap those values in. */
hibits ^= (hibits ^ new_hibits ) & -(uint64_t)indicator;
lobits ^= (lobits ^ new_lobits ) & -(uint64_t)indicator;
shift_up ^= (shift_up ^ new_shift_up ) & -(size_t) indicator;
}
/*
* So now we know the most significant 32 bits of d are at the top
* of hibits. Approximate the reciprocal of those bits.
*/
lobits = (uint64_t)recip_approx_32(hibits >> 32) << 32;
hibits = 0;
/*
* And shift that up by as many bits as the input was shifted up
* just now, so that the product of this approximation and the
* actual input will be close to a fixed power of two regardless
* of where the MSB was.
*
* I do this in another log n individual passes, partly in case
* the CPU's register-controlled shift operation isn't
* time-constant, and also in case the compiler code-generates
* uint64_t shifts out of a variable number of smaller-word shift
* instructions, e.g. by splitting up into cases.
*/
for (size_t i = BIGNUM_INT_BITS_BITS; i-- > 0;) {
size_t sl = 1 << i; /* left shift count */
size_t sr = 64 - sl; /* complementary right-shift count */
/* Should we shift up? */
unsigned indicator = 1 & (shift_up >> i);
/* If we do, what will we get? */
uint64_t new_hibits = (hibits << sl) | (lobits >> sr);
uint64_t new_lobits = lobits << sl;
/* Conditionally swap those values in. */
hibits ^= (hibits ^ new_hibits ) & -(uint64_t)indicator;
lobits ^= (lobits ^ new_lobits ) & -(uint64_t)indicator;
}
/*
* The product of the 128-bit value now in hibits:lobits with the
* 128-bit value we originally retrieved in the same variables
* will be in the vicinity of 2^191. So we'll take log2(R) to be
* 191, plus a multiple of BIGNUM_INT_BITS large enough to allow R
* to hold the combined sizes of n and d.
*/
size_t log2_R;
{
size_t max_log2_n = (n->nw + d->nw) * BIGNUM_INT_BITS;
log2_R = max_log2_n + 3;
log2_R -= size_t_min(191, log2_R);
log2_R = (log2_R + BIGNUM_INT_BITS - 1) & ~(BIGNUM_INT_BITS - 1);
log2_R += 191;
}
/* Number of words in a bignum capable of holding numbers the size
* of twice R. */
size_t rw = ((log2_R+2) + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS;
/*
* Now construct our full-sized starting reciprocal approximation.
*/
mp_int *r_approx = mp_make_sized(rw);
size_t output_bit_index;
{
/* Where in the input number did the input 128-bit value come from? */
size_t input_bit_index =
(hiword_index * BIGNUM_INT_BITS) - (128 - BIGNUM_INT_BITS);
/* So how far do we need to shift our 64-bit output, if the
* product of those two fixed-size values is 2^191 and we want
* to make it 2^log2_R instead? */
output_bit_index = log2_R - 191 - input_bit_index;
/* If we've done all that right, it should be a whole number
* of words. */
assert(output_bit_index % BIGNUM_INT_BITS == 0);
size_t output_word_index = output_bit_index / BIGNUM_INT_BITS;
mp_add_integer_into_shifted_by_words(
r_approx, r_approx, lobits, output_word_index);
mp_add_integer_into_shifted_by_words(
r_approx, r_approx, hibits,
output_word_index + 64 / BIGNUM_INT_BITS);
}
/*
* Make the constant 2*R, which we'll need in the iteration.
*/
mp_int *two_R = mp_make_sized(rw);
mp_add_integer_into_shifted_by_words(
two_R, two_R, (BignumInt)1 << ((log2_R+1) % BIGNUM_INT_BITS),
(log2_R+1) / BIGNUM_INT_BITS);
/*
* Scratch space.
*/
mp_int *dr = mp_make_sized(rw + d->nw);
mp_int *diff = mp_make_sized(size_t_max(rw, dr->nw));
mp_int *product = mp_make_sized(rw + diff->nw);
size_t scratchsize = size_t_max(
mp_mul_scratchspace(dr->nw, r_approx->nw, d->nw),
mp_mul_scratchspace(product->nw, r_approx->nw, diff->nw));
mp_int *scratch = mp_make_sized(scratchsize);
mp_int product_shifted = mp_make_alias(
product, log2_R / BIGNUM_INT_BITS, product->nw);
/*
* Initial error estimate: the 32-bit output of recip_approx_32
* differs by less than 2048 (== 2^11) from the true top 32 bits
* of the reciprocal, so the relative error is at most 2^11
* divided by the 32-bit reciprocal, which at worst is 2^11/2^31 =
* 2^-20. So even in the worst case, we have 20 good bits of
* reciprocal to start with.
