pe.old/SSHBN.C
/*
* Bignum routines for RSA and DH and stuff.
*/
#include <stdio.h>
#include <assert.h>
#include <stdlib.h>
#include <string.h>
#include "misc.h"
/*
* Usage notes:
* * Do not call the DIVMOD_WORD macro with expressions such as array
* subscripts, as some implementations object to this (see below).
* * Note that none of the division methods below will cope if the
* quotient won't fit into BIGNUM_INT_BITS. Callers should be careful
* to avoid this case.
* If this condition occurs, in the case of the x86 DIV instruction,
* an overflow exception will occur, which (according to a correspondent)
* will manifest on Windows as something like
* 0xC0000095: Integer overflow
* The C variant won't give the right answer, either.
*/
#if defined __GNUC__ && defined __i386__
typedef unsigned long BignumInt;
typedef unsigned long long BignumDblInt;
#define BIGNUM_INT_MASK 0xFFFFFFFFUL
#define BIGNUM_TOP_BIT 0x80000000UL
#define BIGNUM_INT_BITS 32
#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
#define DIVMOD_WORD(q, r, hi, lo, w) \
__asm__("div %2" : \
"=d" (r), "=a" (q) : \
"r" (w), "d" (hi), "a" (lo))
#elif defined _MSC_VER && defined _M_IX86
typedef unsigned __int32 BignumInt;
typedef unsigned __int64 BignumDblInt;
#define BIGNUM_INT_MASK 0xFFFFFFFFUL
#define BIGNUM_TOP_BIT 0x80000000UL
#define BIGNUM_INT_BITS 32
#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
/* Note: MASM interprets array subscripts in the macro arguments as
* assembler syntax, which gives the wrong answer. Don't supply them.
* <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */
#define DIVMOD_WORD(q, r, hi, lo, w) do { \
__asm mov edx, hi \
__asm mov eax, lo \
__asm div w \
__asm mov r, edx \
__asm mov q, eax \
} while(0)
#elif defined _LP64
/* 64-bit architectures can do 32x32->64 chunks at a time */
typedef unsigned int BignumInt;
typedef unsigned long BignumDblInt;
#define BIGNUM_INT_MASK 0xFFFFFFFFU
#define BIGNUM_TOP_BIT 0x80000000U
#define BIGNUM_INT_BITS 32
#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
#define DIVMOD_WORD(q, r, hi, lo, w) do { \
BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
q = n / w; \
r = n % w; \
} while (0)
#elif defined _LLP64
/* 64-bit architectures in which unsigned long is 32 bits, not 64 */
typedef unsigned long BignumInt;
typedef unsigned long long BignumDblInt;
#define BIGNUM_INT_MASK 0xFFFFFFFFUL
#define BIGNUM_TOP_BIT 0x80000000UL
#define BIGNUM_INT_BITS 32
#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
#define DIVMOD_WORD(q, r, hi, lo, w) do { \
BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
q = n / w; \
r = n % w; \
} while (0)
#else
/* Fallback for all other cases */
typedef unsigned short BignumInt;
typedef unsigned long BignumDblInt;
#define BIGNUM_INT_MASK 0xFFFFU
#define BIGNUM_TOP_BIT 0x8000U
#define BIGNUM_INT_BITS 16
#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2)
#define DIVMOD_WORD(q, r, hi, lo, w) do { \
BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \
q = n / w; \
r = n % w; \
} while (0)
#endif
#define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
#define BIGNUM_INTERNAL
typedef BignumInt *Bignum;
#include "ssh.h"
BignumInt bnZero[1] = { 0 };
BignumInt bnOne[2] = { 1, 1 };
/*
* The Bignum format is an array of `BignumInt'. The first
* element of the array counts the remaining elements. The
* remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_
* significant digit first. (So it's trivial to extract the bit
* with value 2^n for any n.)
*
* All Bignums in this module are positive. Negative numbers must
* be dealt with outside it.
*
* INVARIANT: the most significant word of any Bignum must be
* nonzero.
*/
Bignum Zero = bnZero, One = bnOne;
static Bignum newbn(int length)
{
Bignum b = snewn(length + 1, BignumInt);
if (!b)
abort(); /* FIXME */
memset(b, 0, (length + 1) * sizeof(*b));
b[0] = length;
return b;
}
void bn_restore_invariant(Bignum b)
{
while (b[0] > 1 && b[b[0]] == 0)
b[0]--;
}
Bignum copybn(Bignum orig)
{
Bignum b = snewn(orig[0] + 1, BignumInt);
if (!b)
abort(); /* FIXME */
memcpy(b, orig, (orig[0] + 1) * sizeof(*b));
return b;
}
void freebn(Bignum b)
{
/*
* Burn the evidence, just in case.
*/
memset(b, 0, sizeof(b[0]) * (b[0] + 1));
sfree(b);
}
Bignum bn_power_2(int n)
{
Bignum ret = newbn(n / BIGNUM_INT_BITS + 1);
bignum_set_bit(ret, n, 1);
return ret;
}
/*
* Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all
* big-endian arrays of 'len' BignumInts. Returns a BignumInt carried
* off the top.
*/
static BignumInt internal_add(const BignumInt *a, const BignumInt *b,
BignumInt *c, int len)
{
int i;
BignumDblInt carry = 0;
for (i = len-1; i >= 0; i--) {
carry += (BignumDblInt)a[i] + b[i];
c[i] = (BignumInt)carry;
carry >>= BIGNUM_INT_BITS;
}
return (BignumInt)carry;
}
/*
* Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are
* all big-endian arrays of 'len' BignumInts. Any borrow from the top
* is ignored.
*/
static void internal_sub(const BignumInt *a, const BignumInt *b,
BignumInt *c, int len)
{
int i;
BignumDblInt carry = 1;
for (i = len-1; i >= 0; i--) {
carry += (BignumDblInt)a[i] + (b[i] ^ BIGNUM_INT_MASK);
c[i] = (BignumInt)carry;
carry >>= BIGNUM_INT_BITS;
}
}
/*
* Compute c = a * b.
* Input is in the first len words of a and b.
* Result is returned in the first 2*len words of c.
*
* 'scratch' must point to an array of BignumInt of size at least
* mul_compute_scratch(len). (This covers the needs of internal_mul
* and all its recursive calls to itself.)
