/*********************************************************************************** * FourQlib: a high-performance crypto library based on the elliptic curve FourQ * * Copyright (c) Microsoft Corporation. All rights reserved. * * Abstract: internal header file * * This code is based on the paper "FourQ: four-dimensional decompositions on a * Q-curve over the Mersenne prime" by Craig Costello and Patrick Longa, in Advances * in Cryptology - ASIACRYPT, 2015. * Preprint available at http://eprint.iacr.org/2015/565. ************************************************************************************/ #ifndef __FOURQ_INTERNAL_H__ #define __FOURQ_INTERNAL_H__ // For C++ #ifdef __cplusplus extern "C" { #endif #include "FourQ_api.h" // Extended datatype definition typedef uint64_t uint128_t[2]; // Basic parameters for fixed-base scalar multiplication #define NBITS_ORDER_PLUS_ONE 246+1 #define E_FIXEDBASE (NBITS_ORDER_PLUS_ONE + W_FIXEDBASE*V_FIXEDBASE - 1)/(W_FIXEDBASE*V_FIXEDBASE) #define D_FIXEDBASE E_FIXEDBASE*V_FIXEDBASE #define L_FIXEDBASE D_FIXEDBASE*W_FIXEDBASE #define NPOINTS_FIXEDBASE V_FIXEDBASE*(1 << (W_FIXEDBASE-1)) #define VPOINTS_FIXEDBASE (1 << (W_FIXEDBASE-1)) #if (NBITS_ORDER_PLUS_ONE-L_FIXEDBASE == 0) // This parameter selection is not supported #error -- "Unsupported parameter selection for fixed-base scalar multiplication" #endif // Basic parameters for double scalar multiplication #define NPOINTS_DOUBLEMUL_WP (1 << (WP_DOUBLEBASE-2)) #define NPOINTS_DOUBLEMUL_WQ (1 << (WQ_DOUBLEBASE-2)) // FourQ's point representations typedef struct { f2elm_t x; f2elm_t y; f2elm_t z; f2elm_t t; } point_extedwards; // Point representation in extended twisted Edwards coordinates. typedef point_extedwards point_extedwards_t[1]; typedef struct { f2elm_t x; f2elm_t y; f2elm_t z; f2elm_t ta; f2elm_t tb; } point_extproj; // Point representation in extended coordinates. typedef point_extproj point_extproj_t[1]; typedef struct { f2elm_t xy; f2elm_t yx; f2elm_t z2; f2elm_t t2; } point_extproj_precomp; // Point representation in extended coordinates (for precomputed points). typedef point_extproj_precomp point_extproj_precomp_t[1]; typedef struct { f2elm_t xy; f2elm_t yx; f2elm_t t2; } point_precomp; // Point representation in extended affine coordinates (for precomputed points). typedef point_precomp point_precomp_t[1]; /********************** Constant-time unsigned comparisons ***********************/ // The following functions return 1 (TRUE) if condition is true, 0 (FALSE) otherwise static __inline unsigned int is_digit_nonzero_ct(digit_t x) { // Is x != 0? return (unsigned int)((x | (0-x)) >> (RADIX-1)); } static __inline unsigned int is_digit_zero_ct(digit_t x) { // Is x = 0? return (unsigned int)(1 ^ is_digit_nonzero_ct(x)); } static __inline unsigned int is_digit_lessthan_ct(digit_t x, digit_t y) { // Is x < y? return (unsigned int)((x ^ ((x ^ y) | ((x - y) ^ y))) >> (RADIX-1)); } /********************** Macros for digit operations **********************/ // Digit multiplication #define MUL(multiplier, multiplicand, hi, lo) \ digit_x_digit((multiplier), (multiplicand), &(lo)); // Digit addition with carry #define ADDC(carryIn, addend1, addend2, carryOut, sumOut) \ { digit_t tempReg = (addend1) + (digit_t)(carryIn); \ (sumOut) = (addend2) + tempReg; \ (carryOut) = (is_digit_lessthan_ct(tempReg, (digit_t)(carryIn)) | is_digit_lessthan_ct((sumOut), tempReg)); } // Digit subtraction with borrow #define SUBC(borrowIn, minuend, subtrahend, borrowOut, differenceOut) \ { digit_t tempReg = (minuend) - (subtrahend); \ unsigned int borrowReg = (is_digit_lessthan_ct((minuend), (subtrahend)) | ((borrowIn) & is_digit_zero_ct(tempReg))); \ (differenceOut) = tempReg - (digit_t)(borrowIn); \ (borrowOut) = borrowReg; } // Shift right with flexible datatype #define SHIFTR(highIn, lowIn, shift, shiftOut, DigitSize) \ (shiftOut) = ((lowIn) >> (shift)) ^ ((highIn) << (DigitSize - (shift))); // 64x64-bit multiplication #define MUL128(multiplier, multiplicand, product) \ mp_mul((digit_t*)&(multiplier), (digit_t*)&(multiplicand), (digit_t*)&(product), NWORDS_FIELD/2); // 128-bit addition, inputs < 2^127 #define ADD128(addend1, addend2, addition) \ mp_add((digit_t*)(addend1), (digit_t*)(addend2), (digit_t*)(addition), NWORDS_FIELD); // 128-bit addition with output carry #define ADC128(addend1, addend2, carry, addition) \ (carry) = mp_add((digit_t*)(addend1), (digit_t*)(addend2), (digit_t*)(addition), NWORDS_FIELD); /**************** Function prototypes ****************/ /************* Arithmetic functions modulo the curve order **************/ // Converting to Montgomery representation void to_Montgomery(const digit_t* ma, digit_t* c); // Converting from Montgomery to standard representation void from_Montgomery(const digit_t* a, digit_t* mc); // 256-bit Montgomery multiplication modulo the curve order void Montgomery_multiply_mod_order(const digit_t* ma, const digit_t* mb, digit_t* mc); // Addition modulo the curve order, c = a+b mod order void add_mod_order(const digit_t* a, const digit_t* b, digit_t* c); // Subtraction modulo the curve order, c = a-b mod order void subtract_mod_order(const digit_t* a, const digit_t* b, digit_t* c); // Reduction modulo the order using Montgomery arithmetic internally void modulo_order(digit_t* a, digit_t* c); /************* Multiprecision functions **************/ // Check if multiprecision element is zero bool is_zero_ct(digit_t* a, unsigned int nwords); // Multiprecision addition, c = a+b. Returns the carry bit unsigned int mp_add(digit_t* a, digit_t* b, digit_t* c, unsigned int nwords); // Schoolbook multiprecision multiply, c = a*b void mp_mul(const digit_t* a, const digit_t* b, digit_t* c, const unsigned int nwords); // Multiprecision subtraction, c = a-b. Returns the borrow bit unsigned int subtract(const digit_t* a, const digit_t* b, digit_t* c, const unsigned int nwords); // Clear "nwords" integer-size digits from memory extern void clear_words(void* mem, unsigned int nwords); /************ Field arithmetic functions *************/ // Copy of a field element, c = a void fpcopy1271(felm_t a, felm_t c); // Field negation, a = -a mod p void fpneg1271(felm_t a); // Integer multiplication, c = a*b void mul1271_a(felm_t a, felm_t b, felm_t c); // Modular correction, a = a mod p void mod1271(felm_t a); void mod1271_a(felm_t a, felm_t c); // Field addition, c = a+b mod p void fpadd1271(felm_t a, felm_t b, felm_t c); void fpadd1271_a(felm_t a, felm_t b, felm_t c); // Field subtraction, c = a-b mod p void fpsub1271(felm_t a, felm_t b, felm_t c); void fpsub1271_a(felm_t a, felm_t b, felm_t c); // Field division by two, c = a/2 mod p void fpdiv1271(felm_t a); // Field multiplication, c = a*b mod p void fpmul1271(felm_t a, felm_t b, felm_t c); // Field squaring, c = a^2 mod p void fpsqr1271(felm_t a, felm_t c); // Field inversion, af = a^-1 = a^(p-2) mod p void fpinv1271(felm_t a); // Exponentiation over GF(p), af = a^(125-1) void fpexp1251(felm_t a, felm_t af); /************ Quadratic extension field arithmetic functions *************/ // Zeroing a quadratic extension field element, a=0 void fp2zero1271(f2elm_t a); // Copy quadratic extension field element, c = a void fp2copy1271(f2elm_t a, f2elm_t c); // Quadratic extension field negation, a = -a in GF((2^127-1)^2) void fp2neg1271(f2elm_t a); // Quadratic extension field addition, c = a+b in GF((2^127-1)^2) void fp2add1271(f2elm_t a, f2elm_t b, f2elm_t c); void fp2add1271_a(f2elm_t a, f2elm_t b, f2elm_t c); // Quadratic extension field subtraction, c = a-b in GF((2^127-1)^2) void fp2sub1271(f2elm_t a, f2elm_t b, f2elm_t c); void fp2sub1271_a(f2elm_t a, f2elm_t b, f2elm_t c); // Quadratic extension field multiplication, c = a*b in GF((2^127-1)^2) void fp2mul1271(f2elm_t a, f2elm_t b, f2elm_t c); void fp2mul1271_a(f2elm_t a, f2elm_t b, f2elm_t c); // Quadratic extension field squaring, c = a^2 in GF((2^127-1)^2) void fp2sqr1271(f2elm_t a, f2elm_t c); void fp2sqr1271_a(f2elm_t a, f2elm_t c); // Quadratic extension field inversion, af = a^-1 = a^(p-2) in GF((2^127-1)^2) void