/* * Minimal code for RSA support from LibTomMath 0.41 * http://libtom.org/ * http://libtom.org/files/ltm-0.41.tar.bz2 * This library was released in public domain by Tom St Denis. * * The combination in this file may not use all of the optimized algorithms * from LibTomMath and may be considerable slower than the LibTomMath with its * default settings. The main purpose of having this version here is to make it * easier to build bignum.c wrapper without having to install and build an * external library. * * If CONFIG_INTERNAL_LIBTOMMATH is defined, bignum.c includes this * libtommath.c file instead of using the external LibTomMath library. */ #ifndef CHAR_BIT #define CHAR_BIT 8 #endif #define BN_MP_INVMOD_C #define BN_S_MP_EXPTMOD_C /* Note: #undef in tommath_superclass.h; this would * require BN_MP_EXPTMOD_FAST_C instead */ #define BN_S_MP_MUL_DIGS_C #define BN_MP_INVMOD_SLOW_C #define BN_S_MP_SQR_C #define BN_S_MP_MUL_HIGH_DIGS_C /* Note: #undef in tommath_superclass.h; this * would require other than mp_reduce */ #ifdef LTM_FAST /* Use faster div at the cost of about 1 kB */ #define BN_MP_MUL_D_C /* Include faster exptmod (Montgomery) at the cost of about 2.5 kB in code */ #define BN_MP_EXPTMOD_FAST_C #define BN_MP_MONTGOMERY_SETUP_C #define BN_FAST_MP_MONTGOMERY_REDUCE_C #define BN_MP_MONTGOMERY_CALC_NORMALIZATION_C #define BN_MP_MUL_2_C /* Include faster sqr at the cost of about 0.5 kB in code */ #define BN_FAST_S_MP_SQR_C /* About 0.25 kB of code, but ~1.7kB of stack space! */ #define BN_FAST_S_MP_MUL_DIGS_C #else /* LTM_FAST */ #define BN_MP_DIV_SMALL #define BN_MP_INIT_MULTI_C #define BN_MP_CLEAR_MULTI_C #define BN_MP_ABS_C #endif /* LTM_FAST */ /* Current uses do not require support for negative exponent in exptmod, so we * can save about 1.5 kB in leaving out invmod. */ #define LTM_NO_NEG_EXP /* from tommath.h */ #define OPT_CAST(x) #ifdef __x86_64__ typedef unsigned long mp_digit; typedef unsigned long mp_word __attribute__((mode(TI))); #define DIGIT_BIT 60 #define MP_64BIT #else typedef unsigned long mp_digit; typedef u64 mp_word; #define DIGIT_BIT 28 #define MP_28BIT #endif #define XMALLOC os_malloc #define XFREE os_free #define XREALLOC os_realloc #define MP_MASK ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1)) #define MP_LT -1 /* less than */ #define MP_EQ 0 /* equal to */ #define MP_GT 1 /* greater than */ #define MP_ZPOS 0 /* positive integer */ #define MP_NEG 1 /* negative */ #define MP_OKAY 0 /* ok result */ #define MP_MEM -2 /* out of mem */ #define MP_VAL -3 /* invalid input */ #define MP_YES 1 /* yes response */ #define MP_NO 0 /* no response */ typedef int mp_err; /* define this to use lower memory usage routines (exptmods mostly) */ #define MP_LOW_MEM /* default precision */ #ifndef MP_PREC #ifndef MP_LOW_MEM #define MP_PREC 32 /* default digits of precision */ #else #define MP_PREC 8 /* default digits of precision */ #endif #endif /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */ #define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1)) /* the infamous mp_int structure */ typedef struct { int used, alloc, sign; mp_digit *dp; } mp_int; /* ---> Basic Manipulations <--- */ #define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO) #define mp_iseven(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO) #define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO) /* prototypes for copied functions */ #define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) static int s_mp_exptmod(mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode); static int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs); static int s_mp_sqr(mp_int * a, mp_int * b); static int s_mp_mul_high_digs(mp_int * a, mp_int * b, mp_int * c, int digs); #ifdef BN_FAST_S_MP_MUL_DIGS_C static int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs); #endif #ifdef BN_MP_INIT_MULTI_C static int mp_init_multi(mp_int *mp, ...); #endif #ifdef BN_MP_CLEAR_MULTI_C static void mp_clear_multi(mp_int *mp, ...); #endif static int mp_lshd(mp_int * a, int b); static void mp_set(mp_int * a, mp_digit b); static void mp_clamp(mp_int * a); static void mp_exch(mp_int * a, mp_int * b); static void mp_rshd(mp_int * a, int b); static void mp_zero(mp_int * a); static int mp_mod_2d(mp_int * a, int b, mp_int * c); static int mp_div_2d(mp_int * a, int b, mp_int * c, mp_int * d); static int mp_init_copy(mp_int * a, mp_int * b); static int mp_mul_2d(mp_int * a, int b, mp_int * c); #ifndef LTM_NO_NEG_EXP static int mp_div_2(mp_int * a, mp_int * b); static int mp_invmod(mp_int * a, mp_int * b, mp_int * c); static int mp_invmod_slow(mp_int * a, mp_int * b, mp_int * c); #endif /* LTM_NO_NEG_EXP */ static int mp_copy(mp_int * a, mp_int * b); static int mp_count_bits(mp_int * a); static int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d); static int mp_mod(mp_int * a, mp_int * b, mp_int * c); static int mp_grow(mp_int * a, int size); static int mp_cmp_mag(mp_int * a, mp_int * b); #ifdef BN_MP_ABS_C static int mp_abs(mp_int * a, mp_int * b); #endif static int mp_sqr(mp_int * a, mp_int * b); static int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d); static int mp_reduce_2k_setup_l(mp_int *a, mp_int *d); static int mp_2expt(mp_int * a, int b); static int mp_reduce_setup(mp_int * a, mp_int * b); static int mp_reduce(mp_int * x, mp_int * m, mp_int * mu); static int mp_init_size(mp_int * a, int size); #ifdef BN_MP_EXPTMOD_FAST_C static int mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode); #endif /* BN_MP_EXPTMOD_FAST_C */ #ifdef BN_FAST_S_MP_SQR_C static int fast_s_mp_sqr (mp_int * a, mp_int * b); #endif /* BN_FAST_S_MP_SQR_C */ #ifdef BN_MP_MUL_D_C static int mp_mul_d (mp_int * a, mp_digit b, mp_int * c); #endif /* BN_MP_MUL_D_C */ /* functions from bn_.c */ /* reverse an array, used for radix code */ static void bn_reverse (unsigned char *s, int len) { int ix, iy; unsigned char t; ix = 0; iy = len - 1; while (ix < iy) { t = s[ix]; s[ix] = s[iy]; s[iy] = t; ++ix; --iy; } } /* low level addition, based on HAC pp.594, Algorithm 14.7 */ static int s_mp_add (mp_int * a, mp_int * b, mp_int * c) { mp_int *x; int olduse, res, min, max; /* find sizes, we let |a| <= |b| which means we have to sort * them. "x" will point to the input with the most digits */ if (a->used > b->used) { min = b->used; max = a->used; x = a; } else { min = a->used; max = b->used; x = b; } /* init result */ if (c->alloc < max + 1) { if ((res = mp_grow (c, max + 1)) != MP_OKAY) { return res; } } /* get old used digit count and set new one */ olduse = c->used; c->used = max + 1; { register mp_digit u, *tmpa, *tmpb, *tmpc; register int i; /* alias for digit pointers */ /* first input */ tmpa = a->dp; /* second input */ tmpb = b->dp; /* destination */ tmpc = c->dp; /* zero the carry */ u = 0; for (i = 0; i < min; i++) { /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ *tmpc = *tmpa++ + *tmpb++ + u; /* U = carry bit of T[i] */ u = *tmpc >> ((mp_digit)DIGIT_BIT); /* take away carry bit from T[i] */ *tmpc++ &= MP_MASK; } /* now copy higher words if any, that is in A+B * if A or B has more digits add those in */ if (min != max) { for (; i < max; i++) { /* T[i] = X[i] + U */ *tmpc = x->dp[i] + u; /* U = carry bit of T[i] */ u = *tmpc >> ((mp_digit)DIGIT_BIT); /* take away carry bit from T[i] */ *tmpc++ &= MP_MASK; } } /* add carry */ *tmpc++ = u; /* clear digits above oldused */ for (i = c->used; i < olduse; i++) { *tmpc++ = 0; } } mp_clamp (c); return MP_OKAY; } /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ static int s_mp_sub (mp_int * a, mp_int * b, mp_int * c) { int olduse, res, min, max; /* find sizes */ min = b->used; max = a->used; /* init result */ if (c->alloc < max) { if ((res = mp_grow (c, max)) != MP_OKAY) { return res; } } olduse = c->used; c->used = max; { register mp_digit u, *tmpa, *tmpb, *tmpc; register int i; /* alias for digit pointers */ tmpa = a->dp; tmpb = b->dp; tmpc = c->dp; /* set carry to zero */ u = 0; for (i = 0; i < min; i++) { /* T[i] = A[i] - B[i] - U */ *tmpc = *tmpa++ - *tmpb++ - u; /* U = carry bit of T[i] * Note this saves performing an AND operation since * if a carry does occur it will propagate all the way to the * MSB. As a result a single shift is enough to get the carry */ u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); /* Clear carry from T[i] */ *tmpc++ &= MP_MASK; } /* now