*/
#define KARATSUBA_THRESHOLD 50
static int mul_compute_scratch(int len)
{
int ret = 0;
while (len > KARATSUBA_THRESHOLD) {
int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
int midlen = botlen + 1;
ret += 4*midlen;
len = midlen;
}
return ret;
}
static void internal_mul(const BignumInt *a, const BignumInt *b,
BignumInt *c, int len, BignumInt *scratch)
{
if (len > KARATSUBA_THRESHOLD) {
int i;
/*
* Karatsuba divide-and-conquer algorithm. 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.
*/
int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
int midlen = botlen + 1;
BignumDblInt carry;
#ifdef KARA_DEBUG
int i;
#endif
/*
* The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping
* in the output array, so we can compute them immediately in
* place.
*/
#ifdef KARA_DEBUG
printf("a1,a0 = 0x");
for (i = 0; i < len; i++) {
if (i == toplen) printf(", 0x");
printf("%0*x", BIGNUM_INT_BITS/4, a[i]);
}
printf("\n");
printf("b1,b0 = 0x");
for (i = 0; i < len; i++) {
if (i == toplen) printf(", 0x");
printf("%0*x", BIGNUM_INT_BITS/4, b[i]);
}
printf("\n");
#endif
/* a_1 b_1 */
internal_mul(a, b, c, toplen, scratch);
#ifdef KARA_DEBUG
printf("a1b1 = 0x");
for (i = 0; i < 2*toplen; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, c[i]);
}
printf("\n");
#endif
/* a_0 b_0 */
internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch);
#ifdef KARA_DEBUG
printf("a0b0 = 0x");
for (i = 0; i < 2*botlen; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]);
}
printf("\n");
#endif
/* Zero padding. midlen exceeds toplen by at most 2, so just
* zero the first two words of each input and the rest will be
* copied over. */
scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0;
for (i = 0; i < toplen; i++) {
scratch[midlen - toplen + i] = a[i]; /* a_1 */
scratch[2*midlen - toplen + i] = b[i]; /* b_1 */
}
/* compute a_1 + a_0 */
scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen);
#ifdef KARA_DEBUG
printf("a1plusa0 = 0x");
for (i = 0; i < midlen; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]);
}
printf("\n");
#endif
/* compute b_1 + b_0 */
scratch[midlen] = internal_add(scratch+midlen+1, b+toplen,
scratch+midlen+1, botlen);
#ifdef KARA_DEBUG
printf("b1plusb0 = 0x");
for (i = 0; i < midlen; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]);
}
printf("\n");
#endif
/*
* Now we can do the third multiplication.
*/
internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen,
scratch + 4*midlen);
#ifdef KARA_DEBUG
printf("a1plusa0timesb1plusb0 = 0x");
for (i = 0; i < 2*midlen; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]);
}
printf("\n");
#endif
/*
* Now we can reuse the first half of 'scratch' to compute the
* sum of the outer two coefficients, to subtract from that
* product to obtain the middle one.
*/
scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0;
for (i = 0; i < 2*toplen; i++)
scratch[2*midlen - 2*toplen + i] = c[i];
scratch[1] = internal_add(scratch+2, c + 2*toplen,
scratch+2, 2*botlen);
#ifdef KARA_DEBUG
printf("a1b1plusa0b0 = 0x");
for (i = 0; i < 2*midlen; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]);
}
printf("\n");
#endif
internal_sub(scratch + 2*midlen, scratch,
scratch + 2*midlen, 2*midlen);
#ifdef KARA_DEBUG
printf("a1b0plusa0b1 = 0x");
for (i = 0; i < 2*midlen; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]);
}
printf("\n");
#endif
/*
* And now all we need to do is to add that middle coefficient
* back into the output. We may have to propagate a carry
* further up the output, but we can be sure it won't
* propagate right the way off the top.
*/
carry = internal_add(c + 2*len - botlen - 2*midlen,
scratch + 2*midlen,
c + 2*len - botlen - 2*midlen, 2*midlen);
i = 2*len - botlen - 2*midlen - 1;
while (carry) {
assert(i >= 0);
carry += c[i];
c[i] = (BignumInt)carry;
carry >>= BIGNUM_INT_BITS;
i--;
}
#ifdef KARA_DEBUG
printf("ab = 0x");
for (i = 0; i < 2*len; i++) {
printf("%0*x", BIGNUM_INT_BITS/4, c[i]);
}
printf("\n");
#endif
} else {
int i;
BignumInt carry;
BignumDblInt t;
const BignumInt *ap, *bp;
BignumInt *cp, *cps;
/*
* Multiply in the ordinary O(N^2) way.
*/
for (i = 0; i < 2 * len; i++)
c[i] = 0;
for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) {
carry = 0;
for (cp = cps, bp = b + len; cp--, bp-- > b ;) {
t = (MUL_WORD(*ap, *bp) + carry) + *cp;
*cp = (BignumInt) t;
carry = (BignumInt)(t >> BIGNUM_INT_BITS);
}
*cp = carry;
}
}
}
/*
* Variant form of internal_mul used for the initial step of
* Montgomery reduction. Only bothers outputting 'len' words
* (everything above that is thrown away).
*/
static void internal_mul_low(const BignumInt *a, const BignumInt *b,
BignumInt *c, int len, BignumInt *scratch)
{
if (len > KARATSUBA_THRESHOLD) {
int i;
/*
* Karatsuba-aware version of internal_mul_low. As before, we
* express each input value as a shifted combination of two
* halves:
*
* a = a_1 D + a_0
* b = b_1 D + b_0
*
* Then the full product is, as before,
*
* ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
*
* Provided we choose D on the large side (so that a_0 and b_0
* are _at least_ as long as a_1 and b_1), we don't need the
* topmost term at all, and we only need half of the middle
* term. So there's no point in doing the proper Karatsuba
* optimisation which computes the middle term using the top
* one, because we'd take as long computing the top one as
* just computing the middle one directly.
*
* So instead, we do a much more obvious thing: we call the
* fully optimised internal_mul to compute a_0 b_0, and we
* recursively call ourself to compute the _bottom halves_ of
* a_1 b_0 and a_0 b_1, each of which we add into the result
* in the obvious way.
*
* In other words, there's no actual Karatsuba _optimisation_
* in this function; the only benefit in doing it this way is
* that we call internal_mul proper for a large part of the
* work, and _that_ can optimise its operation.
