/***************************************************************************************** * 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 papers: * [1] "FourQ: four-dimensional decompositions on a Q-curve over the Mersenne prime" * by Craig Costello and Patrick Longa, ASIACRYPT2015 (http://eprint.iacr.org/2015/565). * [2] "FourQNEON: Faster Elliptic Curve Scalar Multiplications on ARM Processors" * by Patrick Longa, SAC2016 (http://eprint.iacr.org/2016/645). ******************************************************************************************/ #ifndef __FOURQ_INTERNAL_H__ #define __FOURQ_INTERNAL_H__ // For C++ #ifdef __cplusplus extern "C" { #endif #include "FourQ_api.h" // Basic parameters for variable-base scalar multiplication (without using endomorphisms) #define NPOINTS_VARBASE (1 << (W_VARBASE-2)) #define t_VARBASE ((NBITS_ORDER_PLUS_ONE+W_VARBASE-2)/(W_VARBASE-1)) // Basic parameters for fixed-base scalar multiplication #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 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]; // FourQ's vectorized point representations typedef struct { v2elm_t x; v2elm_t y; v2elm_t z; v2elm_t ta; v2elm_t tb; } vpoint_extproj; // Point representation in extended coordinates. typedef vpoint_extproj vpoint_extproj_t[1]; typedef struct { v2elm_t xy; v2elm_t yx; v2elm_t z2; v2elm_t t2; } vpoint_extproj_precomp; // Point representation in extended coordinates (for precomputed points). typedef vpoint_extproj_precomp vpoint_extproj_precomp_t[1]; typedef struct { v2elm_t xy; v2elm_t yx; v2elm_t t2; } vpoint_precomp; // Point representation in extended affine coordinates (for precomputed points). typedef vpoint_precomp vpoint_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 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))); /**************** 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 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 *************/ // Field negation, a = -a mod p void vneg1271(velm_t a); // Field negation over GF(2^127-1) of second element of a GF(p^2) element void v2neg1271_felm(v2elm_t a); // Modular correction, a = a mod p void vmod1271(velm_t a, velm_t c); // Field addition, c = a+b mod p void vadd1271(velm_t a, velm_t b, velm_t c); // Field subtraction, c = a-b mod p void vsub1271(velm_t a, velm_t b, velm_t c); // Field division by two, c = a/2 mod p void vdiv1271(uint32_t* a); // Field squaring, c = a^2 mod p void vsqr1271(velm_t a, velm_t c); void vsqr1271_a(uint32_t* a, uint32_t* c); // Field multiplication, c = a*b mod p void vmul1271(velm_t a, velm_t b, velm_t c); void vmul1271_a(uint32_t* a, uint32_t* b, uint32_t* c); // Field inversion, af = a^-1 = a^(p-2) mod p void vinv1271(velm_t a); // Exponentiation over GF(p), af = a^(125-1) void vexp1251(velm_t a, velm_t af); // Conversion functions void from_std_to_ext(f2elm_t a, v2elm_t c); void from_ext_to_std(v2elm_t a, f2elm_t c); /************ Quadratic extension field arithmetic functions *************/ // Convert GF(p^2) element in 23/23/26/26/26/26/26/26/26/26-bit vector representation to // two field elements in 23/26/26/26/26-bit vector representation void from_v2_to_v(v2elm_t a, velm_t c0, velm_t c1); // Convert two field elements in 23/26/26/26/26-bit vector representation to GF(p^2) element in // 