monty::monty( const vlong &M )
{
m = M;
N = 0; R = 1; while ( R < M ) { R += R; N += 1; }
R1 = modinv( R-m, m );
n1 = R - modinv( m, R );
}
int nCr_ferma(int n, int r, unsigned int p) {
if (r == 0)
return 1;
int fac[n + 1];
fac[0] = 1;
for (int i = 1; i <= n; ++i) {
fac[i] = fac[i-1] * i % p;
}
return (fac[n] * modinv(fac[r], p) % p * modinv(fac[n-r], p) % p) % p;
}
vlong private_key::decrypt( const vlong& cipher )
{
// Calculate values for performing decryption
// These could be cached, but the calculation is quite fast
vlong d = modinv( e, (p-(vlong)1)*(q-(vlong)1) );
vlong u = modinv( p, q );
vlong dp = d % (p-(vlong)1);
vlong dq = d % (q-(vlong)1);
// Apply chinese remainder theorem
vlong a = modexp( cipher % p, dp, p );
vlong b = modexp( cipher % q, dq, q );
if ( b < a ) b += q;
return a + p * ( ((b-a)*u) % q );
}
int RSAKey::Generate( int bits )
{
Bignum pm1, qm1, phi_n;
/*
* We don't generate e; we just use a standard one always.
*/
this->exponent = bignum_from_long(RSA_EXPONENT);
/*
* Generate p and q: primes with combined length `bits', not
* congruent to 1 modulo e. (Strictly speaking, we wanted (p-1)
* and e to be coprime, and (q-1) and e to be coprime, but in
* general that's slightly more fiddly to arrange. By choosing
* a prime e, we can simplify the criterion.)
*/
this->p = primegen(bits / 2, RSA_EXPONENT, 1, NULL, 1);
this->q = primegen(bits - bits / 2, RSA_EXPONENT, 1, NULL, 2);
/*
* Ensure p > q, by swapping them if not.
*/
if (bignum_cmp(this->p, this->q) < 0)
swap( p, q );
/*
* Now we have p, q and e. All we need to do now is work out
* the other helpful quantities: n=pq, d=e^-1 mod (p-1)(q-1),
* and (q^-1 mod p).
*/
this->modulus = bigmul(this->p, this->q);
pm1 = copybn(this->p);
decbn(pm1);
qm1 = copybn(this->q);
decbn(qm1);
phi_n = bigmul(pm1, qm1);
freebn(pm1);
freebn(qm1);
this->private_exponent = modinv(this->exponent, phi_n);
this->iqmp = modinv(this->q, this->p);
/*
* Clean up temporary numbers.
*/
freebn(phi_n);
return 1;
}
//创建公私钥
void rsa_create_key(vlong& e, vlong& d , vlong& m )
{
unsigned char prand[2][128],tc;
UINT i,j, nCount = 64;
srand((unsigned)time(NULL));
for(i=0; i<2; i++)
{
for(j=0; j<nCount; j++)
{
tc = (char)(0x41+rand()%0xAF);
prand[i][j] = tc;
}
prand[i][j]=0;
}
// Choose primes
prime_factory pf;
vlong s1 = from_str((const char*)prand[0]);
vlong s2 = from_str((const char*)prand[1]);
vlong p = pf.find_prime( s1 );
vlong q = pf.find_prime( s2 );
// Calculate public key
m = p*q;
e = 65537;
while ( gcd(p-1,e) != 1 || gcd(q-1,e) != 1 ) e += 2;
// Calculate private key
d = modinv( e, (p-1)*(q-1) );
}
//创建公私钥
void rsa_create_key(RSA_KEY& pub, RSA_PRIVATE_KEY& pri)
{
unsigned char prand[2][128],tc;
UINT i,j, nCount = 64;
srand((unsigned)time(NULL));
for(i=0; i<2; i++)
{
for(j=0; j<nCount; j++)
{
tc = (char)(0x41+rand()%0xAF);
prand[i][j] = tc;
}
prand[i][j]=0;
}
// Choose primes
prime_factory pf;
vlong s1 = from_str((const char*)prand[0]);
vlong s2 = from_str((const char*)prand[1]);
vlong tp = pf.find_prime( s1 );
vlong tq = pf.find_prime( s2 );
if ( tq > tp ) //make sure p > q
{
vlong tmp = tp;
tp = tq;
tq = tmp;
}
// Calculate public key
pub.modulus = tp*tq;
pub.key = 65537;
while ( gcd(tp-1,pub.key) != 1 || gcd(tq-1,pub.key) != 1 ) pub.key += 2;
// Calculate private key
pri.p=tp;
pri.q=tq;
pri.dp=modinv(pub.key,tp-1);
pri.dq=modinv(pub.key,tq-1);
pri.qinv=modinv(tq,tp);
}
static unsigned char *dss_sign(void *key, char *data, int datalen, int *siglen)
{
struct dss_key *dss = (struct dss_key *) key;
Bignum k, gkp, hash, kinv, hxr, r, s;
unsigned char digest[20];
unsigned char *bytes;
int nbytes, i;
SHA_Simple(data, datalen, digest);
k = dss_gen_k("DSA deterministic k generator", dss->q, dss->x,
digest, sizeof(digest));
kinv = modinv(k, dss->q); /* k^-1 mod q */
assert(kinv);
/*
* Now we have k, so just go ahead and compute the signature.
