BIGNUM *BN_generate_prime(BIGNUM *ret, int bits, int safe,
const BIGNUM *add, const BIGNUM *rem,
void (*callback)(int,int,void *), void *cb_arg)
{
BIGNUM *rnd=NULL;
BIGNUM t;
int found=0;
int i,j,c1=0;
BN_CTX *ctx;
int checks = BN_prime_checks_for_size(bits);
ctx=BN_CTX_new();
if (ctx == NULL) goto err;
if (ret == NULL)
{
if ((rnd=BN_new()) == NULL) goto err;
}
else
rnd=ret;
BN_init(&t);
loop:
/* make a random number and set the top and bottom bits */
if (add == NULL)
{
if (!probable_prime(rnd,bits)) goto err;
}
else
{
if (safe)
{
if (!probable_prime_dh_safe(rnd,bits,add,rem,ctx))
goto err;
}
else
{
if (!probable_prime_dh(rnd,bits,add,rem,ctx))
goto err;
}
}
/* if (BN_mod_word(rnd,(BN_ULONG)3) == 1) goto loop; */
if (callback != NULL) callback(0,c1++,cb_arg);
if (!safe)
{
i=BN_is_prime_fasttest(rnd,checks,callback,ctx,cb_arg,0);
if (i == -1) goto err;
if (i == 0) goto loop;
}
else
{
/* for "safe prime" generation,
* check that (p-1)/2 is prime.
* Since a prime is odd, We just
* need to divide by 2 */
if (!BN_rshift1(&t,rnd)) goto err;
for (i=0; i<checks; i++)
{
j=BN_is_prime_fasttest(rnd,1,callback,ctx,cb_arg,0);
if (j == -1) goto err;
if (j == 0) goto loop;
j=BN_is_prime_fasttest(&t,1,callback,ctx,cb_arg,0);
if (j == -1) goto err;
if (j == 0) goto loop;
if (callback != NULL) callback(2,c1-1,cb_arg);
/* We have a safe prime test pass */
}
}
/* we have a prime :-) */
found = 1;
err:
if (!found && (ret == NULL) && (rnd != NULL)) BN_free(rnd);
BN_free(&t);
if (ctx != NULL) BN_CTX_free(ctx);
return(found ? rnd : NULL);
}
int BN_is_prime_fasttest(const BIGNUM *a, int checks,
void (*callback)(int,int,void *),
BN_CTX *ctx_passed, void *cb_arg,
int do_trial_division)
{
int i, j, ret = -1;
int k;
BN_CTX *ctx = NULL;
BIGNUM *A1, *A1_odd, *check; /* taken from ctx */
BN_MONT_CTX *mont = NULL;
const BIGNUM *A = NULL;
if (BN_cmp(a, BN_value_one()) <= 0)
return 0;
if (checks == BN_prime_checks)
checks = BN_prime_checks_for_size(BN_num_bits(a));
/* first look for small factors */
if (!BN_is_odd(a))
return 0;
if (do_trial_division)
{
for (i = 1; i < NUMPRIMES; i++)
if (BN_mod_word(a, primes[i]) == 0)
return 0;
if (callback != NULL) callback(1, -1, cb_arg);
}
if (ctx_passed != NULL)
ctx = ctx_passed;
else
if ((ctx=BN_CTX_new()) == NULL)
goto err;
BN_CTX_start(ctx);
/* A := abs(a) */
if (a->neg)
{
BIGNUM *t;
if ((t = BN_CTX_get(ctx)) == NULL) goto err;
BN_copy(t, a);
t->neg = 0;
A = t;
}
else
A = a;
A1 = BN_CTX_get(ctx);
A1_odd = BN_CTX_get(ctx);
check = BN_CTX_get(ctx);
if (check == NULL) goto err;
/* compute A1 := A - 1 */
if (!BN_copy(A1, A))
goto err;
if (!BN_sub_word(A1, 1))
goto err;
if (BN_is_zero(A1))
{
ret = 0;
goto err;
}
/* write A1 as A1_odd * 2^k */
k = 1;
while (!BN_is_bit_set(A1, k))
k++;
if (!BN_rshift(A1_odd, A1, k))
goto err;
/* Montgomery setup for computations mod A */
mont = BN_MONT_CTX_new();