23/23/26/26/26/26/26/26/26/26-bit vector representation void from_v_to_v2(velm_t a0, velm_t a1, v2elm_t c); // Zeroing a quadratic extension field element, a=0 void v2zero1271(v2elm_t a); // Copy quadratic extension field element, c = a void fp2copy1271(f2elm_t a, f2elm_t c); void v2copy1271(v2elm_t a, v2elm_t c); // Quadratic extension field negation, a = -a in GF((2^127-1)^2) void v2neg1271(v2elm_t a); // Quadratic extension field addition, c = a+b in GF((2^127-1)^2) void v2add1271(v2elm_t a, v2elm_t b, v2elm_t c); void v2add1271_a(uint32_t* a, uint32_t* b, uint32_t* c); // Quadratic extension field subtraction, c = a-b in GF((2^127-1)^2) void v2sub1271(v2elm_t a, v2elm_t b, v2elm_t c); void v2sub1271_a(uint32_t* a, uint32_t* b, uint32_t* c); // Quadratic extension field addition followed by subtraction over GF(2^127-1) void v2dblsub1271(v2elm_t a, v2elm_t b, v2elm_t c); void v2dblsub1271_a(uint32_t* a, uint32_t* b, uint32_t* c); // Quadratic extension field addition and subtraction over GF(2^127-1) void v2addsub1271(v2elm_t a, v2elm_t b, v2elm_t c, v2elm_t d); void v2addsub1271_a(uint32_t* a, uint32_t* b, uint32_t* c, uint32_t* d); // Modular correction over GF(p^2) void v2mod1271(v2elm_t a, v2elm_t c); // Quadratic extension field multiplication, c = a*b in GF((2^127-1)^2) void v2mul1271(v2elm_t a, v2elm_t b, v2elm_t c); void v2mul1271_a(uint32_t* a, uint32_t* b, uint32_t* c); // Quadratic extension field squaring, c = a^2 in GF((2^127-1)^2) void v2sqr1271(v2elm_t a, v2elm_t c); void v2sqr1271_a(uint32_t* a, uint32_t* c); // Vectorized GF(p^2) squaring/addition in GF((2^127-1)^2) void v2sqradd1271(v2elm_t a, v2elm_t c, v2elm_t d, v2elm_t e, v2elm_t f); void v2sqradd1271_a(uint32_t* a, uint32_t* c, uint32_t* d, uint32_t* e, uint32_t* f); // Vectorized GF(p^2) squaring/addition/subtraction in GF((2^127-1)^2) void v2sqraddsub1271(v2elm_t a, v2elm_t c, v2elm_t d, v2elm_t e, v2elm_t f, v2elm_t g); void v2sqraddsub1271_a(uint32_t* a, uint32_t* c, uint32_t* d, uint32_t* e, uint32_t* f, uint32_t* g); // Vectorized GF(p^2) multiplication/addition in GF((2^127-1)^2) void v2muladd1271(v2elm_t a, v2elm_t b, v2elm_t c, v2elm_t d, v2elm_t e, v2elm_t f); void v2muladd1271_a(uint32_t* a, uint32_t* b, uint32_t* c, uint32_t* d, uint32_t* e, uint32_t* f); // Vectorized GF(p^2) multiplication/subtraction in GF((2^127-1)^2) void v2mulsub1271(v2elm_t a, v2elm_t b, v2elm_t c, v2elm_t d, v2elm_t e, v2elm_t f); void v2mulsub1271_a(uint32_t* a, uint32_t* b, uint32_t* c, uint32_t* d, uint32_t* e, uint32_t* f); // Vectorized GF(p^2) multiplication/addition/subtraction in GF((2^127-1)^2) void v2muladdsub1271(v2elm_t a, v2elm_t b, v2elm_t c, v2elm_t d, v2elm_t e, v2elm_t f, v2elm_t g); void v2muladdsub1271_a(uint32_t* a, uint32_t* b, uint32_t* c, uint32_t* d, uint32_t* e, uint32_t* f, uint32_t* g); // Vectorized GF(p^2) multiplication/addition/subtraction in GF((2^127-1)^2) void v2muldlbsub1271(v2elm_t a, v2elm_t b, v2elm_t c, v2elm_t d, v2elm_t e, v2elm_t f); void v2muldblsub1271_a(uint32_t* a, uint32_t* b, uint32_t* c, uint32_t* d, uint32_t* e, uint32_t* f); // Vectorized GF(p^2) inversion, af = a^-1 = a^(p-2) in GF((2^127-1)^2) void v2inv1271(v2elm_t a); /************ Curve and recoding functions *************/ // Normalize projective twisted Edwards point Q = (X,Y,Z) -> P = (x,y) void