*/
int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
/*
* Scratch space for the various bits and pieces we're going
* to be adding together: we need botlen*2 words for a_0 b_0
* (though we may end up throwing away its topmost word), and
* toplen words for each of a_1 b_0 and a_0 b_1. That adds up
* to exactly 2*len.
*/
/* a_0 b_0 */
internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen,
scratch + 2*len);
/* a_1 b_0 */
internal_mul_low(a, b + len - toplen, scratch + toplen, toplen,
scratch + 2*len);
/* a_0 b_1 */
internal_mul_low(a + len - toplen, b, scratch, toplen,
scratch + 2*len);
/* Copy the bottom half of the big coefficient into place */
for (i = 0; i < botlen; i++)
c[toplen + i] = scratch[2*toplen + botlen + i];
/* Add the two small coefficients, throwing away the returned carry */
internal_add(scratch, scratch + toplen, scratch, toplen);
/* And add that to the large coefficient, leaving the result in c. */
internal_add(scratch, scratch + 2*toplen + botlen - toplen,
c, toplen);
} else {
int i;
BignumInt carry;
BignumDblInt t;
const BignumInt *ap, *bp;
BignumInt *cp, *cps;
/*
* Multiply in the ordinary O(N^2) way.
*/
for (i = 0; i < len; i++)
c[i] = 0;
for (cps = c + len, ap = a + len; ap-- > a; cps--) {
carry = 0;
for (cp = cps, bp = b + len; bp--, cp-- > c ;) {
t = (MUL_WORD(*ap, *bp) + carry) + *cp;
*cp = (BignumInt) t;
carry = (BignumInt)(t >> BIGNUM_INT_BITS);
}
}
}
}
/*
* Montgomery reduction. Expects x to be a big-endian array of 2*len
* BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *
* BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array
* a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=
* x' < n.
*
* 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts
* each, containing respectively n and the multiplicative inverse of
* -n mod r.
*
* 'tmp' is an array of BignumInt used as scratch space, of length at
* least 3*len + mul_compute_scratch(len).
*/
static void monty_reduce(BignumInt *x, const BignumInt *n,
const BignumInt *mninv, BignumInt *tmp, int len)
{
int i;
BignumInt carry;
/*
* Multiply x by (-n)^{-1} mod r. This gives us a value m such
* that mn is congruent to -x mod r. Hence, mn+x is an exact
* multiple of r, and is also (obviously) congruent to x mod n.
*/
internal_mul_low(x + len, mninv, tmp, len, tmp + 3*len);
/*
* Compute t = (mn+x)/r in ordinary, non-modular, integer
* arithmetic. By construction this is exact, and is congruent mod
* n to x * r^{-1}, i.e. the answer we want.
*
* The following multiply leaves that answer in the _most_
* significant half of the 'x' array, so then we must shift it
* down.
*/
internal_mul(tmp, n, tmp+len, len, tmp + 3*len);
carry = internal_add(x, tmp+len, x, 2*len);
for (i = 0; i < len; i++)
x[len + i] = x[i], x[i] = 0;
/*
* Reduce t mod n. This doesn't require a full-on division by n,
* but merely a test and single optional subtraction, since we can
* show that 0 <= t < 2n.
*
* Proof:
* + we computed m mod r, so 0 <= m < r.
* + so 0 <= mn < rn, obviously
* + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn
* + yielding 0 <= (mn+x)/r < 2n as required.
*/
if (!carry) {
for (i = 0; i < len; i++)
if (x[len + i] != n[i])
break;
}
if (carry || i >= len || x[len + i] > n[i])
internal_sub(x+len, n, x+len, len);
}
static void internal_add_shifted(BignumInt *number,
unsigned n, int shift)
{
int word = 1 + (shift / BIGNUM_INT_BITS);
int bshift = shift % BIGNUM_INT_BITS;
BignumDblInt addend;
addend = (BignumDblInt)n << bshift;
while (addend) {
addend += number[word];
number[word] = (BignumInt) addend & BIGNUM_INT_MASK;
addend >>= BIGNUM_INT_BITS;
word++;
}
}
/*
* Compute a = a % m.
* Input in first alen words of a and first mlen words of m.
* Output in first alen words of a
* (of which first alen-mlen words will be zero).
* The MSW of m MUST have its high bit set.
* Quotient is accumulated in the `quotient' array, which is a Bignum
* rather than the internal bigendian format. Quotient parts are shifted
* left by `qshift' before adding into quot.
*/
static void internal_mod(BignumInt *a, int alen,
BignumInt *m, int mlen,
BignumInt *quot, int qshift)
{
BignumInt m0, m1;
unsigned int h;
int i, k;
m0 = m[0];
if (mlen > 1)
m1 = m[1];
else
m1 = 0;
for (i = 0; i <= alen - mlen; i++) {
BignumDblInt t;
unsigned int q, r, c, ai1;
if (i == 0) {
h = 0;
} else {
h = a[i - 1];
a[i - 1] = 0;
}
if (i == alen - 1)
ai1 = 0;
else
ai1 = a[i + 1];
/* Find q = h:a[i] / m0 */
if (h >= m0) {
/*
* Special case.
*
* To illustrate it, suppose a BignumInt is 8 bits, and
* we are dividing (say) A1:23:45:67 by A1:B2:C3. Then
* our initial division will be 0xA123 / 0xA1, which
* will give a quotient of 0x100 and a divide overflow.
* However, the invariants in this division algorithm
* are not violated, since the full number A1:23:... is
* _less_ than the quotient prefix A1:B2:... and so the
* following correction loop would have sorted it out.
*
* In this situation we set q to be the largest
* quotient we _can_ stomach (0xFF, of course).