if (mont == NULL)
goto err;
if (!BN_MONT_CTX_set(mont, A, ctx))
goto err;
for (i = 0; i < checks; i++)
{
if (!BN_pseudo_rand_range(check, A1))
goto err;
if (!BN_add_word(check, 1))
goto err;
/* now 1 <= check < A */
j = witness(check, A, A1, A1_odd, k, ctx, mont);
if (j == -1) goto err;
if (j)
{
ret=0;
goto err;
}
if (callback != NULL) callback(1,i,cb_arg);
}
ret=1;
err:
if (ctx != NULL)
{
BN_CTX_end(ctx);
if (ctx_passed == NULL)
BN_CTX_free(ctx);
}
if (mont != NULL)
BN_MONT_CTX_free(mont);
return(ret);
}
int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
{
BIGNUM *t;
int found=0;
int i,j,c1=0;
BN_CTX *ctx;
int checks = BN_prime_checks_for_size(bits);
if (bits < 2)
{
/* There are no prime numbers this small. */
BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
return 0;
}
else if (bits == 2 && safe)
{
/* The smallest safe prime (7) is three bits. */
BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
return 0;
}
ctx=BN_CTX_new();
if (ctx == NULL) goto err;
BN_CTX_start(ctx);
t = BN_CTX_get(ctx);
if(!t) goto err;
loop:
/* make a random number and set the top and bottom bits */
if (add == NULL)
{
if (!probable_prime(ret,bits)) goto err;
}
else
{
if (safe)
{
if (!probable_prime_dh_safe(ret,bits,add,rem,ctx))
goto err;
}
else
{
if (!bn_probable_prime_dh(ret,bits,add,rem,ctx))
goto err;
}
}
/* if (BN_mod_word(ret,(BN_ULONG)3) == 1) goto loop; */
if(!BN_GENCB_call(cb, 0, c1++))
/* aborted */
goto err;
if (!safe)
{
i=BN_is_prime_fasttest_ex(ret,checks,ctx,0,cb);
if (i == -1) goto err;
if (i == 0) goto loop;
}
else
{
/* for "safe prime" generation,
* check that (p-1)/2 is prime.
* Since a prime is odd, We just
* need to divide by 2 */
if (!BN_rshift1(t,ret)) goto err;
for (i=0; i<checks; i++)
{
j=BN_is_prime_fasttest_ex(ret,1,ctx,0,cb);
if (j == -1) goto err;
if (j == 0) goto loop;
j=BN_is_prime_fasttest_ex(t,1,ctx,0,cb);
if (j == -1) goto err;
if (j == 0) goto loop;
if(!BN_GENCB_call(cb, 2, c1-1))
goto err;
/* We have a safe prime test pass */
}
}
/* we have a prime :-) */
found = 1;
err:
if (ctx != NULL)
{
BN_CTX_end(ctx);
BN_CTX_free(ctx);
}
bn_check_top(ret);
return found;
}
int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
int do_trial_division, BN_GENCB *cb)
{
int i, j, ret = -1;
int k;
BN_CTX *ctx = NULL;
BIGNUM *A1, *A1_odd, *check; /* taken from ctx */
BN_MONT_CTX *mont = NULL;
if (BN_cmp(a, BN_value_one()) <= 0)
return 0;
if (checks == BN_prime_checks)
checks = BN_prime_checks_for_size(BN_num_bits(a));
/* first look for small factors */
if (!BN_is_odd(a))
/* a is even => a is prime if and only if a == 2 */
return BN_is_word(a, 2);
if (do_trial_division) {
for (i = 1; i < NUMPRIMES; i++) {
BN_ULONG mod = BN_mod_word(a, primes[i]);
if (mod == (BN_ULONG)-1)
goto err;
if (mod == 0)
return BN_is_word(a, primes[i]);
}
if (!BN_GENCB_call(cb, 1, -1))
goto err;
}
if (ctx_passed != NULL)