fp2inv1271(f2elm_t a); /************ Curve and recoding functions *************/ // Normalize projective twisted Edwards point Q = (X,Y,Z) -> P = (x,y) void eccnorm(point_extedwards_t P, point_t Q); // Normalize two projective points, including full reduction void eccnorm2(point_extedwards_t P, point_t Q, point_extedwards_t R, point_t S); // Conversion from representation (X,Y,Z,Ta,Tb) to (X+Y,Y-X,2Z,2dT), where T = Ta*Tb void R1_to_R2(point_extproj_t P, point_extproj_precomp_t Q); // Point doubling 2P void eccdouble(point_extedwards_t P); // Complete point addition P = P+Q or P = P+P void eccadd(point_extedwards_t P, point_extedwards_t Q, point_extedwards_t R); void eccadd_core(point_extedwards_t P, point_extedwards_t Q, point_extedwards_t R, f2elm_t Ta, f2elm_t Tb); // Psi mapping of a point, P = psi(P) void ecc_psi(point_extedwards_t P); // Phi mapping of a point, P = phi(P) void ecc_phi(point_extedwards_t P); // Scalar decomposition void decompose(uint64_t* k, uint64_t* scalars); // Generate random field element in [0, 2^127-1] void random_felmt(felm_t random); // Scalar randomization for variable-base scalar multiplication void randomize(uint64_t* scalars, unsigned char* r, uint128_t* random_scalars); // Randomization of point coordinates using a random field element void felmt_randomize_point(point_extedwards_t P, felm_t random); // Randomization of all the point coordinates in the precomputed table using a random element in GF(p^2) void randomize_table(point_extedwards_t* Table, felm_t random); // Recoding sub-scalars for use in the variable-base scalar multiplication void recode(uint128_t* scalars, unsigned int* digits); // Computes the fixed window representation of scalar void fixed_window_recode(uint64_t* scalar, unsigned int* digits, unsigned int* sign_masks); // Convert scalar to odd if even using the prime subgroup order r void conversion_to_odd(digit_t* k, digit_t* k_odd); // Co-factor clearing void cofactor_clearing(point_extedwards_t P); // Reduction modulo the order using Montgomery arithmetic void modulo_order(digit_t* a, digit_t* c); // Generation of the precomputation table used by the variable-base scalar multiplication ecc_mul_SCA_secure() void ecc_precomp(point_extedwards_t P, point_extedwards_t R, point_extedwards_t *Table); // Constant-time table lookup to extract an extended twisted Edwards point (X:Y:Z:T) from the precomputed table void table_lookup_1x16(point_extedwards_t* table, point_extedwards_t P, unsigned int digit); // Modular correction of input coordinates and conversion to representation (X,Y,Z,Ta,Tb) void point_setup(point_t P, point_extedwards_t Q); // Point validation: check if point lies on the curve bool ecc_point_validate(point_extedwards_t P); // Output error/success message for a given ECCRYPTO_STATUS const char* FourQ_get_error_message(ECCRYPTO_STATUS Status); // Generation of the precomputation table used internally by the double scalar multiplication function ecc_mul_double() void ecc_precomp_double(point_extedwards_t P, point_extedwards_t* Table, unsigned int npoints); // Computes wNAF recoding of a scalar void wNAF_recode(uint64_t scalar, unsigned int w, int* digits); // Encode point P void encode(point_t P, unsigned char* Pencoded); // Decode point P ECCRYPTO_STATUS decode(const unsigned char* Pencoded, point_t P); /************ Functions based on macros *************/ // Copy extended projective point Q = (X:Y:Z:T) to P #define ecccopy(Q, P); fp2copy1271((Q)->x, (P)->x); \ fp2copy1271((Q)->y, (P)->y); \ fp2copy1271((Q)->z, (P)->z); \ fp2copy1271((Q)->t, (P)->t); // Copy extended projective point Q = (X+Y,Y-X,2Z,2dT) to P #define ecccopy_precomp(Q, P); fp2copy1271((Q)->xy, (P)->xy); \ fp2copy1271((Q)->yx, (P)->yx); \ fp2copy1271((Q)->z2, (P)->z2); \ fp2copy1271((Q)->t2, (P)->t2); // Copy extended affine point Q = (x+y,y-x,2dt) to P #define ecccopy_precomp_fixed_base(Q, P); fp2copy1271((Q)->xy, (P)->xy); \ fp2copy1271((Q)->yx, (P)->yx); \ fp2copy1271((Q)->t2, (P)->t2); #ifdef __cplusplus } #endif #endif