if(mpz_cmp_ui(t1,3)==0)//Algorithm 3.36, a specialization of algorithm 3.34
{
//Calculate R = a^((P+1)/4)
mpz_add_ui(t1,P,1);//t1 = P - 1
mpz_divexact_ui(t3,t1,4);//t3 = t1 / 4
number_theory_exp_modp(R,a,t3,P);//R = a^t3 mod P
}else{//Algorithm 3.37, a specialization of algorithm 3.34
//If P mod 8 = 5
mpz_mod_ui(t1,P,8);
if(mpz_cmp_ui(t1,5)==0)
{
//Initialize d
mpz_td;mpz_init(d);
//Calculate d = a^((P-1)/4)
mpz_sub_ui(t1,P,1);//t1 = P - 1
mpz_divexact_ui(t3,t1,4);//t3 = t1 / 4
number_theory_exp_modp(d,a,t3,P);//d = a^t3 mod P
//If d = 1
if(mpz_cmp_ui(d,1)==0)
{
//Calculate R = a^((P+3)/8)
mpz_add_ui(t1,P,3);//t1 = P - 3
mpz_divexact_ui(t3,t1,8);//t3 = t1 / 8
number_theory_exp_modp(R,a,t3,P);//R = a^t3 mod P
}else{
//If d = P - 1
mpz_sub_ui(t1,P,1);
if(mpz_cmp(d,t1)==0){
//Calculate R = 2a*(4a)^((P-5)/8)
mpz_mul_ui(t1,a,4);//t1 = 4*a
mpz_mod(t4,t1,P);//t4 = t1 mod P
mpz_sub_ui(t1,P,5);//t1 = P - 5
mpz_divexact_ui(t3,t1,8);//t3 = t1 / 8
number_theory_exp_modp(t1,t4,t3,P);//t1 = (t4)^t3 mod P
mpz_mul_ui(t2,a,2);//t2 = 2*a
mpz_mod(t3,t2,P);//t3 = t2 mod P
mpz_mul(t2,t1,t3);//t2 = t1*t2
mpz_mod(R,t2,P);//R = t2 mod P
}
}
//Clear d
mpz_clear(d);
}else{//Algorithm 3.34
//Select b random quadratic nonresidue
mpz_tb;mpz_init(b);
gmp_randstate_trstate;//Initialize random algorithm
gmp_randinit_default(rstate);
do
mpz_urandomm(b,rstate,P);
while(number_theory_legendre(b,P)!=-1);
gmp_randclear(rstate);
//Find s and t, such as p-1 = 2^s*t, where t is odd
mpz_sub_ui(t1,P,1);//t1 = p-1
unsignedlongints=mpz_scan1(t1,0);
/* Scans the binary representation of t1 for 1 from behind, this gives us the
* number of times t1 can be devided with 2 before it gives an odd. This bit
* manipulation ought to be faster than repeated division by 2.
* Example:
* prime = 113 binary = 1110001
* prime - 1 = 112 binary = 1110000
* 112 / 2^4 = 7, 7 is odd.