ctx = ctx_passed;
else if ((ctx = BN_CTX_new()) == NULL)
goto err;
BN_CTX_start(ctx);
A1 = BN_CTX_get(ctx);
A1_odd = BN_CTX_get(ctx);
check = BN_CTX_get(ctx);
if (check == NULL)
goto err;
/* compute A1 := a - 1 */
if (!BN_copy(A1, a))
goto err;
if (!BN_sub_word(A1, 1))
goto err;
if (BN_is_zero(A1)) {
ret = 0;
goto err;
}
/* write A1 as A1_odd * 2^k */
k = 1;
while (!BN_is_bit_set(A1, k))
k++;
if (!BN_rshift(A1_odd, A1, k))
goto err;
/* Montgomery setup for computations mod a */
mont = BN_MONT_CTX_new();
if (mont == NULL)
goto err;
if (!BN_MONT_CTX_set(mont, a, ctx))
goto err;
for (i = 0; i < checks; i++) {
if (!BN_priv_rand_range(check, A1))
goto err;
if (!BN_add_word(check, 1))
goto err;
/* now 1 <= check < a */
j = witness(check, a, A1, A1_odd, k, ctx, mont);
if (j == -1)
goto err;
if (j) {
ret = 0;
goto err;
}
if (!BN_GENCB_call(cb, 1, i))
goto err;
}
ret = 1;
err:
if (ctx != NULL) {
BN_CTX_end(ctx);
if (ctx_passed == NULL)
BN_CTX_free(ctx);
}
BN_MONT_CTX_free(mont);
return ret;
}
int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe, const BIGNUM *add,
const BIGNUM *rem, BN_GENCB *cb) {
BIGNUM *t;
int found = 0;
int i, j, c1 = 0;
BN_CTX *ctx;
int checks = BN_prime_checks_for_size(bits);
if (bits < 2) {
/* There are no prime numbers this small. */
OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);
return 0;
} else if (bits == 2 && safe) {
/* The smallest safe prime (7) is three bits. */
OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);
return 0;
}
ctx = BN_CTX_new();
if (ctx == NULL) {
goto err;
}
BN_CTX_start(ctx);
t = BN_CTX_get(ctx);
if (!t) {
goto err;
}
loop:
/* make a random number and set the top and bottom bits */
if (add == NULL) {
if (!probable_prime(ret, bits)) {
goto err;
}
} else {
if (safe) {
if (!probable_prime_dh_safe(ret, bits, add, rem, ctx)) {
goto err;
}
} else {
if (!probable_prime_dh(ret, bits, add, rem, ctx)) {
goto err;
}
}
}
if (!BN_GENCB_call(cb, BN_GENCB_GENERATED, c1++)) {
/* aborted */
goto err;
}
if (!safe) {
i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
if (i == -1) {
goto err;
} else if (i == 0) {
goto loop;
}
} else {
/* for "safe prime" generation, check that (p-1)/2 is prime. Since a prime
* is odd, We just need to divide by 2 */
if (!BN_rshift1(t, ret)) {
goto err;
}
for (i = 0; i < checks; i++) {
j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, NULL);
if (j == -1) {
goto err;
} else if (j == 0) {
goto loop;
}
j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, NULL);
if (j == -1) {
goto err;
} else if (j == 0) {
goto loop;
}
if (!BN_GENCB_call(cb, i, c1 - 1)) {
goto err;
}
/* We have a safe prime test pass */
}
}
/* we have a prime :-) */
found = 1;
err:
if (ctx != NULL) {
BN_CTX_end(ctx);
BN_CTX_free(ctx);
}
return found;
}
int BN_enhanced_miller_rabin_primality_test(
enum bn_primality_result_t *out_result, const BIGNUM *w, int iterations,