copy higher words if any, e.g. if A has more digits than B */ for (; i < max; i++) { /* T[i] = A[i] - U */ *tmpc = *tmpa++ - u; /* U = carry bit of T[i] */ u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); /* Clear carry from T[i] */ *tmpc++ &= MP_MASK; } /* clear digits above used (since we may not have grown result above) */ for (i = c->used; i < olduse; i++) { *tmpc++ = 0; } } mp_clamp (c); return MP_OKAY; } /* init a new mp_int */ static int mp_init (mp_int * a) { int i; /* allocate memory required and clear it */ a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); if (a->dp == NULL) { return MP_MEM; } /* set the digits to zero */ for (i = 0; i < MP_PREC; i++) { a->dp[i] = 0; } /* set the used to zero, allocated digits to the default precision * and sign to positive */ a->used = 0; a->alloc = MP_PREC; a->sign = MP_ZPOS; return MP_OKAY; } /* clear one (frees) */ static void mp_clear (mp_int * a) { int i; /* only do anything if a hasn't been freed previously */ if (a->dp != NULL) { /* first zero the digits */ for (i = 0; i < a->used; i++) { a->dp[i] = 0; } /* free ram */ XFREE(a->dp); /* reset members to make debugging easier */ a->dp = NULL; a->alloc = a->used = 0; a->sign = MP_ZPOS; } } /* high level addition (handles signs) */ static int mp_add (mp_int * a, mp_int * b, mp_int * c) { int sa, sb, res; /* get sign of both inputs */ sa = a->sign; sb = b->sign; /* handle two cases, not four */ if (sa == sb) { /* both positive or both negative */ /* add their magnitudes, copy the sign */ c->sign = sa; res = s_mp_add (a, b, c); } else { /* one positive, the other negative */ /* subtract the one with the greater magnitude from */ /* the one of the lesser magnitude. The result gets */ /* the sign of the one with the greater magnitude. */ if (mp_cmp_mag (a, b) == MP_LT) { c->sign = sb; res = s_mp_sub (b, a, c); } else { c->sign = sa; res = s_mp_sub (a, b, c); } } return res; } /* high level subtraction (handles signs) */ static int mp_sub (mp_int * a, mp_int * b, mp_int * c) { int sa, sb, res; sa = a->sign; sb = b->sign; if (sa != sb) { /* subtract a negative from a positive, OR */ /* subtract a positive from a negative. */ /* In either case, ADD their magnitudes, */ /* and use the sign of the first number. */ c->sign = sa; res = s_mp_add (a, b, c); } else { /* subtract a positive from a positive, OR */ /* subtract a negative from a negative. */ /* First, take the difference between their */ /* magnitudes, then... */ if (mp_cmp_mag (a, b) != MP_LT) { /* Copy the sign from the first */ c->sign = sa; /* The first has a larger or equal magnitude */ res = s_mp_sub (a, b, c); } else { /* The result has the *opposite* sign from */ /* the first number. */ c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; /* The second has a larger magnitude */ res = s_mp_sub (b, a, c); } } return res; } /* high level multiplication (handles sign) */ static int mp_mul (mp_int * a, mp_int * b, mp_int * c) { int res, neg; neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; /* use Toom-Cook? */ #ifdef BN_MP_TOOM_MUL_C if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) { res = mp_toom_mul(a, b, c); } else #endif #ifdef BN_MP_KARATSUBA_MUL_C /* use Karatsuba? */ if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) { res = mp_karatsuba_mul (a, b, c); } else #endif { /* can we use the fast multiplier? * * The fast multiplier can be used if the output will * have less than MP_WARRAY digits and the number of * digits won't affect carry propagation */ #ifdef BN_FAST_S_MP_MUL_DIGS_C int digs = a->used + b->used + 1; if ((digs < MP_WARRAY) && MIN(a->used, b->used) <= (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { res = fast_s_mp_mul_digs (a, b, c, digs); } else #endif #ifdef BN_S_MP_MUL_DIGS_C res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ #else #error mp_mul could fail res = MP_VAL; #endif } c->sign = (c->used > 0) ? neg : MP_ZPOS; return res; } /* d = a * b (mod c) */ static int mp_mulmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) { int res; mp_int t; if ((res = mp_init (&t)) != MP_OKAY) { return res; } if ((res = mp_mul (a, b, &t)) != MP_OKAY) { mp_clear (&t); return res; } res = mp_mod (&t, c, d); mp_clear (&t); return res; } /* c = a mod b, 0 <= c < b */ static int mp_mod (mp_int * a, mp_int * b, mp_int * c) { mp_int t; int res; if ((res = mp_init (&t)) != MP_OKAY) { return res; } if ((res = mp_div (a, b, NULL, &t)) != MP_OKAY) { mp_clear (&t); return res; } if (t.sign != b->sign) { res = mp_add (b, &t, c); } else { res = MP_OKAY; mp_exch (&t, c); } mp_clear (&t); return res; } /* this is a shell function that calls either the normal or Montgomery * exptmod functions. Originally the call to the montgomery code was * embedded in the normal function but that wasted a lot of stack space * for nothing (since 99% of the time the Montgomery code would be called) */ static int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) { int dr; /* modulus P must be positive */ if (P->sign == MP_NEG) { return MP_VAL; } /* if exponent X is negative we have to recurse */ if (X->sign == MP_NEG) { #ifdef LTM_NO_NEG_EXP return MP_VAL; #else /* LTM_NO_NEG_EXP */ #ifdef BN_MP_INVMOD_C mp_int tmpG, tmpX; int err; /* first compute 1/G mod P */ if ((err = mp_init(&tmpG)) != MP_OKAY) { return err; } if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { mp_clear(&tmpG); return err; } /* now get |X| */ if ((err = mp_init(&tmpX)) != MP_OKAY) { mp_clear(&tmpG); return err; } if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { mp_clear_multi(&tmpG, &tmpX, NULL); return err; } /* and now compute (1/G)**|X| instead of G**X [X < 0] */ err = mp_exptmod(&tmpG, &tmpX, P, Y); mp_clear_multi(&tmpG, &tmpX, NULL); return err; #else #error mp_exptmod would always fail /* no invmod */ return MP_VAL; #endif #endif /* LTM_NO_NEG_EXP */ } /* modified diminished radix reduction */ #if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defined(BN_S_MP_EXPTMOD_C) if (mp_reduce_is_2k_l(P) == MP_YES) { return s_mp_exptmod(G, X, P, Y, 1); } #endif #ifdef BN_MP_DR_IS_MODULUS_C /* is it a DR modulus? */ dr = mp_dr_is_modulus(P); #else /* default to no */ dr = 0; #endif #ifdef BN_MP_REDUCE_IS_2K_C /* if not, is it a unrestricted DR modulus? */ if (dr == 0) { dr = mp_reduce_is_2k(P) << 1; } #endif /* if the modulus is odd or dr != 0 use the montgomery method */ #ifdef BN_MP_EXPTMOD_FAST_C if (mp_isodd (P) == 1 || dr != 0) { return mp_exptmod_fast (G, X, P, Y, dr); } else { #endif #ifdef BN_S_MP_EXPTMOD_C /* otherwise use the generic Barrett reduction technique */ return s_mp_exptmod (G, X, P, Y, 0); #else #error mp_exptmod could fail /* no exptmod for evens */ return MP_VAL; #endif #ifdef BN_MP_EXPTMOD_FAST_C } #endif if (dr == 0) { /* avoid compiler warnings about possibly unused variable */ } } /* compare two ints (signed)*/ static int mp_cmp (mp_int * a, mp_int * b) { /* compare based on sign */ if (a->sign != b->sign) { if (a->sign == MP_NEG) { return MP_LT; } else { return MP_GT; } } /* compare digits */ if (a->sign == MP_NEG) { /* if negative compare opposite direction */ return mp_cmp_mag(b, a); } else { return mp_cmp_mag(a, b); } } /* compare a digit */ static int mp_cmp_d(mp_int * a, mp_digit b) { /* compare based on sign */ if (a->sign == MP_NEG) { return MP_LT; } /* compare based on magnitude */ if (a->used > 1) { return MP_GT; } /* compare the only digit of a to b */ if (a->dp[0] > b) { return MP_GT; } else if (a->dp[0] < b) { return MP_LT; } else { return MP_EQ; } } #ifndef LTM_NO_NEG_EXP /* hac 14.61, pp608 */ static int mp_invmod (mp_int * a, mp_int * b, mp_int * c) { /* b cannot be negative */ if (b->sign == MP_NEG || mp_iszero(b) == 1) { return MP_VAL; } #ifdef BN_FAST_MP_INVMOD_C /* if the modulus is odd we can use a faster routine instead */ if (mp_isodd (b) == 1) { return fast_mp_invmod (a, b, c); } #endif #ifdef BN_MP_INVMOD_SLOW_C return mp_invmod_slow(a, b, c); #endif #ifndef BN_FAST_MP_INVMOD_C #ifndef BN_MP_INVMOD_SLOW_C #error mp_invmod would always fail #endif #endif return MP_VAL; } #endif /* LTM_NO_NEG_EXP */ /* get the size for an unsigned equivalent */ static int mp_unsigned_bin_size (mp_int * a) { int size = mp_count_bits (a); return (size / 8 + ((size & 7) != 0 ? 