*/
gkp = modpow(dss->g, k, dss->p); /* g^k mod p */
r = bigmod(gkp, dss->q); /* r = (g^k mod p) mod q */
freebn(gkp);
hash = bignum_from_bytes(digest, 20);
hxr = bigmuladd(dss->x, r, hash); /* hash + x*r */
s = modmul(kinv, hxr, dss->q); /* s = k^-1 * (hash + x*r) mod q */
freebn(hxr);
freebn(kinv);
freebn(k);
freebn(hash);
/*
* Signature blob is
*
* string "ssh-dss"
* string two 20-byte numbers r and s, end to end
*
* i.e. 4+7 + 4+40 bytes.
*/
nbytes = 4 + 7 + 4 + 40;
bytes = snewn(nbytes, unsigned char);
PUT_32BIT(bytes, 7);
memcpy(bytes + 4, "ssh-dss", 7);
PUT_32BIT(bytes + 4 + 7, 40);
for (i = 0; i < 20; i++) {
bytes[4 + 7 + 4 + i] = bignum_byte(r, 19 - i);
bytes[4 + 7 + 4 + 20 + i] = bignum_byte(s, 19 - i);
}
freebn(r);
freebn(s);
*siglen = nbytes;
return bytes;
}
int main()
{
for (int m = 1; m < 10000; ++m) {
for (int a = 0; a < 2*m; ++a) {
if (gcd(a,m) % m == 1) {
assert((a * modinv(a,m)) % m == 1);
}
}
}
return 0;
}
void solve(int n, int m, int k){
long long sum = 0;
long long sdi2 = m;
for(int i=1;i<=k;i++){
sum+= (sdi2*pnr(n,i))%PRIME;;
long long den = i+1;
long long num = m-i;
sdi2 = (sdi2*num)%PRIME;
sdi2 = (sdi2*modinv(den,PRIME))%PRIME;
}
printf("%ld\n",(sum+PRIME)%PRIME);
}
static unsigned char *dss_sign(void *key, char *data, int datalen, int *siglen)
{
/*
* The basic DSS signing algorithm is:
*
* - invent a random k between 1 and q-1 (exclusive).
* - Compute r = (g^k mod p) mod q.
* - Compute s = k^-1 * (hash + x*r) mod q.
*
* This has the dangerous properties that:
*
* - if an attacker in possession of the public key _and_ the
* signature (for example, the host you just authenticated
* to) can guess your k, he can reverse the computation of s
* and work out x = r^-1 * (s*k - hash) mod q. That is, he
* can deduce the private half of your key, and masquerade
* as you for as long as the key is still valid.
*
* - since r is a function purely of k and the public key, if
* the attacker only has a _range of possibilities_ for k
* it's easy for him to work through them all and check each
* one against r; he'll never be unsure of whether he's got
* the right one.
*
* - if you ever sign two different hashes with the same k, it
* will be immediately obvious because the two signatures
* will have the same r, and moreover an attacker in
* possession of both signatures (and the public key of
* course) can compute k = (hash1-hash2) * (s1-s2)^-1 mod q,
* and from there deduce x as before.