*/
size_t good_bits = 31 - 11;
size_t good_bits_needed = BIGNUM_INT_BITS * n->nw + 4; /* add a few */
/*
* Now do Newton-Raphson iterations until we have reason to think
* they're not converging any more.
*/
while (good_bits < good_bits_needed) {
/*
* Compute the next iterate.
*/
mp_mul_internal(dr, r_approx, d, *scratch);
mp_sub_into(diff, two_R, dr);
mp_mul_internal(product, r_approx, diff, *scratch);
mp_rshift_fixed_into(r_approx, &product_shifted,
log2_R % BIGNUM_INT_BITS);
/*
* Adjust the error estimate.
*/
good_bits = good_bits * 2 - 1;
}
mp_free(dr);
mp_free(diff);
mp_free(product);
mp_free(scratch);
/*
* Now we've got our reciprocal, we can compute the quotient, by
* multiplying in n and then shifting down by log2_R bits.
*/
mp_int *quotient_full = mp_mul(r_approx, n);
mp_int quotient_alias = mp_make_alias(
quotient_full, log2_R / BIGNUM_INT_BITS, quotient_full->nw);
mp_int *quotient = mp_make_sized(n->nw);
mp_rshift_fixed_into(quotient, &quotient_alias, log2_R % BIGNUM_INT_BITS);
/*
* Next, compute the remainder.
*/
mp_int *remainder = mp_make_sized(d->nw);
mp_mul_into(remainder, quotient, d);
mp_sub_into(remainder, n, remainder);
/*
* Finally, two conditional subtractions to fix up any remaining
* rounding error. (I _think_ one should be enough, but this
* routine isn't time-critical enough to take chances.)
*/
unsigned q_correction = 0;
for (unsigned iter = 0; iter < 2; iter++) {
unsigned need_correction = mp_cmp_hs(remainder, d);
mp_cond_sub_into(remainder, remainder, d, need_correction);
q_correction += need_correction;
}
mp_add_integer_into(quotient, quotient, q_correction);
/*
* Now we should have a perfect answer, i.e. 0 <= r < d.
*/
assert(!mp_cmp_hs(remainder, d));
if (q_out)
mp_copy_into(q_out, quotient);
if (r_out)
mp_copy_into(r_out, remainder);
mp_free(r_approx);
mp_free(two_R);
mp_free(quotient_full);
mp_free(quotient);
mp_free(remainder);
}
mp_int *mp_div(mp_int *n, mp_int *d)
{
mp_int *q = mp_make_sized(n->nw);
mp_divmod_into(n, d, q, NULL);
return q;
}
mp_int *mp_mod(mp_int *n, mp_int *d)
{
mp_int *r = mp_make_sized(d->nw);
mp_divmod_into(n, d, NULL, r);
return r;
}
mp_int *mp_nthroot(mp_int *y, unsigned n, mp_int *remainder_out)
{
/*
* Allocate scratch space.
*/
mp_int **alloc, **powers, **newpowers, *scratch;
size_t nalloc = 2*(n+1)+1;
alloc = snewn(nalloc, mp_int *);
for (size_t i = 0; i < nalloc; i++)
alloc[i] = mp_make_sized(y->nw + 1);
powers = alloc;
newpowers = alloc + (n+1);
scratch = alloc[2*n+2];
/*
* We're computing the rounded-down nth root of y, i.e. the
* maximal x such that x^n <= y. We try to add 2^i to it for each
* possible value of i, starting from the largest one that might
* fit (i.e. such that 2^{n*i} fits in the size of y) downwards to
* i=0.
*
* We track all the smaller powers of x in the array 'powers'. In
* each iteration, if we update x, we update all of those values
* to match.
*/
mp_copy_integer_into(powers[0], 1);
for (size_t s = mp_max_bits(y) / n + 1; s-- > 0 ;) {
/*
* Let b = 2^s. We need to compute the powers (x+b)^i for each
* i, starting from our recorded values of x^i.
*/
for (size_t i = 0; i < n+1; i++) {
/*
* (x+b)^i = x^i
* + (i choose 1) x^{i-1} b
* + (i choose 2) x^{i-2} b^2
* + ...