1 : 0)); } #ifndef LTM_NO_NEG_EXP /* hac 14.61, pp608 */ static int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c) { mp_int x, y, u, v, A, B, C, D; int res; /* b cannot be negative */ if (b->sign == MP_NEG || mp_iszero(b) == 1) { return MP_VAL; } /* init temps */ if ((res = mp_init_multi(&x, &y, &u, &v, &A, &B, &C, &D, NULL)) != MP_OKAY) { return res; } /* x = a, y = b */ if ((res = mp_mod(a, b, &x)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_copy (b, &y)) != MP_OKAY) { goto LBL_ERR; } /* 2. [modified] if x,y are both even then return an error! */ if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) { res = MP_VAL; goto LBL_ERR; } /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ if ((res = mp_copy (&x, &u)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_copy (&y, &v)) != MP_OKAY) { goto LBL_ERR; } mp_set (&A, 1); mp_set (&D, 1); top: /* 4. while u is even do */ while (mp_iseven (&u) == 1) { /* 4.1 u = u/2 */ if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { goto LBL_ERR; } /* 4.2 if A or B is odd then */ if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) { /* A = (A+y)/2, B = (B-x)/2 */ if ((res = mp_add (&A, &y, &A)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { goto LBL_ERR; } } /* A = A/2, B = B/2 */ if ((res = mp_div_2 (&A, &A)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { goto LBL_ERR; } } /* 5. while v is even do */ while (mp_iseven (&v) == 1) { /* 5.1 v = v/2 */ if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { goto LBL_ERR; } /* 5.2 if C or D is odd then */ if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) { /* C = (C+y)/2, D = (D-x)/2 */ if ((res = mp_add (&C, &y, &C)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { goto LBL_ERR; } } /* C = C/2, D = D/2 */ if ((res = mp_div_2 (&C, &C)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { goto LBL_ERR; } } /* 6. if u >= v then */ if (mp_cmp (&u, &v) != MP_LT) { /* u = u - v, A = A - C, B = B - D */ if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { goto LBL_ERR; } } else { /* v - v - u, C = C - A, D = D - B */ if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { goto LBL_ERR; } } /* if not zero goto step 4 */ if (mp_iszero (&u) == 0) goto top; /* now a = C, b = D, gcd == g*v */ /* if v != 1 then there is no inverse */ if (mp_cmp_d (&v, 1) != MP_EQ) { res = MP_VAL; goto LBL_ERR; } /* if its too low */ while (mp_cmp_d(&C, 0) == MP_LT) { if ((res = mp_add(&C, b, &C)) != MP_OKAY) { goto LBL_ERR; } } /* too big */ while (mp_cmp_mag(&C, b) != MP_LT) { if ((res = mp_sub(&C, b, &C)) != MP_OKAY) { goto LBL_ERR; } } /* C is now the inverse */ mp_exch (&C, c); res = MP_OKAY; LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL); return res; } #endif /* LTM_NO_NEG_EXP */ /* compare maginitude of two ints (unsigned) */ static int mp_cmp_mag (mp_int * a, mp_int * b) { int n; mp_digit *tmpa, *tmpb; /* compare based on # of non-zero digits */ if (a->used > b->used) { return MP_GT; } if (a->used < b->used) { return MP_LT; } /* alias for a */ tmpa = a->dp + (a->used - 1); /* alias for b */ tmpb = b->dp + (a->used - 1); /* compare based on digits */ for (n = 0; n < a->used; ++n, --tmpa, --tmpb) { if (*tmpa > *tmpb) { return MP_GT; } if (*tmpa < *tmpb) { return MP_LT; } } return MP_EQ; } /* reads a unsigned char array, assumes the msb is stored first [big endian] */ static int mp_read_unsigned_bin (mp_int * a, const unsigned char *b, int c) { int res; /* make sure there are at least two digits */ if (a->alloc < 2) { if ((res = mp_grow(a, 2)) != MP_OKAY) { return res; } } /* zero the int */ mp_zero (a); /* read the bytes in */ while (c-- > 0) { if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) { return res; } #ifndef MP_8BIT a->dp[0] |= *b++; a->used += 1; #else a->dp[0] = (*b & MP_MASK); a->dp[1] |= ((*b++ >> 7U) & 1); a->used += 2; #endif } mp_clamp (a); return MP_OKAY; } /* store in unsigned [big endian] format */ static int mp_to_unsigned_bin (mp_int * a, unsigned char *b) { int x, res; mp_int t; if ((res = mp_init_copy (&t, a)) != MP_OKAY) { return res; } x = 0; while (mp_iszero (&t) == 0) { #ifndef MP_8BIT b[x++] = (unsigned char) (t.dp[0] & 255); #else b[x++] = (unsigned char) (t.dp[0] | ((t.dp[1] & 0x01) << 7)); #endif if ((res = mp_div_2d (&t, 8, &t, NULL)) != MP_OKAY) { mp_clear (&t); return res; } } bn_reverse (b, x); mp_clear (&t); return MP_OKAY; } /* shift right by a certain bit count (store quotient in c, optional remainder in d) */ static int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) { mp_digit D, r, rr; int x, res; mp_int t; /* if the shift count is <= 0 then we do no work */ if (b <= 0) { res = mp_copy (a, c); if (d != NULL) { mp_zero (d); } return res; } if ((res = mp_init (&t)) != MP_OKAY) { return res; } /* get the remainder */ if (d != NULL) { if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) { mp_clear (&t); return res; } } /* copy */ if ((res = mp_copy (a, c)) != MP_OKAY) { mp_clear (&t); return res; } /* shift by as many digits in the bit count */ if (b >= (int)DIGIT_BIT) { mp_rshd (c, b / DIGIT_BIT); } /* shift any bit count < DIGIT_BIT */ D = (mp_digit) (b % DIGIT_BIT); if (D != 0) { register mp_digit *tmpc, mask, shift; /* mask */ mask = (((mp_digit)1) << D) - 1; /* shift for lsb */ shift = DIGIT_BIT - D; /* alias */ tmpc = c->dp + (c->used - 1); /* carry */ r = 0; for (x = c->used - 1; x >= 0; x--) { /* get the lower bits of this word in a temp */ rr = *tmpc & mask; /* shift the current word and mix in the carry bits from the previous word */ *tmpc = (*tmpc >> D) | (r << shift); --tmpc; /* set the carry to the carry bits of the current word found above */ r = rr; } } mp_clamp (c); if (d != NULL) { mp_exch (&t, d); } mp_clear (&t); return MP_OKAY; } static int mp_init_copy (mp_int * a, mp_int * b) { int res; if ((res = mp_init (a)) != MP_OKAY) { return res; } return mp_copy (b, a); } /* set to zero */ static void mp_zero (mp_int * a) { int n; mp_digit *tmp; a->sign = MP_ZPOS; a->used = 0; tmp = a->dp; for (n = 0; n < a->alloc; n++) { *tmp++ = 0; } } /* copy, b = a */ static int mp_copy (mp_int * a, mp_int * b) { int res, n; /* if dst == src do nothing */ if (a == b) { return MP_OKAY; } /* grow dest */ if (b->alloc < a->used) { if ((res = mp_grow (b, a->used)) != MP_OKAY) { return res; } } /* zero b and copy the parameters over */ { register mp_digit *tmpa, *tmpb; /* pointer aliases */ /* source */ tmpa = a->dp; /* destination */ tmpb = b->dp; /* copy all the digits */ for (n = 0; n < a->used; n++) { *tmpb++ = *tmpa++; } /* clear high digits */ for (; n < b->used; n++) { *tmpb++ = 0; } } /* copy used count and sign */ b->used = a->used; b->sign = a->sign; return MP_OKAY; } /* shift right a certain amount of digits */ static void mp_rshd (mp_int * a, int b) { int x; /* if b <= 0 then ignore it */ if (b <= 0) { return; } /* if b > used then simply zero it and return */ if (a->used <= b) { mp_zero (a); return; } { register mp_digit *bottom, *top; /* shift the digits down */ /* bottom */ bottom = a->dp; /* top [offset into digits] */ top = a->dp + b; /* this is implemented as a sliding window where * the window is b-digits long and digits from * the top of the window are copied to the bottom * * e.g. b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> /\ | ----> \-------------------/ ----> */ for (x = 0; x < (a->used - b); x++) { *bottom++ = *top++; } /* zero the top digits */ for (; x < a->used; x++) { *bottom++ = 0; } } /* remove excess digits */ a->used -= b; } /* swap the elements of two integers, for cases where you can't simply swap the * mp_int pointers around */ static void mp_exch (mp_int * a, mp_int * b) { mp_int t; t = *a; *a = *b; *b = t; } /* trim unused digits * * This is used to ensure that leading zero digits are * trimed and the leading "used" digit will be non-zero * Typically very fast. Also fixes the sign if there * are no more leading digits */ static void mp_clamp (mp_int * a) { /* decrease used while the most significant digit is * zero. */ while (a->used > 0 && a->dp[a->used - 1] == 0) { --(a->used); } /* reset the sign flag if used == 0 */ if (a->used == 0) { a->sign = MP_ZPOS; } } /* grow as required */ static int mp_grow (mp_int * a, int size) { int i; mp_digit *tmp; /* if the alloc size is smaller alloc more ram */ if (a->alloc < size) { /* ensure there are always at least MP_PREC digits extra on top */ size += (MP_PREC * 2) - (size % MP_PREC); /* reallocate the array a->dp * * We store the return in a temporary variable * in case the operation failed we don't want * to overwrite the dp member of a. */ tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); if (tmp == NULL) { /* reallocation failed but "a" is still valid [can be freed] */ return MP_MEM; } /* reallocation succeeded so set a->dp */ a->dp = tmp; /* zero excess digits */ i = a->alloc; a->alloc = size; for (; i < a->alloc; i++) { a->dp[i] = 0; } } return MP_OKAY; } #ifdef BN_MP_ABS_C /* b = |a| * * Simple function copies the input and fixes the sign to positive */ static int mp_abs (mp_int * a, mp_int * b) { int res; /* copy a to b */ if (a != b) { if ((res = mp_copy (a, b)) != MP_OKAY) { return res; } } /* force the sign of b to positive */ b->sign = MP_ZPOS; return MP_OKAY; } #endif /* set to a digit */ static void mp_set (mp_int * a, mp_digit b) { mp_zero (a); a->dp[0] = b & MP_MASK; a->used = (a->dp[0] != 0) ? 