fp_generic.c

/********************************************************************************************
* SIDH: an efficient supersingular isogeny cryptography library
*
* Abstract: portable modular arithmetic for P503
*********************************************************************************************/

#include "../P503_internal.h"
#include "../../internal.h"


// Global constants
extern const uint64_t p503[NWORDS64_FIELD];
extern const uint64_t p503p1[NWORDS64_FIELD]; 
extern const uint64_t p503x2[NWORDS64_FIELD];  
extern const uint64_t p503x4[NWORDS64_FIELD];


__inline void mp_sub503_p2(const digit_t* a, const digit_t* b, digit_t* c)
{ // Multiprecision subtraction with correction with 2*p, c = a-b+2p.
    unsigned int i, borrow = 0;

    for (i = 0; i < NWORDS_FIELD; i++) {
        SUBC(borrow, a[i], b[i], borrow, c[i]); 
    }

    borrow = 0;
    for (i = 0; i < NWORDS_FIELD; i++) {
        ADDC(borrow, c[i], ((digit_t*)p503x2)[i], borrow, c[i]); 
    }
} 


__inline void mp_sub503_p4(const digit_t* a, const digit_t* b, digit_t* c)
{ // Multiprecision subtraction with correction with 4*p, c = a-b+4p.
    unsigned int i, borrow = 0;

    for (i = 0; i < NWORDS_FIELD; i++) {
        SUBC(borrow, a[i], b[i], borrow, c[i]); 
    }

    borrow = 0;
    for (i = 0; i < NWORDS_FIELD; i++) {
        ADDC(borrow, c[i], ((digit_t*)p503x4)[i], borrow, c[i]); 
    }
} 


__inline void fpadd503(const digit_t* a, const digit_t* b, digit_t* c)
{ // Modular addition, c = a+b mod p503.
  // Inputs: a, b in [0, 2*p503-1] 
  // Output: c in [0, 2*p503-1] 
    unsigned int i, carry = 0;
    digit_t mask;

    for (i = 0; i < NWORDS_FIELD; i++) {
        ADDC(carry, a[i], b[i], carry, c[i]); 
    }

    carry = 0;
    for (i = 0; i < NWORDS_FIELD; i++) {
        SUBC(carry, c[i], ((digit_t*)p503x2)[i], carry, c[i]); 
    }
    mask = 0 - (digit_t)carry;

    carry = 0;
    for (i = 0; i < NWORDS_FIELD; i++) {
        ADDC(carry, c[i], ((digit_t*)p503x2)[i] & mask, carry, c[i]); 
    }
} 


__inline void fpsub503(const digit_t* a, const digit_t* b, digit_t* c)
{ // Modular subtraction, c = a-b mod p503.
  // Inputs: a, b in [0, 2*p503-1] 
  // Output: c in [0, 2*p503-1] 
    unsigned int i, borrow = 0;
    digit_t mask;

    for (i = 0; i < NWORDS_FIELD; i++) {
        SUBC(borrow, a[i], b[i], borrow, c[i]); 
    }
    mask = 0 - (digit_t)borrow;

    borrow = 0;
    for (i = 0; i < NWORDS_FIELD; i++) {
        ADDC(borrow, c[i], ((digit_t*)p503x2)[i] & mask, borrow, c[i]); 
    }
}


__inline void fpneg503(digit_t* a)
{ // Modular negation, a = -a mod p503.
  // Input/output: a in [0, 2*p503-1] 
    unsigned int i, borrow = 0;

    for (i = 0; i < NWORDS_FIELD; i++) {
        SUBC(borrow, ((digit_t*)p503x2)[i], a[i], borrow, a[i]); 
    }
}


void fpdiv2_503(const digit_t* a, digit_t* c)
{ // Modular division by two, c = a/2 mod p503.
  // Input : a in [0, 2*p503-1] 
  // Output: c in [0, 2*p503-1] 
    unsigned int i, carry = 0;
    digit_t mask;
        
    mask = 0 - (digit_t)(a[0] & 1);    // If a is odd compute a+p503
    for (i = 0; i < NWORDS_FIELD; i++) {
        ADDC(carry, a[i], ((digit_t*)p503)[i] & mask, carry, c[i]); 
    }

    mp_shiftr1(c, NWORDS_FIELD);
} 


void fpcorrection503(digit_t* a)
{ // Modular correction to reduce field element a in [0, 2*p503-1] to [0, p503-1].
    unsigned int i, borrow = 0;
    digit_t mask;

    for (i = 0; i < NWORDS_FIELD; i++) {
        SUBC(borrow, a[i], ((digit_t*)p503)[i], borrow, a[i]); 
    }
    mask = 0 - (digit_t)borrow;

    borrow = 0;
    for (i = 0; i < NWORDS_FIELD; i++) {
        ADDC(borrow, a[i], ((digit_t*)p503)[i] & mask, borrow, a[i]); 
    }
}