* + b^i
*/
uint16_t binom = 1; /* coefficient of b^i */
mp_copy_into(newpowers[i], powers[i]);
for (size_t j = 0; j < i; j++) {
/* newpowers[i] += binom * powers[j] * 2^{(i-j)*s} */
mp_mul_integer_into(scratch, powers[j], binom);
mp_lshift_fixed_into(scratch, scratch, (i-j) * s);
mp_add_into(newpowers[i], newpowers[i], scratch);
uint32_t binom_mul = binom;
binom_mul *= (i-j);
binom_mul /= (j+1);
assert(binom_mul < 0x10000);
binom = binom_mul;
}
}
/*
* Now, is the new value of x^n still <= y? If so, update.
*/
unsigned newbit = mp_cmp_hs(y, newpowers[n]);
for (size_t i = 0; i < n+1; i++)
mp_select_into(powers[i], powers[i], newpowers[i], newbit);
}
if (remainder_out)
mp_sub_into(remainder_out, y, powers[n]);
mp_int *root = mp_new(mp_max_bits(y) / n);
mp_copy_into(root, powers[1]);
for (size_t i = 0; i < nalloc; i++)
mp_free(alloc[i]);
sfree(alloc);
return root;
}
mp_int *mp_modmul(mp_int *x, mp_int *y, mp_int *modulus)
{
mp_int *product = mp_mul(x, y);
mp_int *reduced = mp_mod(product, modulus);
mp_free(product);
return reduced;
}
mp_int *mp_modadd(mp_int *x, mp_int *y, mp_int *modulus)
{
mp_int *sum = mp_add(x, y);
mp_int *reduced = mp_mod(sum, modulus);
mp_free(sum);
return reduced;
}
mp_int *mp_modsub(mp_int *x, mp_int *y, mp_int *modulus)
{
mp_int *diff = mp_make_sized(size_t_max(x->nw, y->nw));
mp_sub_into(diff, x, y);
unsigned negate = mp_cmp_hs(y, x);
mp_cond_negate(diff, diff, negate);
mp_int *residue = mp_mod(diff, modulus);
mp_cond_negate(residue, residue, negate);
/* If we've just negated the residue, then it will be < 0 and need
* the modulus adding to it to make it positive - *except* if the
* residue was zero when we negated it. */
unsigned make_positive = negate & ~mp_eq_integer(residue, 0);
mp_cond_add_into(residue, residue, modulus, make_positive);
mp_free(diff);
return residue;
}
static mp_int *mp_modadd_in_range(mp_int *x, mp_int *y, mp_int *modulus)
{
mp_int *sum = mp_make_sized(modulus->nw);
unsigned carry = mp_add_into_internal(sum, x, y);
mp_cond_sub_into(sum, sum, modulus, carry | mp_cmp_hs(sum, modulus));
return sum;
}
static mp_int *mp_modsub_in_range(mp_int *x, mp_int *y, mp_int *modulus)
{
mp_int *diff = mp_make_sized(modulus->nw);
mp_sub_into(diff, x, y);
mp_cond_add_into(diff, diff, modulus, 1 ^ mp_cmp_hs(x, y));
return diff;
}
mp_int *monty_add(MontyContext *mc, mp_int *x, mp_int *y)
{
return mp_modadd_in_range(x, y, mc->m);
}
mp_int *monty_sub(MontyContext *mc, mp_int *x, mp_int *y)
{
return mp_modsub_in_range(x, y, mc->m);
}
void mp_min_into(mp_int *r, mp_int *x, mp_int *y)
{
mp_select_into(r, x, y, mp_cmp_hs(x, y));
}
void mp_max_into(mp_int *r, mp_int *x, mp_int *y)
{
mp_select_into(r, y, x, mp_cmp_hs(x, y));
}
mp_int *mp_min(mp_int *x, mp_int *y)
{
mp_int *r = mp_make_sized(size_t_min(x->nw, y->nw));
mp_min_into(r, x, y);
return r;
}
mp_int *mp_max(mp_int *x, mp_int *y)
{
mp_int *r = mp_make_sized(size_t_max(x->nw, y->nw));
mp_max_into(r, x, y);
return r;
}
mp_int *mp_power_2(size_t power)
{
mp_int *x = mp_new(power + 1);
mp_set_bit(x, power, 1);
return x;
}
struct ModsqrtContext {
mp_int *p; /* the prime */
MontyContext *mc; /* for doing arithmetic mod p */
/* Decompose p-1 as 2^e k, for positive integer e and odd k */
size_t e;
mp_int *k;
mp_int *km1o2; /* (k-1)/2 */
/* The user-provided value z which is not a quadratic residue mod
* p, and its kth power. Both in Montgomery form. */
mp_int *z, *zk;
};
ModsqrtContext *modsqrt_new(mp_int *p, mp_int *any_nonsquare_mod_p)
{
ModsqrtContext *sc = snew(ModsqrtContext);
memset(sc, 0, sizeof(ModsqrtContext));