1 : 0; } #ifndef LTM_NO_NEG_EXP /* b = a/2 */ static int mp_div_2(mp_int * a, mp_int * b) { int x, res, oldused; /* copy */ if (b->alloc < a->used) { if ((res = mp_grow (b, a->used)) != MP_OKAY) { return res; } } oldused = b->used; b->used = a->used; { register mp_digit r, rr, *tmpa, *tmpb; /* source alias */ tmpa = a->dp + b->used - 1; /* dest alias */ tmpb = b->dp + b->used - 1; /* carry */ r = 0; for (x = b->used - 1; x >= 0; x--) { /* get the carry for the next iteration */ rr = *tmpa & 1; /* shift the current digit, add in carry and store */ *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); /* forward carry to next iteration */ r = rr; } /* zero excess digits */ tmpb = b->dp + b->used; for (x = b->used; x < oldused; x++) { *tmpb++ = 0; } } b->sign = a->sign; mp_clamp (b); return MP_OKAY; } #endif /* LTM_NO_NEG_EXP */ /* shift left by a certain bit count */ static int mp_mul_2d (mp_int * a, int b, mp_int * c) { mp_digit d; int res; /* copy */ if (a != c) { if ((res = mp_copy (a, c)) != MP_OKAY) { return res; } } if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) { if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) { return res; } } /* shift by as many digits in the bit count */ if (b >= (int)DIGIT_BIT) { if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) { return res; } } /* shift any bit count < DIGIT_BIT */ d = (mp_digit) (b % DIGIT_BIT); if (d != 0) { register mp_digit *tmpc, shift, mask, r, rr; register int x; /* bitmask for carries */ mask = (((mp_digit)1) << d) - 1; /* shift for msbs */ shift = DIGIT_BIT - d; /* alias */ tmpc = c->dp; /* carry */ r = 0; for (x = 0; x < c->used; x++) { /* get the higher bits of the current word */ rr = (*tmpc >> shift) & mask; /* shift the current word and OR in the carry */ *tmpc = ((*tmpc << d) | r) & MP_MASK; ++tmpc; /* set the carry to the carry bits of the current word */ r = rr; } /* set final carry */ if (r != 0) { c->dp[(c->used)++] = r; } } mp_clamp (c); return MP_OKAY; } #ifdef BN_MP_INIT_MULTI_C static int mp_init_multi(mp_int *mp, ...) { mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ int n = 0; /* Number of ok inits */ mp_int* cur_arg = mp; va_list args; va_start(args, mp); /* init args to next argument from caller */ while (cur_arg != NULL) { if (mp_init(cur_arg) != MP_OKAY) { /* Oops - error! Back-track and mp_clear what we already succeeded in init-ing, then return error. */ va_list clean_args; /* end the current list */ va_end(args); /* now start cleaning up */ cur_arg = mp; va_start(clean_args, mp); while (n--) { mp_clear(cur_arg); cur_arg = va_arg(clean_args, mp_int*); } va_end(clean_args); return MP_MEM; } n++; cur_arg = va_arg(args, mp_int*); } va_end(args); return res; /* Assumed ok, if error flagged above. */ } #endif #ifdef BN_MP_CLEAR_MULTI_C static void mp_clear_multi(mp_int *mp, ...) { mp_int* next_mp = mp; va_list args; va_start(args, mp); while (next_mp != NULL) { mp_clear(next_mp); next_mp = va_arg(args, mp_int*); } va_end(args); } #endif /* shift left a certain amount of digits */ static int mp_lshd (mp_int * a, int b) { int x, res; /* if its less than zero return */ if (b <= 0) { return MP_OKAY; } /* grow to fit the new digits */ if (a->alloc < a->used + b) { if ((res = mp_grow (a, a->used + b)) != MP_OKAY) { return res; } } { register mp_digit *top, *bottom; /* increment the used by the shift amount then copy upwards */ a->used += b; /* top */ top = a->dp + a->used - 1; /* base */ bottom = a->dp + a->used - 1 - b; /* much like mp_rshd this is implemented using a sliding window * except the window goes the otherway around. Copying from * the bottom to the top. see bn_mp_rshd.c for more info. */ for (x = a->used - 1; x >= b; x--) { *top-- = *bottom--; } /* zero the lower digits */ top = a->dp; for (x = 0; x < b; x++) { *top++ = 0; } } return MP_OKAY; } /* returns the number of bits in an int */ static int mp_count_bits (mp_int * a) { int r; mp_digit q; /* shortcut */ if (a->used == 0) { return 0; } /* get number of digits and add that */ r = (a->used - 1) * DIGIT_BIT; /* take the last digit and count the bits in it */ q = a->dp[a->used - 1]; while (q > ((mp_digit) 0)) { ++r; q >>= ((mp_digit) 1); } return r; } /* calc a value mod 2**b */ static int mp_mod_2d (mp_int * a, int b, mp_int * c) { int x, res; /* if b is <= 0 then zero the int */ if (b <= 0) { mp_zero (c); return MP_OKAY; } /* if the modulus is larger than the value than return */ if (b >= (int) (a->used * DIGIT_BIT)) { res = mp_copy (a, c); return res; } /* copy */ if ((res = mp_copy (a, c)) != MP_OKAY) { return res; } /* zero digits above the last digit of the modulus */ for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x++) { c->dp[x] = 0; } /* clear the digit that is not completely outside/inside the modulus */ c->dp[b / DIGIT_BIT] &= (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1)); mp_clamp (c); return MP_OKAY; } #ifdef BN_MP_DIV_SMALL /* slower bit-bang division... also smaller */ static int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) { mp_int ta, tb, tq, q; int res, n, n2; /* is divisor zero ? */ if (mp_iszero (b) == 1) { return MP_VAL; } /* if a < b then q=0, r = a */ if (mp_cmp_mag (a, b) == MP_LT) { if (d != NULL) { res = mp_copy (a, d); } else { res = MP_OKAY; } if (c != NULL) { mp_zero (c); } return res; } /* init our temps */ if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL)) != MP_OKAY) { return res; } mp_set(&tq, 1); n = mp_count_bits(a) - mp_count_bits(b); if (((res = mp_abs(a, &ta)) != MP_OKAY) || ((res = mp_abs(b, &tb)) != MP_OKAY) || ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) { goto LBL_ERR; } while (n-- >= 0) { if (mp_cmp(&tb, &ta) != MP_GT) { if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) { goto LBL_ERR; } } if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) { goto LBL_ERR; } } /* now q == quotient and ta == remainder */ n = a->sign; n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG); if (c != NULL) { mp_exch(c, &q); c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; } if (d != NULL) { mp_exch(d, &ta); d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; } LBL_ERR: mp_clear_multi(&ta, &tb, &tq, &q, NULL); return res; } #else /* integer signed division. * c*b + d == a [e.g. a/b, c=quotient, d=remainder] * HAC pp.598 Algorithm 14.20 * * Note that the description in HAC is horribly * incomplete. For example, it doesn't consider * the case where digits are removed from 'x' in * the inner loop. It also doesn't consider the * case that y has fewer than three digits, etc.. * * The overall algorithm is as described as * 14.20 from HAC but fixed to treat these cases. */ static int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) { mp_int q, x, y, t1, t2; int res, n, t, i, norm, neg; /* is divisor zero ? */ if (mp_iszero (b) == 1) { return MP_VAL; } /* if a < b then q=0, r = a */ if (mp_cmp_mag (a, b) == MP_LT) { if (d != NULL) { res = mp_copy (a, d); } else { res = MP_OKAY; } if (c != NULL) { mp_zero (c); } return res; } if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) { return res; } q.used = a->used + 2; if ((res = mp_init (&t1)) != MP_OKAY) { goto LBL_Q; } if ((res = mp_init (&t2)) != MP_OKAY) { goto LBL_T1; } if ((res = mp_init_copy (&x, a)) != MP_OKAY) { goto LBL_T2; } if ((res = mp_init_copy (&y, b)) != MP_OKAY) { goto LBL_X; } /* fix the sign */ neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; x.sign = y.sign = MP_ZPOS; /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ norm = mp_count_bits(&y) % DIGIT_BIT; if (norm < (int)(DIGIT_BIT-1)) { norm = (DIGIT_BIT-1) - norm; if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) { goto LBL_Y; } if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) { goto LBL_Y; } } else { norm = 0; } /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ n = x.used - 1; t = y.used - 1; /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */ if ((res = mp_lshd (&y, n - t)) != MP_OKAY) { /* y = y*b**{n-t} */ goto LBL_Y; } while (mp_cmp (&x, &y) != MP_LT) { ++(q.dp[n - t]); if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) { goto LBL_Y; } } /* reset y by shifting it back down */ mp_rshd (&y, n - t); /* step 3. for i from n down to (t + 1) */ for (i = n; i >= (t + 1); i--) { if (i > x.used) { continue; } /* step 3.1 if xi == yt then set q{i-t-1} to b-1, * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */ if (x.dp[i] == y.dp[t]) { q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); } else { mp_word tmp; tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); tmp |= ((mp_word) x.dp[i - 1]); tmp /= ((mp_word) y.dp[t]); if (tmp > (mp_word) MP_MASK) tmp = MP_MASK; q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); } /* while (q{i-t-1} * (yt * b + y{t-1})) > xi * b**2 + xi-1 * b + xi-2 do q{i-t-1} -= 1; */ q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK; do { q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK; /* find left hand */ mp_zero (&t1); t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1]; t1.dp[1] = y.dp[t]; t1.used = 2; if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) { goto LBL_Y; } /* find right hand */ t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2]; t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1]; t2.dp[2] = x.dp[i]; t2.used = 3; } while (mp_cmp_mag(&t1, &t2) == MP_GT); /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */ if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) { goto LBL_Y; } if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { goto LBL_Y; } if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) { goto LBL_Y; } /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */ if (x.sign == MP_NEG) { if ((res = mp_copy (&y, &t1)) != MP_OKAY) { goto LBL_Y; } if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { goto LBL_Y; } if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) { goto LBL_Y; } q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; } } /* now q is the quotient and x is the remainder * [which we have to normalize] */ /* get sign before writing to c */ x.sign = x.used == 0 ? MP_ZPOS : a->sign; if (c != NULL) { mp_clamp (&q); mp_exch (&q, c); c->sign = neg; } if (d != NULL) { mp_div_2d (&x, norm, &x, NULL); mp_exch (&x, d); } res = MP_OKAY; LBL_Y:mp_clear (&y); LBL_X:mp_clear (&x); LBL_T2:mp_clear (&t2); LBL_T1:mp_clear (&t1); LBL_Q:mp_clear (&q); return res; } #endif #ifdef MP_LOW_MEM #define TAB_SIZE 32 #else #define TAB_SIZE 256 #endif static int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) { mp_int M[TAB_SIZE], res, mu; mp_digit buf; int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; int (*redux)(mp_int*,mp_int*,mp_int*); /* find window size */ x = mp_count_bits (X); if (x <= 7) { winsize = 2; } else if (x <= 36) { winsize = 3; } else if (x <= 140) { winsize = 4; } else if (x <= 450) { winsize = 5; } else if (x <= 1303) { winsize = 6; } else if (x <= 3529) { winsize = 7; } else { winsize = 8; } #ifdef MP_LOW_MEM if (winsize > 5) { winsize = 5; } #endif /* init M array */ /* init first cell */ if ((err = mp_init(&M[1])) != MP_OKAY) { return err; } /* now init the second half of the array */ for (x = 1<<(winsize-1); x < (1 << winsize); x++) { if ((err = mp_init(&M[x])) != MP_OKAY) { for (y = 1<<(winsize-1); y < x; y++) { mp_clear (&M[y]); } mp_clear(&M[1]); return err; } } /* create mu, used for Barrett reduction */ if ((err = mp_init (&mu)) != MP_OKAY) { goto LBL_M; } if (redmode == 0) { if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) { goto LBL_MU; } redux = mp_reduce; } else { if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) { goto LBL_MU; } redux = mp_reduce_2k_l; } /* create M table * * The M table contains powers of the base, * e.g. M[x] = G**x mod P * * The first half of the table is not * computed though accept for M[0] and M[1] */ if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) { goto LBL_MU; } /* compute the value at M[1<<(winsize-1)] by squaring * M[1] (winsize-1) times */ if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { goto LBL_MU; } for (x = 0; x < (winsize - 1); x++) { /* square it */ if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != MP_OKAY) { goto LBL_MU; } /* reduce modulo P */ if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) { goto LBL_MU; } } /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) */ for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { goto LBL_MU; } if ((err = redux (&M[x], P, &mu)) != MP_OKAY) { goto LBL_MU; } } /* setup result */ if ((err = mp_init (&res)) != MP_OKAY) { goto LBL_MU; } mp_set (&res, 1); /* set initial mode and bit cnt */ mode = 0; bitcnt = 1; buf = 0; digidx = X->used - 1; bitcpy = 0; bitbuf = 0; for (;;) { /* grab next digit as required */ if (--bitcnt == 0) { /* if digidx == -1 we are out of digits */ if (digidx == -1) { break; } /* read next digit and reset the bitcnt */ buf = X->dp[digidx--]; bitcnt = (int) DIGIT_BIT; } /* grab the next msb from the exponent */ y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; buf <<= (mp_digit)1; /* if the bit is zero and mode == 0 then we ignore it * These represent the leading zero bits before the first 1 bit * in the exponent. Technically this opt is not required but it * does lower the # of trivial squaring/reductions used */ if (mode == 0 && y == 0) { continue; } /* if the bit is zero and mode == 1 then we square */ if (mode == 1 && y == 0) { if ((err = mp_sqr (&res, &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, &mu)) != MP_OKAY) { goto LBL_RES; } continue; } /* else we add it to the window */ bitbuf |= (y << (winsize - ++bitcpy)); mode = 2; if (bitcpy == winsize) { /* ok window is filled so square as required and multiply */ /* square first */ for (x = 0; x < winsize; x++) { if ((err = mp_sqr (&res, &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, &mu)) != MP_OKAY) { goto LBL_RES; } } /* then multiply */ if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, &mu)) != MP_OKAY) { goto LBL_RES; } /* empty window and reset */ bitcpy = 0; bitbuf = 0; mode = 1; } } /* if bits remain then square/multiply */ if (mode == 2 && bitcpy > 0) { /* square then multiply if the bit is set */ for (x = 0; x < bitcpy; x++) { if ((err = mp_sqr (&res, &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, &mu)) != MP_OKAY) { goto LBL_RES; } bitbuf <<= 1; if ((bitbuf & (1 << winsize)) != 0) { /* then multiply */ if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, &mu)) != MP_OKAY) { goto LBL_RES; } } } } mp_exch (&res, Y); err = MP_OKAY; LBL_RES:mp_clear (&res); LBL_MU:mp_clear (&mu); LBL_M: mp_clear(&M[1]); for (x = 1<<(winsize-1); x < (1 << winsize); x++) { mp_clear (&M[x]); } return err; } /* computes b = a*a */ static int mp_sqr (mp_int * a, mp_int * b) { int res; #ifdef BN_MP_TOOM_SQR_C /* use Toom-Cook? */ if (a->used >= TOOM_SQR_CUTOFF) { res = mp_toom_sqr(a, b); /* Karatsuba? */ } else #endif #ifdef BN_MP_KARATSUBA_SQR_C if (a->used >= KARATSUBA_SQR_CUTOFF) { res = mp_karatsuba_sqr (a, b); } else #endif { #ifdef BN_FAST_S_MP_SQR_C /* can we use the fast comba multiplier? */ if ((a->used * 2 + 1) < MP_WARRAY && a->used < (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) { res = fast_s_mp_sqr (a, b); } else #endif #ifdef BN_S_MP_SQR_C res = s_mp_sqr (a, b); #else #error mp_sqr could fail res = MP_VAL; #endif } b->sign = MP_ZPOS; return res; } /* reduces a modulo n where n is of the form 2**p - d This differs from reduce_2k since "d" can be larger than a single digit. */ static int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d) { mp_int q; int p, res; if ((res = mp_init(&q)) != MP_OKAY) { return res; } p = mp_count_bits(n); top: /* q = a/2**p, a = a mod 2**p */ if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { goto ERR; } /* q = q * d */ if ((res = mp_mul(&q, d, &q)) != MP_OKAY) { goto ERR; } /* a = a + q */ if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { goto ERR; } if (mp_cmp_mag(a, n) != MP_LT) { s_mp_sub(a, n, a); goto top; } ERR: mp_clear(&q); return res; } /* determines the setup value */ static int mp_reduce_2k_setup_l(mp_int *a, mp_int *d) { int res; mp_int tmp; if ((res = mp_init(&tmp)) != MP_OKAY) { return res; } if ((res = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) { goto ERR; } if ((res = s_mp_sub(&tmp, a, d)) != MP_OKAY) { goto ERR; } ERR: mp_clear(&tmp); return res; } /* computes a = 2**b * * Simple algorithm which zeroes the int, grows it then just sets one bit * as required. */ static int mp_2expt (mp_int * a, int b) { int res; /* zero a as per default */ mp_zero (a); /* grow a to accommodate the single bit */ if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) { return res; } /* set the used count of where the bit will go */ a->used = b / DIGIT_BIT + 1; /* put the single bit in its place */ a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); return MP_OKAY; } /* pre-calculate the value required for Barrett reduction * For a given modulus "b" it calulates the value required in "a" */ static int mp_reduce_setup (mp_int * a, mp_int * b) { int res; if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) { return res; } return mp_div (a, b, a, NULL); } /* reduces x mod m, assumes 0 < x < m**2, mu is * precomputed via mp_reduce_setup. * From HAC pp.604 Algorithm 14.42 */ static int mp_reduce (mp_int * x, mp_int * m, mp_int * mu) { mp_int q; int res, um = m->used; /* q = x */ if ((res = mp_init_copy (&q, x)) != MP_OKAY) { return res; } /* q1 = x / b**(k-1) */ mp_rshd (&q, um - 1); /* according to HAC this optimization is ok */ if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) { if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) { goto CLEANUP; } } else { #ifdef BN_S_MP_MUL_HIGH_DIGS_C if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) { goto CLEANUP; } #elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) { goto CLEANUP; } #else { #error mp_reduce would always fail res = MP_VAL; goto CLEANUP; } #endif } /* q3 = q2 / b**(k+1) */ mp_rshd (&q, um + 1); /* x = x mod b**(k+1), quick (no division) */ if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) { goto CLEANUP; } /* q = q * m mod b**(k+1), quick (no division) */ if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) { goto CLEANUP; } /* x = x - q */ if ((res = mp_sub (x, &q, x)) != MP_OKAY) { goto CLEANUP; } /* If x < 0, add b**(k+1) to it */ if (mp_cmp_d (x, 0) == MP_LT) { mp_set (&q, 1); if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) { goto CLEANUP; } if ((res = mp_add (x, &q, x)) != MP_OKAY) { goto CLEANUP; } } /* Back off if it's too big */ while (mp_cmp (x, m) != MP_LT) { if ((res = s_mp_sub (x, m, x)) != MP_OKAY) { goto CLEANUP; } } CLEANUP: mp_clear (&q); return res; } /* multiplies |a| * |b| and only computes up to digs digits of result * HAC pp. 595, Algorithm 14.12 Modified so you can control how * many digits of output are created. */ static int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) { mp_int t; int res, pa, pb, ix, iy; mp_digit u; mp_word r; mp_digit tmpx, *tmpt, *tmpy; #ifdef BN_FAST_S_MP_MUL_DIGS_C /* can we use the fast multiplier? */ if (((digs) < MP_WARRAY) && MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { return fast_s_mp_mul_digs (a, b, c, digs); } #endif if ((res = mp_init_size (&t, digs)) != MP_OKAY) { return res; } t.used = digs; /* compute the digits of the product directly */ pa = a->used; for (ix = 0; ix < pa; ix++) { /* set the carry to zero */ u = 0; /* limit ourselves to making digs digits of output */ pb = MIN (b->used, digs - ix); /* setup some aliases */ /* copy of the digit from a used within the nested loop */ tmpx = a->dp[ix]; /* an alias for the destination shifted ix places */ tmpt = t.dp + ix; /* an alias for the digits of b */ tmpy = b->dp; /* compute the columns of the output and propagate the carry */ for (iy = 0; iy < pb; iy++) { /* compute the column as a mp_word */ r = ((mp_word)*tmpt) + ((mp_word)tmpx) * ((mp_word)*tmpy++) + ((mp_word) u); /* the new column is the lower part of the result */ *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); /* get the carry word from the result */ u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); } /* set carry if it is placed below digs */ if (ix + iy < digs) { *tmpt = u; } } mp_clamp (&t); mp_exch (&t, c); mp_clear (&t); return MP_OKAY; } #ifdef BN_FAST_S_MP_MUL_DIGS_C /* Fast (comba) multiplier * * This is the fast column-array [comba] multiplier. It is * designed to compute the columns of the product first * then handle the carries afterwards. This has the effect * of making the nested loops that compute the columns very * simple and schedulable on super-scalar processors. * * This has been modified to produce a variable number of * digits of output so if say only a half-product is required * you don't have to compute the upper half (a feature * required for fast Barrett reduction). * * Based on Algorithm 14.12 on pp.595 of HAC. * */ static int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) { int olduse, res, pa, ix, iz; mp_digit W[MP_WARRAY]; register mp_word _W; /* grow the destination as required */ if (c->alloc < digs) { if ((res = mp_grow (c, digs)) != MP_OKAY) { return res; } } /* number of output digits to produce */ pa = MIN(digs, a->used + b->used); /* clear the carry */ _W = 0; os_memset(W, 0, sizeof(W)); for (ix = 0; ix < pa; ix++) { int tx, ty; int iy; mp_digit *tmpx, *tmpy; /* get offsets into the two bignums */ ty = MIN(b->used-1, ix); tx = ix - ty; /* setup temp aliases */ tmpx = a->dp + tx; tmpy = b->dp + ty; /* this is the number of times the loop will iterrate, essentially while (tx++ < a->used && ty-- >= 0) { ... } */ iy = MIN(a->used-tx, ty+1); /* execute loop */ for (iz = 0; iz < iy; ++iz) { _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); } /* store term */ W[ix] = ((mp_digit)_W) & MP_MASK; /* make next carry */ _W = _W >> ((mp_word)DIGIT_BIT); } /* setup dest */ olduse = c->used; c->used = pa; { register mp_digit *tmpc; tmpc = c->dp; for (ix = 0; ix < pa+1; ix++) { /* now extract the previous digit [below the carry] */ *tmpc++ = W[ix]; } /* clear unused digits [that existed in the old copy of c] */ for (; ix < olduse; ix++) { *tmpc++ = 0; } } mp_clamp (c); return MP_OKAY; } #endif /* BN_FAST_S_MP_MUL_DIGS_C */ /* init an mp_init for a given size */ static int mp_init_size (mp_int * a, int size) { int x; /* pad size so there are always extra digits */ size += (MP_PREC * 2) - (size % MP_PREC); /* alloc mem */ a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); if (a->dp == NULL) { return MP_MEM; } /* set the members */ a->used = 0; a->alloc = size; a->sign = MP_ZPOS; /* zero the digits */ for (x = 0; x < size; x++) { a->dp[x] = 0; } return MP_OKAY; } /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ static int s_mp_sqr (mp_int * a, mp_int * b) { mp_int t; int res, ix, iy, pa; mp_word r; mp_digit u, tmpx, *tmpt; pa = a->used; if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) { return res; } /* default used is maximum possible size */ t.used = 2*pa + 1; for (ix = 0; ix < pa; ix++) { /* first calculate the digit at 2*ix */ /* calculate double precision result */ r = ((mp_word) t.dp[2*ix]) + ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); /* store lower part in result */ t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); /* get the carry */ u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); /* left hand side of A[ix] * A[iy] */ tmpx = a->dp[ix]; /* alias for where to store the results */ tmpt = t.dp + (2*ix + 1); for (iy = ix + 1; iy < pa; iy++) { /* first calculate the product */ r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); /* now calculate the double precision result, note we use * addition instead of *2 since it's easier to optimize */ r = ((mp_word) *tmpt) + r + r + ((mp_word) u); /* store lower part */ *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); /* get carry */ u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); } /* propagate upwards */ while (u != ((mp_digit) 0)) { r = ((mp_word) *tmpt) + ((mp_word) u); *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); } } mp_clamp (&t); mp_exch (&t, b); mp_clear (&t); return MP_OKAY; } /* multiplies |a| * |b| and does not compute the lower digs