BN_CTX *ctx, BN_GENCB *cb) {
/* Enhanced Miller-Rabin is only valid on odd integers greater than 3. */
if (!BN_is_odd(w) || BN_cmp_word(w, 3) <= 0) {
OPENSSL_PUT_ERROR(BN, BN_R_INVALID_INPUT);
return 0;
}
if (iterations == BN_prime_checks) {
iterations = BN_prime_checks_for_size(BN_num_bits(w));
}
int ret = 0;
BN_MONT_CTX *mont = NULL;
BN_CTX_start(ctx);
BIGNUM *w1 = BN_CTX_get(ctx);
if (w1 == NULL ||
!BN_copy(w1, w) ||
!BN_sub_word(w1, 1)) {
goto err;
}
/* Write w1 as m*2^a (Steps 1 and 2). */
int a = 0;
while (!BN_is_bit_set(w1, a)) {
a++;
}
BIGNUM *m = BN_CTX_get(ctx);
if (m == NULL ||
!BN_rshift(m, w1, a)) {
goto err;
}
BIGNUM *b = BN_CTX_get(ctx);
BIGNUM *g = BN_CTX_get(ctx);
BIGNUM *z = BN_CTX_get(ctx);
BIGNUM *x = BN_CTX_get(ctx);
BIGNUM *x1 = BN_CTX_get(ctx);
if (b == NULL ||
g == NULL ||
z == NULL ||
x == NULL ||
x1 == NULL) {
goto err;
}
/* Montgomery setup for computations mod A */
mont = BN_MONT_CTX_new();
if (mont == NULL ||
!BN_MONT_CTX_set(mont, w, ctx)) {
goto err;
}
/* The following loop performs in inner iteration of the Enhanced Miller-Rabin
* Primality test (Step 4). */
for (int i = 1; i <= iterations; i++) {
/* Step 4.1-4.2 */
if (!BN_rand_range_ex(b, 2, w1)) {
goto err;
}
/* Step 4.3-4.4 */
if (!BN_gcd(g, b, w, ctx)) {
goto err;
}
if (BN_cmp_word(g, 1) > 0) {
*out_result = bn_composite;
ret = 1;
goto err;
}
/* Step 4.5 */
if (!BN_mod_exp_mont(z, b, m, w, ctx, mont)) {
goto err;
}
/* Step 4.6 */
if (BN_is_one(z) || BN_cmp(z, w1) == 0) {
goto loop;
}
/* Step 4.7 */
for (int j = 1; j < a; j++) {
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {
goto err;
}
if (BN_cmp(z, w1) == 0) {
goto loop;
}
if (BN_is_one(z)) {
goto composite;
}
}
/* Step 4.8-4.9 */
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {
goto err;
}
/* Step 4.10-4.11 */
if (!BN_is_one(z) && !BN_copy(x, z)) {
goto err;
}
composite:
/* Step 4.12-4.14 */
if (!BN_copy(x1, x) ||
!BN_sub_word(x1, 1) ||
!BN_gcd(g, x1, w, ctx)) {
goto err;
}
if (BN_cmp_word(g, 1) > 0) {
*out_result = bn_composite;
} else {
*out_result = bn_non_prime_power_composite;
}
ret = 1;
goto err;
loop:
/* Step 4.15 */
if (!BN_GENCB_call(cb, 1, i)) {
goto err;
}
}
*out_result = bn_probably_prime;
ret = 1;
err:
BN_MONT_CTX_free(mont);
BN_CTX_end(ctx);
return ret;
}
/*
* Refer to FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test.
* OR C.3.1 Miller-Rabin Probabilistic Primality Test (if enhanced is zero).
* The Step numbers listed in the code refer to the enhanced case.
*
* if enhanced is set, then status returns one of the following:
* BN_PRIMETEST_PROBABLY_PRIME
* BN_PRIMETEST_COMPOSITE_WITH_FACTOR
* BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME
* if enhanced is zero, then status returns either
* BN_PRIMETEST_PROBABLY_PRIME or
* BN_PRIMETEST_COMPOSITE
*
* returns 0 if there was an error, otherwise it returns 1.