*
* - the Bleichenbacher attack on DSA makes use of methods of
* generating k which are significantly non-uniformly
* distributed; in particular, generating a 160-bit random
* number and reducing it mod q is right out.
*
* For this reason we must be pretty careful about how we
* generate our k. Since this code runs on Windows, with no
* particularly good system entropy sources, we can't trust our
* RNG itself to produce properly unpredictable data. Hence, we
* use a totally different scheme instead.
*
* What we do is to take a SHA-512 (_big_) hash of the private
* key x, and then feed this into another SHA-512 hash that
* also includes the message hash being signed. That is:
*
* proto_k = SHA512 ( SHA512(x) || SHA160(message) )
*
* This number is 512 bits long, so reducing it mod q won't be
* noticeably non-uniform. So
*
* k = proto_k mod q
*
* This has the interesting property that it's _deterministic_:
* signing the same hash twice with the same key yields the
* same signature.
*
* Despite this determinism, it's still not predictable to an
* attacker, because in order to repeat the SHA-512
* construction that created it, the attacker would have to
* know the private key value x - and by assumption he doesn't,
* because if he knew that he wouldn't be attacking k!
*
* (This trick doesn't, _per se_, protect against reuse of k.
* Reuse of k is left to chance; all it does is prevent
* _excessively high_ chances of reuse of k due to entropy
* problems.)
*
* Thanks to Colin Plumb for the general idea of using x to
* ensure k is hard to guess, and to the Cambridge University
* Computer Security Group for helping to argue out all the
* fine details.
*/
struct dss_key *dss = (struct dss_key *) key;
SHA512_State ss;
unsigned char digest[20], digest512[64];
Bignum proto_k, k, gkp, hash, kinv, hxr, r, s;
unsigned char *bytes;
int nbytes, i;
SHA_Simple(data, datalen, digest);
/*
* Hash some identifying text plus x.
*/
SHA512_Init(&ss);
SHA512_Bytes(&ss, "DSA deterministic k generator", 30);
sha512_mpint(&ss, dss->x);
SHA512_Final(&ss, digest512);
/*
* Now hash that digest plus the message hash.
*/
SHA512_Init(&ss);
SHA512_Bytes(&ss, digest512, sizeof(digest512));
SHA512_Bytes(&ss, digest, sizeof(digest));
SHA512_Final(&ss, digest512);
memset(&ss, 0, sizeof(ss));
/*
* Now convert the result into a bignum, and reduce it mod q.
*/
proto_k = bignum_from_bytes(digest512, 64);
k = bigmod(proto_k, dss->q);
freebn(proto_k);
memset(digest512, 0, sizeof(digest512));
/*
* Now we have k, so just go ahead and compute the signature.
*/
gkp = modpow(dss->g, k, dss->p); /* g^k mod p */
r = bigmod(gkp, dss->q); /* r = (g^k mod p) mod q */
freebn(gkp);
hash = bignum_from_bytes(digest, 20);
kinv = modinv(k, dss->q); /* k^-1 mod q */
hxr = bigmuladd(dss->x, r, hash); /* hash + x*r */
s = modmul(kinv, hxr, dss->q); /* s = k^-1 * (hash + x*r) mod q */
freebn(hxr);
freebn(kinv);
freebn(hash);
/*
* Signature blob is
*
* string "ssh-dss"
* string two 20-byte numbers r and s, end to end
*
* i.e. 4+7 + 4+40 bytes.
*/
nbytes = 4 + 7 + 4 + 40;
bytes = snewn(nbytes, unsigned char);
PUT_32BIT(bytes, 7);
memcpy(bytes + 4, "ssh-dss", 7);
PUT_32BIT(bytes + 4 + 7, 40);
for (i = 0; i < 20; i++) {
bytes[4 + 7 + 4 + i] = bignum_byte(r, 19 - i);
bytes[4 + 7 + 4 + 20 + i] = bignum_byte(s, 19 - i);
}
freebn(r);
freebn(s);
*siglen = nbytes;
return bytes;
}
static int dss_verifysig(void *key, char *sig, int siglen,
char *data, int datalen)
{
struct dss_key *dss = (struct dss_key *) key;
char *p;
int slen;
char hash[20];
Bignum r, s, w, gu1p, yu2p, gu1yu2p, u1, u2, sha, v;
int ret;
if (!dss->p)
return 0;
#ifdef DEBUG_DSS
{
int i;
printf("sig:");
for (i = 0; i < siglen; i++)
printf(" %02x", (unsigned char) (sig[i]));
printf("\n");
}
#endif
/*
* Commercial SSH (2.0.13) and OpenSSH disagree over the format
* of a DSA signature. OpenSSH is in line with the IETF drafts:
* it uses a string "ssh-dss", followed by a 40-byte string
* containing two 160-bit integers end-to-end. Commercial SSH
* can't be bothered with the header bit, and considers a DSA
* signature blob to be _just_ the 40-byte string containing
* the two 160-bit integers. We tell them apart by measuring
* the length: length 40 means the commercial-SSH bug, anything
* else is assumed to be IETF-compliant.