digits * [meant to get the higher part of the product] */ static int s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) { mp_int t; int res, pa, pb, ix, iy; mp_digit u; mp_word r; mp_digit tmpx, *tmpt, *tmpy; /* can we use the fast multiplier? */ #ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C if (((a->used + b->used + 1) < MP_WARRAY) && MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { return fast_s_mp_mul_high_digs (a, b, c, digs); } #endif if ((res = mp_init_size (&t, a->used + b->used + 1)) != MP_OKAY) { return res; } t.used = a->used + b->used + 1; pa = a->used; pb = b->used; for (ix = 0; ix < pa; ix++) { /* clear the carry */ u = 0; /* left hand side of A[ix] * B[iy] */ tmpx = a->dp[ix]; /* alias to the address of where the digits will be stored */ tmpt = &(t.dp[digs]); /* alias for where to read the right hand side from */ tmpy = b->dp + (digs - ix); for (iy = digs - ix; iy < pb; iy++) { /* calculate the double precision result */ r = ((mp_word)*tmpt) + ((mp_word)tmpx) * ((mp_word)*tmpy++) + ((mp_word) u); /* get the lower part */ *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); /* carry the carry */ u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); } *tmpt = u; } mp_clamp (&t); mp_exch (&t, c); mp_clear (&t); return MP_OKAY; } #ifdef BN_MP_MONTGOMERY_SETUP_C /* setups the montgomery reduction stuff */ static int mp_montgomery_setup (mp_int * n, mp_digit * rho) { mp_digit x, b; /* fast inversion mod 2**k * * Based on the fact that * * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) * => 2*X*A - X*X*A*A = 1 * => 2*(1) - (1) = 1 */ b = n->dp[0]; if ((b & 1) == 0) { return MP_VAL; } x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ x *= 2 - b * x; /* here x*a==1 mod 2**8 */ #if !defined(MP_8BIT) x *= 2 - b * x; /* here x*a==1 mod 2**16 */ #endif #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) x *= 2 - b * x; /* here x*a==1 mod 2**32 */ #endif #ifdef MP_64BIT x *= 2 - b * x; /* here x*a==1 mod 2**64 */ #endif /* rho = -1/m mod b */ *rho = (unsigned long)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; return MP_OKAY; } #endif #ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C /* computes xR**-1 == x (mod N) via Montgomery Reduction * * This is an optimized implementation of montgomery_reduce * which uses the comba method to quickly calculate the columns of the * reduction. * * Based on Algorithm 14.32 on pp.601 of HAC. */ static int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) { int ix, res, olduse; mp_word W[MP_WARRAY]; /* get old used count */ olduse = x->used; /* grow a as required */ if (x->alloc < n->used + 1) { if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) { return res; } } /* first we have to get the digits of the input into * an array of double precision words W[...] */ { register mp_word *_W; register mp_digit *tmpx; /* alias for the W[] array */ _W = W; /* alias for the digits of x*/ tmpx = x->dp; /* copy the digits of a into W[0..a->used-1] */ for (ix = 0; ix < x->used; ix++) { *_W++ = *tmpx++; } /* zero the high words of W[a->used..m->used*2] */ for (; ix < n->used * 2 + 1; ix++) { *_W++ = 0; } } /* now we proceed to zero successive digits * from the least significant upwards */ for (ix = 0; ix < n->used; ix++) { /* mu = ai * m' mod b * * We avoid a double precision multiplication (which isn't required) * by casting the value down to a mp_digit. Note this requires * that W[ix-1] have the carry cleared (see after the inner loop) */ register mp_digit mu; mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); /* a = a + mu * m * b**i * * This is computed in place and on the fly. The multiplication * by b**i is handled by offseting which columns the results * are added to. * * Note the comba method normally doesn't handle carries in the * inner loop In this case we fix the carry from the previous * column since the Montgomery reduction requires digits of the * result (so far) [see above] to work. This is * handled by fixing up one carry after the inner loop. The * carry fixups are done in order so after these loops the * first m->used words of W[] have the carries fixed */ { register int iy; register mp_digit *tmpn; register mp_word *_W; /* alias for the digits of the modulus */ tmpn = n->dp; /* Alias for the columns set by an offset of ix */ _W = W + ix; /* inner loop */ for (iy = 0; iy < n->used; iy++) { *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); } } /* now fix carry for next digit, W[ix+1] */ W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); } /* now we have to propagate the carries and * shift the words downward [all those least * significant digits we zeroed]. */ { register mp_digit *tmpx; register mp_word *_W, *_W1; /* nox fix rest of carries */ /* alias for current word */ _W1 = W + ix; /* alias for next word, where the carry goes */ _W = W + ++ix; for (; ix <= n->used * 2 + 1; ix++) { *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); } /* copy out, A = A/b**n * * The result is A/b**n but instead of converting from an * array of mp_word to mp_digit than calling mp_rshd * we just copy them in the right order */ /* alias for destination word */ tmpx = x->dp; /* alias for shifted double precision result */ _W = W + n->used; for (ix = 0; ix < n->used + 1; ix++) { *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); } /* zero oldused digits, if the input a was larger than * m->used+1 we'll have to clear the digits */ for (; ix < olduse; ix++) { *tmpx++ = 0; } } /* set the max used and clamp */ x->used = n->used + 1; mp_clamp (x); /* if A >= m then A = A - m */ if (mp_cmp_mag (x, n) != MP_LT) { return s_mp_sub (x, n, x); } return MP_OKAY; } #endif #ifdef BN_MP_MUL_2_C /* b = a*2 */ static int mp_mul_2(mp_int * a, mp_int * b) { int x, res, oldused; /* grow to accommodate result */ if (b->alloc < a->used + 1) { if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) { return res; } } oldused = b->used; b->used = a->used; { register mp_digit r, rr, *tmpa, *tmpb; /* alias for source */ tmpa = a->dp; /* alias for dest */ tmpb = b->dp; /* carry */ r = 0; for (x = 0; x < a->used; x++) { /* get what will be the *next* carry bit from the * MSB of the current digit */ rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); /* now shift up this digit, add in the carry [from the previous] */ *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; /* copy the carry that would be from the source * digit into the next iteration */ r = rr; } /* new leading digit? */ if (r != 0) { /* add a MSB which is always 1 at this point */ *tmpb = 1; ++(b->used); } /* now zero any excess digits on the destination * that we didn't write to */ tmpb = b->dp + b->used; for (x = b->used; x < oldused; x++) { *tmpb++ = 0; } } b->sign = a->sign; return MP_OKAY; } #endif #ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C /* * shifts with subtractions when the result is greater than b. * * The method is slightly modified to shift B unconditionally up to just under * the leading bit of b. This saves a lot of multiple precision shifting. */ static int mp_montgomery_calc_normalization (mp_int * a, mp_int * b) { int x, bits, res; /* how many bits of last digit does b use */ bits = mp_count_bits (b) % DIGIT_BIT; if (b->used > 1) { if ((res = mp_2expt (a, (b->used - 1) * DIGIT_BIT + bits - 1)) != MP_OKAY) { return res; } } else { mp_set(a, 1); bits = 1; } /* now compute C = A * B mod b */ for (x = bits - 1; x < (int)DIGIT_BIT; x++) { if ((res = mp_mul_2 (a, a)) != MP_OKAY) { return res; } if (mp_cmp_mag (a, b) != MP_LT) { if ((res = s_mp_sub (a, b, a)) != MP_OKAY) { return res; } } } return MP_OKAY; } #endif #ifdef BN_MP_EXPTMOD_FAST_C /* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85 * * Uses a left-to-right k-ary sliding window to compute the modular exponentiation. * The value of k changes based on the size of the exponent. * * Uses Montgomery or Diminished Radix reduction [whichever appropriate] */ static int mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) { mp_int M[TAB_SIZE], res; mp_digit buf, mp; int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; /* use a pointer to the reduction algorithm. This allows us to use * one of many reduction algorithms without modding the guts of * the code with if statements everywhere. */ int (*redux)(mp_int*,mp_int*,mp_digit); /* find window size */ x = mp_count_bits (X); if (x <= 7) { winsize = 2; } else if (x <= 36) { winsize = 3; } else if (x <= 140) { winsize = 4; } else if (x <= 450) { winsize = 5; } else if (x <= 1303) { winsize = 6; } else if (x <= 3529) { winsize = 7; } else { winsize = 8; } #ifdef MP_LOW_MEM if (winsize > 5) { winsize = 5; } #endif /* init M array */ /* init first cell */ if ((err = mp_init(&M[1])) != MP_OKAY) { return err; } /* now init the second half of the array */ for (x = 1<<(winsize-1); x < (1 << winsize); x++) { if ((err = mp_init(&M[x])) != MP_OKAY) { for (y = 1<<(winsize-1); y < x; y++) { mp_clear (&M[y]); } mp_clear(&M[1]); return err; } } /* determine and setup reduction code */ if (redmode == 0) { #ifdef BN_MP_MONTGOMERY_SETUP_C /* now setup montgomery */ if ((err = mp_montgomery_setup (P, &mp)) != MP_OKAY) { goto LBL_M; } #else err = MP_VAL; goto LBL_M; #endif /* automatically pick the comba one if available (saves quite a few calls/ifs) */ #ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C if (((P->used * 2 + 1) < MP_WARRAY) && P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { redux = fast_mp_montgomery_reduce; } else #endif { #ifdef BN_MP_MONTGOMERY_REDUCE_C /* use slower baseline Montgomery method */ redux = mp_montgomery_reduce; #else err = MP_VAL; goto LBL_M; #endif } } else if (redmode == 1) { #if defined(BN_MP_DR_SETUP_C) && defined(BN_MP_DR_REDUCE_C) /* setup DR reduction for moduli of the form B**k - b */ mp_dr_setup(P, &mp); redux = mp_dr_reduce; #else err = MP_VAL; goto LBL_M; #endif } else { #if defined(BN_MP_REDUCE_2K_SETUP_C) && defined(BN_MP_REDUCE_2K_C) /* setup DR reduction for moduli of the form 2**k - b */ if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) { goto LBL_M; } redux = mp_reduce_2k; #else err = MP_VAL; goto LBL_M; #endif } /* setup result */ if ((err = mp_init (&res)) != MP_OKAY) { goto LBL_M; } /* create M table * * * The first half of the table is not computed though accept for M[0] and M[1] */ if (redmode == 0) { #ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C /* now we need R mod m */ if ((err = mp_montgomery_calc_normalization (&res, P)) != MP_OKAY) { goto LBL_RES; } #else err = MP_VAL; goto LBL_RES; #endif /* now set M[1] to G * R mod m */ if ((err = mp_mulmod (G, &res, P, &M[1])) != MP_OKAY) { goto LBL_RES; } } else { mp_set(&res, 1); if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) { goto LBL_RES; } } /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */ if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { goto LBL_RES; } for (x = 0; x < (winsize - 1); x++) { if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&M[1 << (winsize - 1)], P, mp)) != MP_OKAY) { goto LBL_RES; } } /* create upper table */ for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&M[x], P, mp)) != MP_OKAY) { goto LBL_RES; } } /* set initial mode and bit cnt */ mode = 0; bitcnt = 1; buf = 0; digidx = X->used - 1; bitcpy = 0; bitbuf = 0; for (;;) { /* grab next digit as required */ if (--bitcnt == 0) { /* if digidx == -1 we are out of digits so break */ if (digidx == -1) { break; } /* read next digit and reset bitcnt */ buf = X->dp[digidx--]; bitcnt = (int)DIGIT_BIT; } /* grab the next msb from the exponent */ y = (mp_digit)(buf >> (DIGIT_BIT - 1)) & 1; buf <<= (mp_digit)1; /* if the bit is zero and mode == 0 then we ignore it * These represent the leading zero bits before the first 1 bit * in the exponent. Technically this opt is not required but it * does lower the # of trivial squaring/reductions used */ if (mode == 0 && y == 0) { continue; } /* if the bit is zero and mode == 1 then we square */ if (mode == 1 && y == 0) { if ((err = mp_sqr (&res, &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, mp)) != MP_OKAY) { goto LBL_RES; } continue; } /* else we add it to the window */ bitbuf |= (y << (winsize - ++bitcpy)); mode = 2; if (bitcpy == winsize) { /* ok window is filled so square as required and multiply */ /* square first */ for (x = 0; x < winsize; x++) { if ((err = mp_sqr (&res, &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, mp)) != MP_OKAY) { goto LBL_RES; } } /* then multiply */ if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, mp)) != MP_OKAY) { goto LBL_RES; } /* empty window and reset */ bitcpy = 0; bitbuf = 0; mode = 1; } } /* if bits remain then square/multiply */ if (mode == 2 && bitcpy > 0) { /* square then multiply if the bit is set */ for (x = 0; x < bitcpy; x++) { if ((err = mp_sqr (&res, &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, mp)) != MP_OKAY) { goto LBL_RES; } /* get next bit of the window */ bitbuf <<= 1; if ((bitbuf & (1 << winsize)) != 0) { /* then multiply */ if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { goto LBL_RES; } if ((err = redux (&res, P, mp)) != MP_OKAY) { goto LBL_RES; } } } } if (redmode == 0) { /* fixup result if Montgomery reduction is used * recall that any value in a Montgomery system is * actually multiplied by R mod n. So we have * to reduce one more time to cancel out the factor * of R. */ if ((err = redux(&res, P, mp)) != MP_OKAY) { goto LBL_RES; } } /* swap res with Y */ mp_exch (&res, Y); err = MP_OKAY; LBL_RES:mp_clear (&res); LBL_M: mp_clear(&M[1]); for (x = 1<<(winsize-1); x < (1 << winsize); x++) { mp_clear (&M[x]); } return err; } #endif #ifdef BN_FAST_S_MP_SQR_C /* the jist of squaring... * you do like mult except the offset of the tmpx [one that * starts closer to zero] can't equal the offset of tmpy. * So basically you set up iy like before then you min it with * (ty-tx) so that it never happens. You double all those * you add in the inner loop After that loop you do the squares and add them in. */ static int fast_s_mp_sqr (mp_int * a, mp_int * b) { int olduse, res, pa, ix, iz; mp_digit W[MP_WARRAY], *tmpx; mp_word W1; /* grow the destination as required */ pa = a->used + a->used; if (b->alloc < pa) { if ((res = mp_grow (b, pa)) != MP_OKAY) { return res; } } /* number of output digits to produce */ W1 = 0; for (ix = 0; ix < pa; ix++) { int tx, ty, iy; mp_word _W; mp_digit *tmpy; /* clear counter */ _W = 0; /* get offsets into the two bignums */ ty = MIN(a->used-1, ix); tx = ix - ty; /* setup temp aliases */ tmpx = a->dp + tx; tmpy = a->dp + ty; /* this is the number of times the loop will iterrate, essentially while (tx++ < a->used && ty-- >= 0) { ... } */ iy = MIN(a->used-tx, ty+1); /* now for squaring tx can never equal ty * we halve the distance since they approach at a rate of 2x * and we have to round because odd cases need to be executed */ iy = MIN(iy, (ty-tx+1)>>1); /* execute loop */ for (iz = 0; iz < iy; iz++) { _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); } /* double the inner product and add carry */ _W = _W + _W + W1; /* even columns have the square term in them */ if ((ix&1) == 0) { _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); } /* store it */ W[ix] = (mp_digit)(_W & MP_MASK); /* make next carry */ W1 = _W >> ((mp_word)DIGIT_BIT); } /* setup dest */ olduse = b->used; b->used = a->used+a->used; { mp_digit *tmpb; tmpb = b->dp; for (ix = 0; ix < pa; ix++) { *tmpb++ = W[ix] & MP_MASK; } /* clear unused digits [that existed in the old copy of c] */ for (; ix < olduse; ix++) { *tmpb++ = 0; } } mp_clamp (b); return MP_OKAY; } #endif #ifdef BN_MP_MUL_D_C /* multiply by a digit */ static int mp_mul_d (mp_int * a, mp_digit b, mp_int * c) { mp_digit u, *tmpa, *tmpc; mp_word r; int ix, res, olduse; /* make sure c is big enough to hold a*b */ if (c->alloc < a->used + 1) { if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) { return res; } } /* get the original destinations used count */ olduse = c->used; /* set the sign */ c->sign = a->sign; /* alias for a->dp [source] */ tmpa = a->dp; /* alias for c->dp [dest] */ tmpc = c->dp; /* zero carry */ u = 0; /* compute columns */ for (ix = 0; ix < a->used; ix++) { /* compute product and carry sum for this term */ r = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b); /* mask off higher bits to get a single digit */ *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); /* send carry into next iteration */ u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); } /* store final carry [if any] and increment ix offset */ *tmpc++ = u; ++ix; /* now zero digits above the top */ while (ix++ < olduse) { *tmpc++ = 0; } /* set used count */ c->used = a->used + 1; mp_clamp(c); return MP_OKAY; } #endif