*/
q = BIGNUM_INT_MASK;
} else {
/* Macro doesn't want an array subscript expression passed
* into it (see definition), so use a temporary. */
BignumInt tmplo = a[i];
DIVMOD_WORD(q, r, h, tmplo, m0);
/* Refine our estimate of q by looking at
h:a[i]:a[i+1] / m0:m1 */
t = MUL_WORD(m1, q);
if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) {
q--;
t -= m1;
r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */
if (r >= (BignumDblInt) m0 &&
t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--;
}
}
/* Subtract q * m from a[i...] */
c = 0;
for (k = mlen - 1; k >= 0; k--) {
t = MUL_WORD(q, m[k]);
t += c;
c = (unsigned)(t >> BIGNUM_INT_BITS);
if ((BignumInt) t > a[i + k])
c++;
a[i + k] -= (BignumInt) t;
}
/* Add back m in case of borrow */
if (c != h) {
t = 0;
for (k = mlen - 1; k >= 0; k--) {
t += m[k];
t += a[i + k];
a[i + k] = (BignumInt) t;
t = t >> BIGNUM_INT_BITS;
}
q--;
}
if (quot)
internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i));
}
}
/*
* Compute (base ^ exp) % mod, the pedestrian way.
*/
Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod)
{
BignumInt *a, *b, *n, *m, *scratch;
int mshift;
int mlen, scratchlen, i, j;
Bignum base, result;
/*
* The most significant word of mod needs to be non-zero. It
* should already be, but let's make sure.
*/
assert(mod[mod[0]] != 0);
/*
* Make sure the base is smaller than the modulus, by reducing
* it modulo the modulus if not.
*/
base = bigmod(base_in, mod);
/* Allocate m of size mlen, copy mod to m */
/* We use big endian internally */
mlen = mod[0];
m = snewn(mlen, BignumInt);
for (j = 0; j < mlen; j++)
m[j] = mod[mod[0] - j];
/* Shift m left to make msb bit set */
for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
if ((m[0] << mshift) & BIGNUM_TOP_BIT)
break;
if (mshift) {
for (i = 0; i < mlen - 1; i++)
m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
m[mlen - 1] = m[mlen - 1] << mshift;
}
/* Allocate n of size mlen, copy base to n */
n = snewn(mlen, BignumInt);
i = mlen - base[0];
for (j = 0; j < i; j++)
n[j] = 0;
for (j = 0; j < (int)base[0]; j++)
n[i + j] = base[base[0] - j];
/* Allocate a and b of size 2*mlen. Set a = 1 */
a = snewn(2 * mlen, BignumInt);
b = snewn(2 * mlen, BignumInt);
for (i = 0; i < 2 * mlen; i++)
a[i] = 0;
a[2 * mlen - 1] = 1;
/* Scratch space for multiplies */
scratchlen = mul_compute_scratch(mlen);
scratch = snewn(scratchlen, BignumInt);
/* Skip leading zero bits of exp. */
i = 0;
j = BIGNUM_INT_BITS-1;
while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
j--;
if (j < 0) {
i++;
j = BIGNUM_INT_BITS-1;
}
}
/* Main computation */
while (i < (int)exp[0]) {
while (j >= 0) {
internal_mul(a + mlen, a + mlen, b, mlen, scratch);
internal_mod(b, mlen * 2, m, mlen, NULL, 0);
if ((exp[exp[0] - i] & (1 << j)) != 0) {
internal_mul(b + mlen, n, a, mlen, scratch);
internal_mod(a, mlen * 2, m, mlen, NULL, 0);
} else {
BignumInt *t;
t = a;
a = b;
b = t;
}
j--;
}
i++;
j = BIGNUM_INT_BITS-1;
}
/* Fixup result in case the modulus was shifted */
if (mshift) {
for (i = mlen - 1; i < 2 * mlen - 1; i++)
a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
a[2 * mlen - 1] = a[2 * mlen - 1] << mshift;
internal_mod(a, mlen * 2, m, mlen, NULL, 0);
for (i = 2 * mlen - 1; i >= mlen; i--)
a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
}
/* Copy result to buffer */
result = newbn(mod[0]);
for (i = 0; i < mlen; i++)
result[result[0] - i] = a[i + mlen];
while (result[0] > 1 && result[result[0]] == 0)
result[0]--;
/* Free temporary arrays */
for (i = 0; i < 2 * mlen; i++)
a[i] = 0;
sfree(a);
for (i = 0; i < scratchlen; i++)
scratch[i] = 0;
sfree(scratch);
for (i = 0; i < 2 * mlen; i++)
b[i] = 0;
sfree(b);
for (i = 0; i < mlen; i++)
m[i] = 0;
sfree(m);
for (i = 0; i < mlen; i++)
n[i] = 0;
sfree(n);
freebn(base);
return result;
}
/*
* Compute (base ^ exp) % mod. Uses the Montgomery multiplication
* technique where possible, falling back to modpow_simple otherwise.
*/
Bignum modpow(Bignum base_in, Bignum exp, Bignum mod)
{
BignumInt *a, *b, *x, *n, *mninv, *scratch;
int len, scratchlen, i, j;
Bignum base, base2, r, rn, inv, result;
/*
* The most significant word of mod needs to be non-zero. It
* should already be, but let's make sure.
*/
assert(mod[mod[0]] != 0);
/*
* mod had better be odd, or we can't do Montgomery multiplication
* using a power of two at all.
*/
if (!(mod[1] & 1))
return modpow_simple(base_in, exp, mod);
/*
* Make sure the base is smaller than the modulus, by reducing
* it modulo the modulus if not.
*/
base = bigmod(base_in, mod);
/*
* Compute the inverse of n mod r, for monty_reduce. (In fact we
* want the inverse of _minus_ n mod r, but we'll sort that out
* below.)
*/
len = mod[0];
r = bn_power_2(BIGNUM_INT_BITS * len);
inv = modinv(mod, r);
/*
* Multiply the base by r mod n, to get it into Montgomery
* representation.
*/
base2 = modmul(base, r, mod);
freebn(base);
base = base2;
rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */
freebn(r); /* won't need this any more */
/*
* Set up internal arrays of the right lengths, in big-endian
* format, containing the base, the modulus, and the modulus's
* inverse.