sc->p = mp_copy(p);
sc->mc = monty_new(sc->p);
sc->z = monty_import(sc->mc, any_nonsquare_mod_p);
/* Find the lowest set bit in p-1. Since this routine expects p to
* be non-secret (typically a well-known standard elliptic curve
* parameter), for once we don't need clever bit tricks. */
for (sc->e = 1; sc->e < BIGNUM_INT_BITS * p->nw; sc->e++)
if (mp_get_bit(p, sc->e))
break;
sc->k = mp_rshift_fixed(p, sc->e);
sc->km1o2 = mp_rshift_fixed(sc->k, 1);
/* Leave zk to be filled in lazily, since it's more expensive to
* compute. If this context turns out never to be needed, we can
* save the bulk of the setup time this way. */
return sc;
}
static void modsqrt_lazy_setup(ModsqrtContext *sc)
{
if (!sc->zk)
sc->zk = monty_pow(sc->mc, sc->z, sc->k);
}
void modsqrt_free(ModsqrtContext *sc)
{
monty_free(sc->mc);
mp_free(sc->p);
mp_free(sc->z);
mp_free(sc->k);
mp_free(sc->km1o2);
if (sc->zk)
mp_free(sc->zk);
sfree(sc);
}
mp_int *mp_modsqrt(ModsqrtContext *sc, mp_int *x, unsigned *success)
{
mp_int *mx = monty_import(sc->mc, x);
mp_int *mroot = monty_modsqrt(sc, mx, success);
mp_free(mx);
mp_int *root = monty_export(sc->mc, mroot);
mp_free(mroot);
return root;
}
/*
* Modular square root, using an algorithm more or less similar to
* Tonelli-Shanks but adapted for constant time.
*
* The basic idea is to write p-1 = k 2^e, where k is odd and e > 0.
* Then the multiplicative group mod p (call it G) has a sequence of
* e+1 nested subgroups G = G_0 > G_1 > G_2 > ... > G_e, where each
* G_i is exactly half the size of G_{i-1} and consists of all the
* squares of elements in G_{i-1}. So the innermost group G_e has
* order k, which is odd, and hence within that group you can take a
* square root by raising to the power (k+1)/2.
*
* Our strategy is to iterate over these groups one by one and make
* sure the number x we're trying to take the square root of is inside
* each one, by adjusting it if it isn't.
*
* Suppose g is a primitive root of p, i.e. a generator of G_0. (We
* don't actually need to know what g _is_; we just imagine it for the
* sake of understanding.) Then G_i consists of precisely the (2^i)th
* powers of g, and hence, you can tell if a number is in G_i if
* raising it to the power k 2^{e-i} gives 1. So the conceptual
* algorithm goes: for each i, test whether x is in G_i by that
* method. If it isn't, then the previous iteration ensured it's in
* G_{i-1}, so it will be an odd power of g^{2^{i-1}}, and hence
* multiplying by any other odd power of g^{2^{i-1}} will give x' in
* G_i. And we have one of those, because our non-square z is an odd
* power of g, so z^{2^{i-1}} is an odd power of g^{2^{i-1}}.
*
* (There's a special case in the very first iteration, where we don't
* have a G_{i-1}. If it turns out that x is not even in G_1, that
* means it's not a square, so we set *success to 0. We still run the
* rest of the algorithm anyway, for the sake of constant time, but we
* don't give a hoot what it returns.)
*
* When we get to the end and have x in G_e, then we can take its
* square root by raising to (k+1)/2. But of course that's not the
* square root of the original input - it's only the square root of
* the adjusted version we produced during the algorithm. To get the
* true output answer we also have to multiply by a power of z,
* namely, z to the power of _half_ whatever we've been multiplying in
* as we go along. (The power of z we multiplied in must have been
* even, because the case in which we would have multiplied in an odd
* power of z is the i=0 case, in which we instead set the failure
* flag.)