*/
if (siglen != 40) { /* bug not present; read admin fields */
getstring(&sig, &siglen, &p, &slen);
if (!p || slen != 7 || memcmp(p, "ssh-dss", 7)) {
return 0;
}
sig += 4, siglen -= 4; /* skip yet another length field */
}
r = get160(&sig, &siglen);
s = get160(&sig, &siglen);
if (!r || !s)
return 0;
/*
* Step 1. w <- s^-1 mod q.
*/
w = modinv(s, dss->q);
/*
* Step 2. u1 <- SHA(message) * w mod q.
*/
SHA_Simple(data, datalen, (unsigned char *)hash);
p = hash;
slen = 20;
sha = get160(&p, &slen);
u1 = modmul(sha, w, dss->q);
/*
* Step 3. u2 <- r * w mod q.
*/
u2 = modmul(r, w, dss->q);
/*
* Step 4. v <- (g^u1 * y^u2 mod p) mod q.
*/
gu1p = modpow(dss->g, u1, dss->p);
yu2p = modpow(dss->y, u2, dss->p);
gu1yu2p = modmul(gu1p, yu2p, dss->p);
v = modmul(gu1yu2p, One, dss->q);
/*
* Step 5. v should now be equal to r.
*/
ret = !bignum_cmp(v, r);
freebn(w);
freebn(sha);
freebn(gu1p);
freebn(yu2p);
freebn(gu1yu2p);
freebn(v);
freebn(r);
freebn(s);
return ret;
}
/*
* This function is a wrapper on modpow(). It has the same effect
* as modpow(), but employs RSA blinding to protect against timing
* attacks.
*/
static Bignum rsa_privkey_op(Bignum input, struct RSAKey *key)
{
Bignum random, random_encrypted, random_inverse;
Bignum input_blinded, ret_blinded;
Bignum ret;
SHA512_State ss;
unsigned char digest512[64];
int digestused = lenof(digest512);
int hashseq = 0;
/*
* Start by inventing a random number chosen uniformly from the
* range 2..modulus-1. (We do this by preparing a random number
* of the right length and retrying if it's greater than the
* modulus, to prevent any potential Bleichenbacher-like
* attacks making use of the uneven distribution within the
* range that would arise from just reducing our number mod n.
* There are timing implications to the potential retries, of
* course, but all they tell you is the modulus, which you
* already knew.)
*
* To preserve determinism and avoid Pageant needing to share
* the random number pool, we actually generate this `random'
* number by hashing stuff with the private key.
*/
while (1) {
int bits, byte, bitsleft, v;
random = copybn(key->modulus);
/*
* Find the topmost set bit. (This function will return its
* index plus one.) Then we'll set all bits from that one
* downwards randomly.
*/
bits = bignum_bitcount(random);
byte = 0;
bitsleft = 0;
while (bits--) {
if (bitsleft <= 0) {
bitsleft = 8;
/*
* Conceptually the following few lines are equivalent to
* byte = random_byte();
*/
if (digestused >= lenof(digest512)) {
unsigned char seqbuf[4];
PUT_32BIT(seqbuf, hashseq);
pSHA512_Init(&ss);
SHA512_Bytes(&ss, "RSA deterministic blinding", 26);
SHA512_Bytes(&ss, seqbuf, sizeof(seqbuf));
sha512_mpint(&ss, key->private_exponent);
pSHA512_Final(&ss, digest512);
hashseq++;
/*
* Now hash that digest plus the signature
* input.