*/
n = snewn(len, BignumInt);
for (j = 0; j < len; j++)
n[len - 1 - j] = mod[j + 1];
mninv = snewn(len, BignumInt);
for (j = 0; j < len; j++)
mninv[len - 1 - j] = (j < (int)inv[0] ? inv[j + 1] : 0);
freebn(inv); /* we don't need this copy of it any more */
/* Now negate mninv mod r, so it's the inverse of -n rather than +n. */
x = snewn(len, BignumInt);
for (j = 0; j < len; j++)
x[j] = 0;
internal_sub(x, mninv, mninv, len);
/* x = snewn(len, BignumInt); */ /* already done above */
for (j = 0; j < len; j++)
x[len - 1 - j] = (j < (int)base[0] ? base[j + 1] : 0);
freebn(base); /* we don't need this copy of it any more */
a = snewn(2*len, BignumInt);
b = snewn(2*len, BignumInt);
for (j = 0; j < len; j++)
a[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0);
freebn(rn);
/* Scratch space for multiplies */
scratchlen = 3*len + mul_compute_scratch(len);
scratch = snewn(scratchlen, BignumInt);
/* Skip leading zero bits of exp. */
i = 0;
j = BIGNUM_INT_BITS-1;
while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
j--;
if (j < 0) {
i++;
j = BIGNUM_INT_BITS-1;
}
}
/* Main computation */
while (i < (int)exp[0]) {
while (j >= 0) {
internal_mul(a + len, a + len, b, len, scratch);
monty_reduce(b, n, mninv, scratch, len);
if ((exp[exp[0] - i] & (1 << j)) != 0) {
internal_mul(b + len, x, a, len, scratch);
monty_reduce(a, n, mninv, scratch, len);
} else {
BignumInt *t;
t = a;
a = b;
b = t;
}
j--;
}
i++;
j = BIGNUM_INT_BITS-1;
}
/*
* Final monty_reduce to get back from the adjusted Montgomery
* representation.
*/
monty_reduce(a, n, mninv, scratch, len);
/* Copy result to buffer */
result = newbn(mod[0]);
for (i = 0; i < len; i++)
result[result[0] - i] = a[i + len];
while (result[0] > 1 && result[result[0]] == 0)
result[0]--;
/* Free temporary arrays */
for (i = 0; i < scratchlen; i++)
scratch[i] = 0;
sfree(scratch);
for (i = 0; i < 2 * len; i++)
a[i] = 0;
sfree(a);
for (i = 0; i < 2 * len; i++)
b[i] = 0;
sfree(b);
for (i = 0; i < len; i++)
mninv[i] = 0;
sfree(mninv);
for (i = 0; i < len; i++)
n[i] = 0;
sfree(n);
for (i = 0; i < len; i++)
x[i] = 0;
sfree(x);
return result;
}
/*
* Compute (p * q) % mod.
* The most significant word of mod MUST be non-zero.
* We assume that the result array is the same size as the mod array.
*/
Bignum modmul(Bignum p, Bignum q, Bignum mod)
{
BignumInt *a, *n, *m, *o, *scratch;
int mshift, scratchlen;
int pqlen, mlen, rlen, i, j;
Bignum result;
/* Allocate m of size mlen, copy mod to m */
/* We use big endian internally */
mlen = mod[0];
m = snewn(mlen, BignumInt);
for (j = 0; j < mlen; j++)
m[j] = mod[mod[0] - j];
/* Shift m left to make msb bit set */
for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
if ((m[0] << mshift) & BIGNUM_TOP_BIT)
break;
if (mshift) {
for (i = 0; i < mlen - 1; i++)
m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
m[mlen - 1] = m[mlen - 1] << mshift;
}
pqlen = (p[0] > q[0] ? p[0] : q[0]);
/* Allocate n of size pqlen, copy p to n */
n = snewn(pqlen, BignumInt);
i = pqlen - p[0];
for (j = 0; j < i; j++)
n[j] = 0;
for (j = 0; j < (int)p[0]; j++)
n[i + j] = p[p[0] - j];
/* Allocate o of size pqlen, copy q to o */
o = snewn(pqlen, BignumInt);
i = pqlen - q[0];
for (j = 0; j < i; j++)
o[j] = 0;
for (j = 0; j < (int)q[0]; j++)
o[i + j] = q[q[0] - j];
/* Allocate a of size 2*pqlen for result */
a = snewn(2 * pqlen, BignumInt);
/* Scratch space for multiplies */
scratchlen = mul_compute_scratch(pqlen);
scratch = snewn(scratchlen, BignumInt);
/* Main computation */
internal_mul(n, o, a, pqlen, scratch);
internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
/* Fixup result in case the modulus was shifted */
if (mshift) {
for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++)
a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift;
internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--)
a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
}
/* Copy result to buffer */
rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2);
result = newbn(rlen);
for (i = 0; i < rlen; i++)
result[result[0] - i] = a[i + 2 * pqlen - rlen];
while (result[0] > 1 && result[result[0]] == 0)
result[0]--;
/* Free temporary arrays */
for (i = 0; i < scratchlen; i++)
scratch[i] = 0;
sfree(scratch);
for (i = 0; i < 2 * pqlen; i++)
a[i] = 0;
sfree(a);
for (i = 0; i < mlen; i++)
m[i] = 0;
sfree(m);
for (i = 0; i < pqlen; i++)
n[i] = 0;
sfree(n);
for (i = 0; i < pqlen; i++)
o[i] = 0;
sfree(o);
return result;
}
/*
* Compute p % mod.
* The most significant word of mod MUST be non-zero.
* We assume that the result array is the same size as the mod array.
* We optionally write out a quotient if `quotient' is non-NULL.
* We can avoid writing out the result if `result' is NULL.
*/
static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient)
{
BignumInt *n, *m;
int mshift;
int plen, mlen, i, j;
/* Allocate m of size mlen, copy mod to m */
/* We use big endian internally */
mlen = mod[0];
m = snewn(mlen, BignumInt);
for (j = 0; j < mlen; j++)
m[j] = mod[mod[0] - j];
/* Shift m left to make msb bit set */
for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
if ((m[0] << mshift) & BIGNUM_TOP_BIT)
break;
if (mshift) {
for (i = 0; i < mlen - 1; i++)
m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
m[mlen - 1] = m[mlen - 1] << mshift;
}
plen = p[0];
/* Ensure plen > mlen */
if (plen <= mlen)
plen = mlen + 1;
/* Allocate n of size plen, copy p to n */
n = snewn(plen, BignumInt);
for (j = 0; j < plen; j++)
n[j] = 0;
for (j = 1; j <= (int)p[0]; j++)
n[plen - j] = p[j];
/* Main computation */
internal_mod(n, plen, m, mlen, quotient, mshift);
/* Fixup result in case the modulus was shifted */
if (mshift) {
for (i = plen - mlen - 1; i < plen - 1; i++)
n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift));
n[plen - 1] = n[plen - 1] << mshift;
internal_mod(n, plen, m, mlen, quotient, 0);
for (i = plen - 1; i >= plen - mlen; i--)
n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift));
}
/* Copy result to buffer */
if (result) {
for (i = 1; i <= (int)result[0]; i++) {
int j = plen - i;
result[i] = j >= 0 ? n[j] : 0;
}
}
/* Free temporary arrays */
for (i = 0; i < mlen; i++)
m[i] = 0;
sfree(m);
for (i = 0; i < plen; i++)
n[i] = 0;
sfree(n);
}
/*
* Decrement a number.