eccnorm(vpoint_extproj_t P, vpoint_t Q); // Conversion from representation (X,Y,Z,Ta,Tb) to (X+Y,Y-X,2Z,2dT), where T = Ta*Tb void R1_to_R2(vpoint_extproj_t P, vpoint_extproj_precomp_t Q); // Conversion from representation (X,Y,Z,Ta,Tb) to (X+Y,Y-X,Z,T), where T = Ta*Tb void R1_to_R3(vpoint_extproj_t P, vpoint_extproj_precomp_t Q); // Conversion from representation (X+Y,Y-X,2Z,2dT) to (2X,2Y,2Z,2dT) void R2_to_R4(vpoint_extproj_precomp_t P, vpoint_extproj_t Q); // Point doubling 2P void eccdouble(vpoint_extproj_t P); // Complete point addition P = P+Q or P = P+P void eccadd(vpoint_extproj_precomp_t Q, vpoint_extproj_t P); void eccadd_core(vpoint_extproj_precomp_t P, vpoint_extproj_precomp_t Q, vpoint_extproj_t R); // Psi mapping of a point, P = psi(P) void ecc_psi(vpoint_extproj_t P); // Phi mapping of a point, P = phi(P) void ecc_phi(vpoint_extproj_t P); // Scalar decomposition void decompose(uint64_t* k, uint64_t* scalars); // 256-bit multiplication with truncation for the scalar decomposition // Outputs 64-bit value c = (uint64_t)((a * b) >> 256) void mul_truncate_a(uint32_t* a, uint32_t* b, uint32_t* c); // Recoding sub-scalars for use in the variable-base scalar multiplication void recode(uint64_t* scalars, 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(vpoint_extproj_t P); // Reduction modulo the order using Montgomery arithmetic void modulo_order(digit_t* a, digit_t* c); // Precomputation function void ecc_precomp(vpoint_extproj_t P, vpoint_extproj_precomp_t *T); // Constant-time table lookup to extract an extended twisted Edwards point (X+Y:Y-X:2Z:2T) from the precomputed table void table_lookup_1x8(vpoint_extproj_precomp_t* table, vpoint_extproj_precomp_t P, unsigned int digit, unsigned int sign_mask); // Modular correction of input coordinates and conversion to representation (X,Y,Z,Ta,Tb) void point_setup(point_t P, vpoint_extproj_t Q); // Point validation: check if point lies on the curve bool ecc_point_validate(vpoint_extproj_t P); // Output error/success message for a given ECCRYPTO_STATUS const char* FourQ_get_error_message(ECCRYPTO_STATUS Status); // Constant-time table lookup to extract a point represented as (x+y,y-x,2t) void table_lookup_fixed_base(vpoint_precomp_t* table, vpoint_precomp_t P, unsigned int digit, unsigned int sign); // Computes the modified LSB-set representation of scalar void mLSB_set_recode(uint64_t* scalar, unsigned int *digits); // Generation of the precomputation table used internally by the double scalar multiplication function ecc_mul_double() void ecc_precomp_double(vpoint_extproj_t P, vpoint_extproj_precomp_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:Ta:Tb) to P #define ecccopy(Q, P); v2copy1271((Q)->x, (P)->x); \ v2copy1271((Q)->y, (P)->y); \ v2copy1271((Q)->z, (P)->z); \ v2copy1271((Q)->ta, (P)->ta); \ v2copy1271((Q)->tb, (P)->tb); // Copy extended affine point Q = (x+y,y-x,2dt) to P #define ecccopy_precomp_fixed_base(Q, P); v2copy1271((Q)->xy, (P)->xy); \ v2copy1271((Q)->yx, (P)->yx); \ v2copy1271((Q)->t2, (P)->t2); // Vectorize extended projective point Q = (X:Y:Z:Ta:Tb) #define point_from_std_to_ext(Q, P); from_std_to_ext(Q->x, P->x); \ from_std_to_ext(Q->y, P->y); \ from_std_to_ext(Q->z, P->z); \ from_std_to_ext(Q->ta, P->ta); \ from_std_to_ext(Q->tb, P->tb); #ifdef __cplusplus } #endif #endif