*/
int bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,
BN_GENCB *cb, int enhanced, int *status)
{
int i, j, a, ret = 0;
BIGNUM *g, *w1, *w3, *x, *m, *z, *b;
BN_MONT_CTX *mont = NULL;
/* w must be odd */
if (!BN_is_odd(w))
return 0;
BN_CTX_start(ctx);
g = BN_CTX_get(ctx);
w1 = BN_CTX_get(ctx);
w3 = BN_CTX_get(ctx);
x = BN_CTX_get(ctx);
m = BN_CTX_get(ctx);
z = BN_CTX_get(ctx);
b = BN_CTX_get(ctx);
if (!(b != NULL
/* w1 := w - 1 */
&& BN_copy(w1, w)
&& BN_sub_word(w1, 1)
/* w3 := w - 3 */
&& BN_copy(w3, w)
&& BN_sub_word(w3, 3)))
goto err;
/* check w is larger than 3, otherwise the random b will be too small */
if (BN_is_zero(w3) || BN_is_negative(w3))
goto err;
/* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */
a = 1;
while (!BN_is_bit_set(w1, a))
a++;
/* (Step 2) m = (w-1) / 2^a */
if (!BN_rshift(m, w1, a))
goto err;
/* Montgomery setup for computations mod a */
mont = BN_MONT_CTX_new();
if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))
goto err;
if (iterations == BN_prime_checks)
iterations = BN_prime_checks_for_size(BN_num_bits(w));
/* (Step 4) */
for (i = 0; i < iterations; ++i) {
/* (Step 4.1) obtain a Random string of bits b where 1 < b < w-1 */
if (!BN_priv_rand_range(b, w3) || !BN_add_word(b, 2)) /* 1 < b < w-1 */
goto err;
if (enhanced) {
/* (Step 4.3) */
if (!BN_gcd(g, b, w, ctx))
goto err;
/* (Step 4.4) */
if (!BN_is_one(g)) {
*status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
ret = 1;
goto err;
}
}
/* (Step 4.5) z = b^m mod w */
if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))
goto err;
/* (Step 4.6) if (z = 1 or z = w-1) */
if (BN_is_one(z) || BN_cmp(z, w1) == 0)
goto outer_loop;
/* (Step 4.7) for j = 1 to a-1 */
for (j = 1; j < a ; ++j) {
/* (Step 4.7.1 - 4.7.2) x = z. z = x^2 mod w */
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
goto err;
/* (Step 4.7.3) */
if (BN_cmp(z, w1) == 0)
goto outer_loop;
/* (Step 4.7.4) */
if (BN_is_one(z))
goto composite;
}
/* At this point z = b^((w-1)/2) mod w */
/* (Steps 4.8 - 4.9) x = z, z = x^2 mod w */
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
goto err;
/* (Step 4.10) */
if (BN_is_one(z))
goto composite;
/* (Step 4.11) x = b^(w-1) mod w */
if (!BN_copy(x, z))
goto err;
composite:
if (enhanced) {
/* (Step 4.1.2) g = GCD(x-1, w) */
if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))
goto err;
/* (Steps 4.1.3 - 4.1.4) */
if (BN_is_one(g))
*status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;
else
*status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
} else {
*status = BN_PRIMETEST_COMPOSITE;
}
ret = 1;
goto err;
outer_loop: ;
/* (Step 4.1.5) */
if (!BN_GENCB_call(cb, 1, i))
goto err;
}
/* (Step 5) */
*status = BN_PRIMETEST_PROBABLY_PRIME;
ret = 1;
err:
BN_clear(g);
BN_clear(w1);
BN_clear(w3);
BN_clear(x);
BN_clear(m);
BN_clear(z);
BN_clear(b);
BN_CTX_end(ctx);
BN_MONT_CTX_free(mont);
return ret;
}