*/
void decbn(Bignum bn)
{
int i = 1;
while (i < (int)bn[0] && bn[i] == 0)
bn[i++] = BIGNUM_INT_MASK;
bn[i]--;
}
Bignum bignum_from_bytes(const unsigned char *data, int nbytes)
{
Bignum result;
int w, i;
w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */
result = newbn(w);
for (i = 1; i <= w; i++)
result[i] = 0;
for (i = nbytes; i--;) {
unsigned char byte = *data++;
result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS);
}
while (result[0] > 1 && result[result[0]] == 0)
result[0]--;
return result;
}
/*
* Read an SSH-1-format bignum from a data buffer. Return the number
* of bytes consumed, or -1 if there wasn't enough data.
*/
int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result)
{
const unsigned char *p = data;
int i;
int w, b;
if (len < 2)
return -1;
w = 0;
for (i = 0; i < 2; i++)
w = (w << 8) + *p++;
b = (w + 7) / 8; /* bits -> bytes */
if (len < b+2)
return -1;
if (!result) /* just return length */
return b + 2;
*result = bignum_from_bytes(p, b);
return p + b - data;
}
/*
* Return the bit count of a bignum, for SSH-1 encoding.
*/
int bignum_bitcount(Bignum bn)
{
int bitcount = bn[0] * BIGNUM_INT_BITS - 1;
while (bitcount >= 0
&& (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--;
return bitcount + 1;
}
/*
* Return the byte length of a bignum when SSH-1 encoded.
*/
int ssh1_bignum_length(Bignum bn)
{
return 2 + (bignum_bitcount(bn) + 7) / 8;
}
/*
* Return the byte length of a bignum when SSH-2 encoded.
*/
int ssh2_bignum_length(Bignum bn)
{
return 4 + (bignum_bitcount(bn) + 8) / 8;
}
/*
* Return a byte from a bignum; 0 is least significant, etc.
*/
int bignum_byte(Bignum bn, int i)
{
if (i >= (int)(BIGNUM_INT_BYTES * bn[0]))
return 0; /* beyond the end */
else
return (bn[i / BIGNUM_INT_BYTES + 1] >>
((i % BIGNUM_INT_BYTES)*8)) & 0xFF;
}
/*
* Return a bit from a bignum; 0 is least significant, etc.
*/
int bignum_bit(Bignum bn, int i)
{
if (i >= (int)(BIGNUM_INT_BITS * bn[0]))
return 0; /* beyond the end */
else
return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1;
}
/*
* Set a bit in a bignum; 0 is least significant, etc.
*/
void bignum_set_bit(Bignum bn, int bitnum, int value)
{
if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0]))
abort(); /* beyond the end */
else {
int v = bitnum / BIGNUM_INT_BITS + 1;
int mask = 1 << (bitnum % BIGNUM_INT_BITS);
if (value)
bn[v] |= mask;
else
bn[v] &= ~mask;
}
}
/*
* Write a SSH-1-format bignum into a buffer. It is assumed the
* buffer is big enough. Returns the number of bytes used.
*/
int ssh1_write_bignum(void *data, Bignum bn)
{
unsigned char *p = data;
int len = ssh1_bignum_length(bn);
int i;
int bitc = bignum_bitcount(bn);
*p++ = (bitc >> 8) & 0xFF;
*p++ = (bitc) & 0xFF;
for (i = len - 2; i--;)
*p++ = bignum_byte(bn, i);
return len;
}
/*
* Compare two bignums. Returns like strcmp.
*/
int bignum_cmp(Bignum a, Bignum b)
{
int amax = a[0], bmax = b[0];
int i = (amax > bmax ? amax : bmax);
while (i) {
BignumInt aval = (i > amax ? 0 : a[i]);
BignumInt bval = (i > bmax ? 0 : b[i]);
if (aval < bval)
return -1;
if (aval > bval)
return +1;
i--;
}
return 0;
}
/*
* Right-shift one bignum to form another.
*/
Bignum bignum_rshift(Bignum a, int shift)
{
Bignum ret;
int i, shiftw, shiftb, shiftbb, bits;
BignumInt ai, ai1;
bits = bignum_bitcount(a) - shift;
ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS);
if (ret) {
shiftw = shift / BIGNUM_INT_BITS;
shiftb = shift % BIGNUM_INT_BITS;
shiftbb = BIGNUM_INT_BITS - shiftb;
ai1 = a[shiftw + 1];
for (i = 1; i <= (int)ret[0]; i++) {
ai = ai1;
ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0);
ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK;
}
}
return ret;
}
/*
* Non-modular multiplication and addition.
*/
Bignum bigmuladd(Bignum a, Bignum b, Bignum addend)
{
int alen = a[0], blen = b[0];
int mlen = (alen > blen ? alen : blen);
int rlen, i, maxspot;
int wslen;
BignumInt *workspace;
Bignum ret;
/* mlen space for a, mlen space for b, 2*mlen for result,
* plus scratch space for multiplication */
wslen = mlen * 4 + mul_compute_scratch(mlen);
workspace = snewn(wslen, BignumInt);
for (i = 0; i < mlen; i++) {
workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0);
workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0);
}
internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,
workspace + 2 * mlen, mlen, workspace + 4 * mlen);
/* now just copy the result back */
rlen = alen + blen + 1;
if (addend && rlen <= (int)addend[0])
rlen = addend[0] + 1;
ret = newbn(rlen);
maxspot = 0;
for (i = 1; i <= (int)ret[0]; i++) {
ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0);
if (ret[i] != 0)
maxspot = i;
}
ret[0] = maxspot;
/* now add in the addend, if any */
if (addend) {
BignumDblInt carry = 0;
for (i = 1; i <= rlen; i++) {
carry += (i <= (int)ret[0] ? ret[i] : 0);
carry += (i <= (int)addend[0] ? addend[i] : 0);
ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
carry >>= BIGNUM_INT_BITS;
if (ret[i] != 0 && i > maxspot)
maxspot = i;
}
}
ret[0] = maxspot;
for (i = 0; i < wslen; i++)
workspace[i] = 0;
sfree(workspace);
return ret;
}
/*
* Non-modular multiplication.