*/
mpz_ui_pow_ui(t2,2,s);//t2 = 2^s
mpz_tt;mpz_init(t);
mpz_divexact(t,t1,t2);//t = t1 / t2
//Computation of a^-1 mod p
mpz_ta_inv;mpz_init(a_inv);
number_theory_inverse(a_inv,a,P);
//Initialize variable for c and d
mpz_tc;mpz_init(c);
mpz_td;mpz_init(d);
//Set c = b^t mod p
number_theory_exp_modp(c,b,t,P);
//Set R = a^((t+1)/2) mod p
mpz_add_ui(t1,t,1);//t1 = t+1
mpz_divexact_ui(t2,t1,2);//t2 = t1 / 2
number_theory_exp_modp(R,a,t2,P);//R = a^t2 mod p
unsignedlonginti;
for(i=1;i<s;i++)
{
//Set d = (R²*a_inv)^(2^(s-i-1)) mod p
number_theory_exp_modp_ui(t1,R,2,P);//t1 = R²
mpz_mul(t2,t1,a_inv);//t2 = t1 * a_inv
mpz_mod(t5,t2,P);//t5 = t2 mod p
mpz_set_ui(t1,s-i-1);//t1 = s-i-1
mpz_set_ui(t2,2);//t2 = 2
number_theory_exp_modp(t3,t2,t1,P);//t3 = t2^t1 mod p
number_theory_exp_modp(d,t5,t3,P);//d = t5^t3 mod p
//If d-(-1) mod p == 0, since d<p then we can use P-1 == d instead
mpz_sub_ui(t1,P,1);
if(mpz_cmp(d,t1)==0)
{
//Set R = R*c mod p
mpz_mul(t1,R,c);//t1 = R*c
mpz_mod(R,t1,P);//R = t1 mod p
}
//Set c = c² mod p
number_theory_exp_modp_ui(t1,c,2,P);//t1 = c² mod p
mpz_set(c,t1);//c = t1
}
//Clear variables
mpz_clear(b);
mpz_clear(t);
mpz_clear(a_inv);
mpz_clear(c);
mpz_clear(d);
}
}
}
//TODO: implement algorithm 3.39
/*Algorithm 3.39 requires operations on the polynomial field Fx over F, and polynomial exponentiation, thus polynomial multiplication and reduction. According to Handbook of applied cryptography this algorithm should be faster than 3.34, when s in p-1 = 2^s*t, where t is odd, is large. But I've decided to settle with the two specializations of 3.34 and algorithm 3.34.*/
//Clear variables
mpz_clear(t1);
mpz_clear(t2);
mpz_clear(t3);
mpz_clear(t4);
mpz_clear(t5);
}
}
/*Calculate the multiplicative inverse of a mod p, using the extended euclidean algorithm
/*Calculate the multiplicative inverse of a mod p, using the extended euclidean algorithm
*Handbook of applied cryptography: Algorithm 2.107
mpz_mod(lastx,t1,P);//We must keep it mod p, so why not just do it where instead of using set
//temp := y
//y := lasty-quotient*y
//lasty := temp
mpz_set(t1,y);
mpz_mul(t2,q,y);
mpz_sub(y,lasty,t2);
mpz_mod(lasty,t1,P);//We must keep it mod p, so why not just do it where instead of using set
//Set b = r
mpz_set(b,r);
}
/*d = a, greatest common divisitor
*lastx = x
*lasty = y
*in d = a*x+b*y
*Thus x is the multiplicative inverse of a mod b
*if d = 1, since otherwise there's no mulitplicative inverse.
*But when b is a prime, a must be coprime thus d=1
*/
//Set the result
mpz_set(R,lastx);
//Clear variables
mpz_clear(a);
mpz_clear(b);
mpz_clear(r);
mpz_clear(q);
mpz_clear(x);
mpz_clear(y);
mpz_clear(lastx);
mpz_clear(lasty);
mpz_clear(t1);
mpz_clear(t2);
#endif
}
}
/*Calculates the legendre symbol of a and p
/*Calculates the legendre symbol of a and p
*Handbook of applied cryptography: Fact 2.146 */
*Handbook of applied cryptography: Fact 2.146 */
intnumber_theory_legendre(mpz_ta,mpz_tp)
intnumber_theory_legendre(mpz_ta,mpz_tp)
{
{
#if EXTERNAL_NUMBER_THEORY_IMPLEMENTATION
//Do this using gmp number theory implementation
returnmpz_legendre(a,p);
#else
//Initializing variables
mpz_tt1;mpz_init(t1);
mpz_tt2;mpz_init(t2);
mpz_tt3;mpz_init(t3);
//Legendre = a ^ ((p-1)/2) mod p
returnmpz_legendre(a,p);
mpz_sub_ui(t1,p,1);//t1 = p - 1
mpz_set_ui(t2,2);//t2 = 2
mpz_divexact(t3,t1,t2);//t3 = t1 / 2
number_theory_exp_modp(t2,a,t3,p);//t2 = a^t3 mod p
//Store return value, so we can release memory
intvalue;
/*Exponentiation modulo a prime, can't give a negativ number, hence -1 can't be the result however if -1 was suppose to be the result, the result must be p-1, therefore we shall check if t2 == t1 since t1 is still p-1