void digit_x_digit(const digit_t a, const digit_t b, digit_t* c)
{ // Digit multiplication, digit * digit -> 2-digit result    
    register digit_t al, ah, bl, bh, temp;
    digit_t albl, albh, ahbl, ahbh, res1, res2, res3, carry;
    digit_t mask_low = (digit_t)(-1) >> (sizeof(digit_t)*4), mask_high = (digit_t)(-1) << (sizeof(digit_t)*4);

    al = a & mask_low;                        // Low part
    ah = a >> (sizeof(digit_t) * 4);          // High part
    bl = b & mask_low;
    bh = b >> (sizeof(digit_t) * 4);

    albl = al*bl;
    albh = al*bh;
    ahbl = ah*bl;
    ahbh = ah*bh;
    c[0] = albl & mask_low;                   // C00

    res1 = albl >> (sizeof(digit_t) * 4);
    res2 = ahbl & mask_low;
    res3 = albh & mask_low;  
    temp = res1 + res2 + res3;
    carry = temp >> (sizeof(digit_t) * 4);
    c[0] ^= temp << (sizeof(digit_t) * 4);    // C01   

    res1 = ahbl >> (sizeof(digit_t) * 4);
    res2 = albh >> (sizeof(digit_t) * 4);
    res3 = ahbh & mask_low;
    temp = res1 + res2 + res3 + carry;
    c[1] = temp & mask_low;                   // C10 
    carry = temp & mask_high; 
    c[1] ^= (ahbh & mask_high) + carry;       // C11
}


void mp_mul(const digit_t* a, const digit_t* b, digit_t* c, const unsigned int nwords)
{ // Multiprecision comba multiply, c = a*b, where lng(a) = lng(b) = nwords.   
    unsigned int i, j;
    digit_t t = 0, u = 0, v = 0, UV[2];
    unsigned int carry = 0;
    
    for (i = 0; i < nwords; i++) {
        for (j = 0; j <= i; j++) {
            MUL(a[j], b[i-j], UV+1, UV[0]); 
            ADDC(0, UV[0], v, carry, v); 
            ADDC(carry, UV[1], u, carry, u); 
            t += carry;
        }
        c[i] = v;
        v = u; 
        u = t;
        t = 0;
    }

    for (i = nwords; i < 2*nwords-1; i++) {
        for (j = i-nwords+1; j < nwords; j++) {
            MUL(a[j], b[i-j], UV+1, UV[0]); 
            ADDC(0, UV[0], v, carry, v); 
            ADDC(carry, UV[1], u, carry, u); 
            t += carry;
        }
        c[i] = v;
        v = u; 
        u = t;
        t = 0;
    }
    c[2*nwords-1] = v; 
}


void rdc_mont(digit_t* ma, digit_t* mc)
{ // Efficient Montgomery reduction using comba and exploiting the special form of the prime p503.
  // mc = ma*R^-1 mod p503x2, where R = 2^512.
  // If ma < 2^512*p503, the output mc is in the range [0, 2*p503-1].
  // ma is assumed to be in Montgomery representation.
    unsigned int i, j, carry, count = p503_ZERO_WORDS;
    digit_t UV[2], t = 0, u = 0, v = 0;

    for (i = 0; i < NWORDS_FIELD; i++) {
        mc[i] = 0;
    }

    for (i = 0; i < NWORDS_FIELD; i++) {
        for (j = 0; j < i; j++) {
            if (j < (i-p503_ZERO_WORDS+1)) { 
                MUL(mc[j], ((digit_t*)p503p1)[i-j], UV+1, UV[0]);
                ADDC(0, UV[0], v, carry, v); 
                ADDC(carry, UV[1], u, carry, u); 
                t += carry; 
            }
        }
        ADDC(0, v, ma[i], carry, v); 
        ADDC(carry, u, 0, carry, u); 
        t += carry; 
        mc[i] = v;
        v = u;
        u = t;
        t = 0;
    }    

    for (i = NWORDS_FIELD; i < 2*NWORDS_FIELD-1; i++) {
        if (count > 0) {
            count -= 1;
        }
        for (j = i-NWORDS_FIELD+1; j < NWORDS_FIELD; j++) {
            if (j < (NWORDS_FIELD-count)) { 
                MUL(mc[j], ((digit_t*)p503p1)[i-j], UV+1, UV[0]);
                ADDC(0, UV[0], v, carry, v); 
                ADDC(carry, UV[1], u, carry, u); 
                t += carry;
            }
        }
        ADDC(0, v, ma[i], carry, v); 
        ADDC(carry, u, 0, carry, u); 
        t += carry; 
        mc[i-NWORDS_FIELD] = v;
        v = u;
        u = t;
        t = 0;
    }
    ADDC(0, v, ma[2*NWORDS_FIELD-1], carry, v); 
    mc[NWORDS_FIELD-1] = v;
}