*/
Bignum bigmul(Bignum a, Bignum b)
{
return bigmuladd(a, b, NULL);
}
/*
* Simple addition.
*/
Bignum bigadd(Bignum a, Bignum b)
{
int alen = a[0], blen = b[0];
int rlen = (alen > blen ? alen : blen) + 1;
int i, maxspot;
Bignum ret;
BignumDblInt carry;
ret = newbn(rlen);
carry = 0;
maxspot = 0;
for (i = 1; i <= rlen; i++) {
carry += (i <= (int)a[0] ? a[i] : 0);
carry += (i <= (int)b[0] ? b[i] : 0);
ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
carry >>= BIGNUM_INT_BITS;
if (ret[i] != 0 && i > maxspot)
maxspot = i;
}
ret[0] = maxspot;
return ret;
}
/*
* Subtraction. Returns a-b, or NULL if the result would come out
* negative (recall that this entire bignum module only handles
* positive numbers).
*/
Bignum bigsub(Bignum a, Bignum b)
{
int alen = a[0], blen = b[0];
int rlen = (alen > blen ? alen : blen);
int i, maxspot;
Bignum ret;
BignumDblInt carry;
ret = newbn(rlen);
carry = 1;
maxspot = 0;
for (i = 1; i <= rlen; i++) {
carry += (i <= (int)a[0] ? a[i] : 0);
carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK);
ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
carry >>= BIGNUM_INT_BITS;
if (ret[i] != 0 && i > maxspot)
maxspot = i;
}
ret[0] = maxspot;
if (!carry) {
freebn(ret);
return NULL;
}
return ret;
}
/*
* Create a bignum which is the bitmask covering another one. That
* is, the smallest integer which is >= N and is also one less than
* a power of two.
*/
Bignum bignum_bitmask(Bignum n)
{
Bignum ret = copybn(n);
int i;
BignumInt j;
i = ret[0];
while (n[i] == 0 && i > 0)
i--;
if (i <= 0)
return ret; /* input was zero */
j = 1;
while (j < n[i])
j = 2 * j + 1;
ret[i] = j;
while (--i > 0)
ret[i] = BIGNUM_INT_MASK;
return ret;
}
/*
* Convert a (max 32-bit) long into a bignum.
*/
Bignum bignum_from_long(unsigned long nn)
{
Bignum ret;
BignumDblInt n = nn;
ret = newbn(3);
ret[1] = (BignumInt)(n & BIGNUM_INT_MASK);
ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK);
ret[3] = 0;
ret[0] = (ret[2] ? 2 : 1);
return ret;
}
/*
* Add a long to a bignum.
*/
Bignum bignum_add_long(Bignum number, unsigned long addendx)
{
Bignum ret = newbn(number[0] + 1);
int i, maxspot = 0;
BignumDblInt carry = 0, addend = addendx;
for (i = 1; i <= (int)ret[0]; i++) {
carry += addend & BIGNUM_INT_MASK;
carry += (i <= (int)number[0] ? number[i] : 0);
addend >>= BIGNUM_INT_BITS;
ret[i] = (BignumInt) carry & BIGNUM_INT_MASK;
carry >>= BIGNUM_INT_BITS;
if (ret[i] != 0)
maxspot = i;
}
ret[0] = maxspot;
return ret;
}
/*
* Compute the residue of a bignum, modulo a (max 16-bit) short.
*/
unsigned short bignum_mod_short(Bignum number, unsigned short modulus)
{
BignumDblInt mod, r;
int i;
r = 0;
mod = modulus;
for (i = number[0]; i > 0; i--)
r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod;
return (unsigned short) r;
}
#ifdef DEBUG
void diagbn(char *prefix, Bignum md)
{
int i, nibbles, morenibbles;
static const char hex[] = "0123456789ABCDEF";
debug(("%s0x", prefix ? prefix : ""));
nibbles = (3 + bignum_bitcount(md)) / 4;
if (nibbles < 1)
nibbles = 1;
morenibbles = 4 * md[0] - nibbles;
for (i = 0; i < morenibbles; i++)
debug(("-"));
for (i = nibbles; i--;)
debug(("%c",
hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]));
if (prefix)
debug(("\n"));
}
#endif
/*
* Simple division.
*/
Bignum bigdiv(Bignum a, Bignum b)
{
Bignum q = newbn(a[0]);
bigdivmod(a, b, NULL, q);
return q;
}
/*
* Simple remainder.
*/
Bignum bigmod(Bignum a, Bignum b)
{
Bignum r = newbn(b[0]);
bigdivmod(a, b, r, NULL);
return r;
}
/*
* Greatest common divisor.
*/
Bignum biggcd(Bignum av, Bignum bv)
{
Bignum a = copybn(av);
Bignum b = copybn(bv);
while (bignum_cmp(b, Zero) != 0) {
Bignum t = newbn(b[0]);
bigdivmod(a, b, t, NULL);
while (t[0] > 1 && t[t[0]] == 0)
t[0]--;
freebn(a);
a = b;
b = t;
}
freebn(b);
return a;
}
/*
* Modular inverse, using Euclid's extended algorithm.
*/
Bignum modinv(Bignum number, Bignum modulus)
{
Bignum a = copybn(modulus);
Bignum b = copybn(number);
Bignum xp = copybn(Zero);
Bignum x = copybn(One);
int sign = +1;
while (bignum_cmp(b, One) != 0) {
Bignum t = newbn(b[0]);
Bignum q = newbn(a[0]);
bigdivmod(a, b, t, q);
while (t[0] > 1 && t[t[0]] == 0)
t[0]--;
freebn(a);
a = b;
b = t;
t = xp;
xp = x;
x = bigmuladd(q, xp, t);
sign = -sign;
freebn(t);
freebn(q);
}
freebn(b);
freebn(a);
freebn(xp);
/* now we know that sign * x == 1, and that x < modulus */
if (sign < 0) {
/* set a new x to be modulus - x */
Bignum newx = newbn(modulus[0]);
BignumInt carry = 0;
int maxspot = 1;
int i;
for (i = 1; i <= (int)newx[0]; i++) {
BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0);
BignumInt bword = (i <= (int)x[0] ? x[i] : 0);
newx[i] = aword - bword - carry;
bword = ~bword;
carry = carry ? (newx[i] >= bword) : (newx[i] > bword);
if (newx[i] != 0)
maxspot = i;
}
newx[0] = maxspot;
freebn(x);
x = newx;
}
/* and return. */
return x;
}
/*
* Render a bignum into decimal. Return a malloced string holding
* the decimal representation.