*/
pSHA512_Init(&ss);
SHA512_Bytes(&ss, digest512, sizeof(digest512));
sha512_mpint(&ss, input);
pSHA512_Final(&ss, digest512);
digestused = 0;
}
byte = digest512[digestused++];
}
v = byte & 1;
byte >>= 1;
bitsleft--;
bignum_set_bit(random, bits, v);
}
/*
* Now check that this number is strictly greater than
* zero, and strictly less than modulus.
*/
if (bignum_cmp(random, Zero) <= 0 ||
bignum_cmp(random, key->modulus) >= 0) {
freebn(random);
continue;
} else {
break;
}
}
/*
* RSA blinding relies on the fact that (xy)^d mod n is equal
* to (x^d mod n) * (y^d mod n) mod n. We invent a random pair
* y and y^d; then we multiply x by y, raise to the power d mod
* n as usual, and divide by y^d to recover x^d. Thus an
* attacker can't correlate the timing of the modpow with the
* input, because they don't know anything about the number
* that was input to the actual modpow.
*
* The clever bit is that we don't have to do a huge modpow to
* get y and y^d; we will use the number we just invented as
* _y^d_, and use the _public_ exponent to compute (y^d)^e = y
* from it, which is much faster to do.
*/
random_encrypted = modpow(random, key->exponent, key->modulus);
random_inverse = modinv(random, key->modulus);
input_blinded = modmul(input, random_encrypted, key->modulus);
ret_blinded = modpow(input_blinded, key->private_exponent, key->modulus);
ret = modmul(ret_blinded, random_inverse, key->modulus);
freebn(ret_blinded);
freebn(input_blinded);
freebn(random_inverse);
freebn(random_encrypted);
freebn(random);
return ret;
}
int
main(int argc, char *argv[])
{
KryState state = STATE_BADARG;
long mod_bits;
if (argc == 3 && !strcmp(argv[1], "-g")) {
// Key generation
state = STATE_KEYGEN;
mod_bits = strtol(argv[2], NULL, 10);
if (mod_bits < 16) {
state = STATE_BADARG;
fprintf(stderr, "Too small key length\n");
}
} else if (argc == 5 && !strcmp(argv[1], "-e")) {
// Encryption
state = STATE_ENCRYPTION;
} else if (argc == 5 && !strcmp(argv[1], "-d")) {
// Decryption
state = STATE_DECRYPTION;
} else if (argc == 5 && !strcmp(argv[1], "-b")) {
// RSA cracking
state = STATE_CRACKING;
}
if (state == STATE_BADARG) {
// Someting is messed up
print_help(argv[0]);
return EXIT_FAILURE;
}
// Prepare pseudo number generator stuff
gmp_randstate_t randstate;
gmp_randinit_default(randstate);
//gmp_randseed_ui(randstate, 1);
gmp_randseed_ui(randstate, time(NULL));
int rc = EXIT_SUCCESS;
mpz_t res;
mpz_init(res);
switch (state) {
case STATE_KEYGEN: {
mpz_t p, q, n, e; // For d use res var
mpz_init(p);
mpz_init(q);
mpz_init(n);
mpz_init(e);
rc = keygen(p, q, n, e, res, mod_bits, randstate);
gmp_printf("0x%Zx 0x%Zx 0x%Zx 0x%Zx 0x%Zx\n", p, q, n, e, res);
mpz_clear(p);
mpz_clear(q);
mpz_clear(n);
mpz_clear(e);
break;
}
case STATE_ENCRYPTION:
// Argv: [2]=e; [3]=n; [4]=m;
rc = encrypt(res, argv[2], argv[3], argv[4]);
gmp_printf("0x%Zx\n", res);
break;
case STATE_DECRYPTION:
// Argv: [2]=d; [3]=n; [4]=c;
rc = decrypt(res, argv[2], argv[3], argv[4]);
gmp_printf("0x%Zx\n", res);
break;
case STATE_CRACKING: {
// Argv: [2]=e; [3]=n; [4]=c;
mpz_t p, q, p_, q_, e, d;
mpz_init(p);
mpz_init(q);
mpz_init(p_);
mpz_init(q_);
mpz_init(e);
mpz_init(d);
rc = factorization(p, q, argv[3]);
if (rc != 0)
goto cracking_cleanup;
// phi = (p-1) * (q-1)
mpz_sub_ui(p_, p, 1L);
mpz_sub_ui(q_, q, 1L);
mpz_mul(res, p_, q_);
// d = e^1 mod phi
mpz_set_str(e, argv[2], 0);
modinv(d, e, res);
rc = decrypt_mpz_d(res, d, argv[3], argv[4]);
gmp_printf("0x%Zx 0x%Zx 0x%Zx\n", p, q, res);
cracking_cleanup:
mpz_clear(p);
mpz_clear(q);
mpz_clear(p_);
mpz_clear(q_);
mpz_clear(e);
mpz_clear(d);
break;
}
case STATE_BADARG:
break;
}
mpz_clear(res);
gmp_randclear(randstate);
return rc;
}
uint64_t moddiv(uint64_t a, uint64_t b, uint64_t m) {
uint64_t i = modinv(b, m);
return a*i%m;
}
int main() {
printf("%d\n", modinv(3, 4));
return 0;
}
int rsa_generate(struct RSAKey *key, struct RSAAux *aux, int bits,
progfn_t pfn, void *pfnparam) {
Bignum pm1, qm1, phi_n;
/*
* Set up the phase limits for the progress report. We do this
* by passing minus the phase number.