*
* The code below is an optimised version of that basic idea, in which
* we _start_ by computing x^k so as to be able to test membership in
* G_i by only a few squarings rather than a full from-scratch modpow
* every time; we also start by computing our candidate output value
* x^{(k+1)/2}. So when the above description says 'adjust x by z^i'
* for some i, we have to adjust our running values of x^k and
* x^{(k+1)/2} by z^{ik} and z^{ik/2} respectively (the latter is safe
* because, as above, i is always even). And it turns out that we
* don't actually have to store the adjusted version of x itself at
* all - we _only_ keep those two powers of it.
*/
mp_int *monty_modsqrt(ModsqrtContext *sc, mp_int *x, unsigned *success)
{
modsqrt_lazy_setup(sc);
mp_int *scratch_to_free = mp_make_sized(3 * sc->mc->rw);
mp_int scratch = *scratch_to_free;
/*
* Compute toret = x^{(k+1)/2}, our starting point for the output
* square root, and also xk = x^k which we'll use as we go along
* for knowing when to apply correction factors. We do this by
* first computing x^{(k-1)/2}, then multiplying it by x, then
* multiplying the two together.
*/
mp_int *toret = monty_pow(sc->mc, x, sc->km1o2);
mp_int xk = mp_alloc_from_scratch(&scratch, sc->mc->rw);
mp_copy_into(&xk, toret);
monty_mul_into(sc->mc, toret, toret, x);
monty_mul_into(sc->mc, &xk, toret, &xk);
mp_int tmp = mp_alloc_from_scratch(&scratch, sc->mc->rw);
mp_int power_of_zk = mp_alloc_from_scratch(&scratch, sc->mc->rw);
mp_copy_into(&power_of_zk, sc->zk);
for (size_t i = 0; i < sc->e; i++) {
mp_copy_into(&tmp, &xk);
for (size_t j = i+1; j < sc->e; j++)
monty_mul_into(sc->mc, &tmp, &tmp, &tmp);
unsigned eq1 = mp_cmp_eq(&tmp, monty_identity(sc->mc));
if (i == 0) {
/* One special case: if x=0, then no power of x will ever
* equal 1, but we should still report success on the
* grounds that 0 does have a square root mod p. */
*success = eq1 | mp_eq_integer(x, 0);
} else {
monty_mul_into(sc->mc, &tmp, toret, &power_of_zk);
mp_select_into(toret, &tmp, toret, eq1);
monty_mul_into(sc->mc, &power_of_zk,
&power_of_zk, &power_of_zk);
monty_mul_into(sc->mc, &tmp, &xk, &power_of_zk);
mp_select_into(&xk, &tmp, &xk, eq1);
}
}
mp_free(scratch_to_free);
return toret;
}
mp_int *mp_random_bits_fn(size_t bits, random_read_fn_t random_read)
{
size_t bytes = (bits + 7) / 8;
uint8_t *randbuf = snewn(bytes, uint8_t);
random_read(randbuf, bytes);
if (bytes)
randbuf[0] &= (2 << ((bits-1) & 7)) - 1;
mp_int *toret = mp_from_bytes_be(make_ptrlen(randbuf, bytes));
smemclr(randbuf, bytes);
sfree(randbuf);
return toret;
}
mp_int *mp_random_upto_fn(mp_int *limit, random_read_fn_t rf)
{
/*
* It would be nice to generate our random numbers in such a way
* as to make every possible outcome literally equiprobable. But
* we can't do that in constant time, so we have to go for a very
* close approximation instead. I'm going to take the view that a
* factor of (1+2^-128) between the probabilities of two outcomes
* is acceptable on the grounds that you'd have to examine so many
* outputs to even detect it.
*/
mp_int *unreduced = mp_random_bits_fn(mp_max_bits(limit) + 128, rf);
mp_int *reduced = mp_mod(unreduced, limit);
mp_free(unreduced);
return reduced;
}
mp_int *mp_random_in_range_fn(mp_int *lo, mp_int *hi, random_read_fn_t rf)
{
mp_int *n_outcomes = mp_sub(hi, lo);
mp_int *addend = mp_random_upto_fn(n_outcomes, rf);
mp_int *result = mp_make_sized(hi->nw);
mp_add_into(result, addend, lo);
mp_free(addend);
mp_free(n_outcomes);
return result;
}
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