*/
char *bignum_decimal(Bignum x)
{
int ndigits, ndigit;
int i, iszero;
BignumDblInt carry;
char *ret;
BignumInt *workspace;
/*
* First, estimate the number of digits. Since log(10)/log(2)
* is just greater than 93/28 (the joys of continued fraction
* approximations...) we know that for every 93 bits, we need
* at most 28 digits. This will tell us how much to malloc.
*
* Formally: if x has i bits, that means x is strictly less
* than 2^i. Since 2 is less than 10^(28/93), this is less than
* 10^(28i/93). We need an integer power of ten, so we must
* round up (rounding down might make it less than x again).
* Therefore if we multiply the bit count by 28/93, rounding
* up, we will have enough digits.
*
* i=0 (i.e., x=0) is an irritating special case.
*/
i = bignum_bitcount(x);
if (!i)
ndigits = 1; /* x = 0 */
else
ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */
ndigits++; /* allow for trailing \0 */
ret = snewn(ndigits, char);
/*
* Now allocate some workspace to hold the binary form as we
* repeatedly divide it by ten. Initialise this to the
* big-endian form of the number.
*/
workspace = snewn(x[0], BignumInt);
for (i = 0; i < (int)x[0]; i++)
workspace[i] = x[x[0] - i];
/*
* Next, write the decimal number starting with the last digit.
* We use ordinary short division, dividing 10 into the
* workspace.
*/
ndigit = ndigits - 1;
ret[ndigit] = '\0';
do {
iszero = 1;
carry = 0;
for (i = 0; i < (int)x[0]; i++) {
carry = (carry << BIGNUM_INT_BITS) + workspace[i];
workspace[i] = (BignumInt) (carry / 10);
if (workspace[i])
iszero = 0;
carry %= 10;
}
ret[--ndigit] = (char) (carry + '0');
} while (!iszero);
/*
* There's a chance we've fallen short of the start of the
* string. Correct if so.
*/
if (ndigit > 0)
memmove(ret, ret + ndigit, ndigits - ndigit);
/*
* Done.
*/
sfree(workspace);
return ret;
}
#ifdef TESTBN
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
/*
* gcc -g -O0 -DTESTBN -o testbn sshbn.c misc.c -I unix -I charset
*
* Then feed to this program's standard input the output of
* testdata/bignum.py .
*/
void modalfatalbox(char *p, ...)
{
va_list ap;
fprintf(stderr, "FATAL ERROR: ");
va_start(ap, p);
vfprintf(stderr, p, ap);
va_end(ap);
fputc('\n', stderr);
exit(1);
}
#define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' )
int main(int argc, char **argv)
{
char *buf;
int line = 0;
int passes = 0, fails = 0;
while ((buf = fgetline(stdin)) != NULL) {
int maxlen = strlen(buf);
unsigned char *data = snewn(maxlen, unsigned char);
unsigned char *ptrs[5], *q;
int ptrnum;
char *bufp = buf;
line++;
q = data;
ptrnum = 0;
while (*bufp && !isspace((unsigned char)*bufp))
bufp++;
if (bufp)
*bufp++ = '\0';
while (*bufp) {
char *start, *end;
int i;
while (*bufp && !isxdigit((unsigned char)*bufp))
bufp++;
start = bufp;
if (!*bufp)
break;
while (*bufp && isxdigit((unsigned char)*bufp))
bufp++;
end = bufp;
if (ptrnum >= lenof(ptrs))
break;
ptrs[ptrnum++] = q;
for (i = -((end - start) & 1); i < end-start; i += 2) {
unsigned char val = (i < 0 ? 0 : fromxdigit(start[i]));
val = val * 16 + fromxdigit(start[i+1]);
*q++ = val;
}
ptrs[ptrnum] = q;
}
if (!strcmp(buf, "mul")) {
Bignum a, b, c, p;
if (ptrnum != 3) {
printf("%d: mul with %d parameters, expected 3\n", line);
exit(1);
}
a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);
b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);
c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);
p = bigmul(a, b);
if (bignum_cmp(c, p) == 0) {
passes++;
} else {
char *as = bignum_decimal(a);
char *bs = bignum_decimal(b);
char *cs = bignum_decimal(c);
char *ps = bignum_decimal(p);
printf("%d: fail: %s * %s gave %s expected %s\n",
line, as, bs, ps, cs);
fails++;
sfree(as);
sfree(bs);
sfree(cs);
sfree(ps);
}
freebn(a);
freebn(b);
freebn(c);
freebn(p);
} else if (!strcmp(buf, "pow")) {
Bignum base, expt, modulus, expected, answer;
if (ptrnum != 4) {
printf("%d: mul with %d parameters, expected 3\n", line);
exit(1);
}
base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);
expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);
modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);
expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]);
answer = modpow(base, expt, modulus);
if (bignum_cmp(expected, answer) == 0) {
passes++;
} else {
char *as = bignum_decimal(base);
char *bs = bignum_decimal(expt);
char *cs = bignum_decimal(modulus);
char *ds = bignum_decimal(answer);
char *ps = bignum_decimal(expected);
printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n",
line, as, bs, cs, ds, ps);
fails++;
sfree(as);
sfree(bs);
sfree(cs);
sfree(ds);
sfree(ps);
}
freebn(base);
freebn(expt);
freebn(modulus);
freebn(expected);
freebn(answer);
} else {
printf("%d: unrecognised test keyword: '%s'\n", line, buf);
exit(1);
}
sfree(buf);
sfree(data);
}
printf("passed %d failed %d total %d\n", passes, fails, passes+fails);
return fails != 0;
}
#endif