*
* For prime generation: our initial filter finds things
* coprime to everything below 2^16. Computing the product of
* (p-1)/p for all prime p below 2^16 gives about 20.33; so
* among B-bit integers, one in every 20.33 will get through
* the initial filter to be a candidate prime.
*
* Meanwhile, we are searching for primes in the region of 2^B;
* since pi(x) ~ x/log(x), when x is in the region of 2^B, the
* prime density will be d/dx pi(x) ~ 1/log(B), i.e. about
* 1/0.6931B. So the chance of any given candidate being prime
* is 20.33/0.6931B, which is roughly 29.34 divided by B.
*
* So now we have this probability P, we're looking at an
* exponential distribution with parameter P: we will manage in
* one attempt with probability P, in two with probability
* P(1-P), in three with probability P(1-P)^2, etc. The
* probability that we have still not managed to find a prime
* after N attempts is (1-P)^N.
*
* We therefore inform the progress indicator of the number B
* (29.34/B), so that it knows how much to increment by each
* time. We do this in 16-bit fixed point, so 29.34 becomes
* 0x1D.57C4.
*/
pfn(pfnparam, -1, -0x1D57C4/(bits/2));
pfn(pfnparam, -2, -0x1D57C4/(bits-bits/2));
pfn(pfnparam, -3, 5);
/*
* We don't generate e; we just use a standard one always.
*/
key->exponent = bignum_from_short(RSA_EXPONENT);
/*
* Generate p and q: primes with combined length `bits', not
* congruent to 1 modulo e. (Strictly speaking, we wanted (p-1)
* and e to be coprime, and (q-1) and e to be coprime, but in
* general that's slightly more fiddly to arrange. By choosing
* a prime e, we can simplify the criterion.)
*/
aux->p = primegen(bits/2, RSA_EXPONENT, 1, 1, pfn, pfnparam);
aux->q = primegen(bits - bits/2, RSA_EXPONENT, 1, 2, pfn, pfnparam);
/*
* Ensure p > q, by swapping them if not.
*/
if (bignum_cmp(aux->p, aux->q) < 0) {
Bignum t = aux->p;
aux->p = aux->q;
aux->q = t;
}
/*
* Now we have p, q and e. All we need to do now is work out
* the other helpful quantities: n=pq, d=e^-1 mod (p-1)(q-1),
* and (q^-1 mod p).
*/
pfn(pfnparam, 3, 1);
key->modulus = bigmul(aux->p, aux->q);
pfn(pfnparam, 3, 2);
pm1 = copybn(aux->p);
decbn(pm1);
qm1 = copybn(aux->q);
decbn(qm1);
phi_n = bigmul(pm1, qm1);
pfn(pfnparam, 3, 3);
freebn(pm1);
freebn(qm1);
key->private_exponent = modinv(key->exponent, phi_n);
pfn(pfnparam, 3, 4);
aux->iqmp = modinv(aux->q, aux->p);
pfn(pfnparam, 3, 5);
/*
* Clean up temporary numbers.
*/
freebn(phi_n);
return 1;
}