diff --git a/donna_32.cpp b/donna_32.cpp index 0879fc4a..f2ace71f 100644 --- a/donna_32.cpp +++ b/donna_32.cpp @@ -38,9 +38,9 @@ using CryptoPP::BigEndian; using CryptoPP::LittleEndian; typedef word32 bignum25519[10]; -typedef word32 bignum25519align16[12]; #define mul32x32_64(a,b) (((word64)(a))*(b)) +#define ALIGN(n) CRYPTOPP_ALIGN_DATA(n) const byte basePoint[32] = {9}; const word32 reduce_mask_25 = (1 << 25) - 1; @@ -66,114 +66,43 @@ curve25519_add(bignum25519 out, const bignum25519 a, const bignum25519 b) { out[8] = a[8] + b[8]; out[9] = a[9] + b[9]; } -inline void -curve25519_add_after_basic(bignum25519 out, const bignum25519 a, const bignum25519 b) { - word32 c; - out[0] = a[0] + b[0] ; c = (out[0] >> 26); out[0] &= reduce_mask_26; - out[1] = a[1] + b[1] + c; c = (out[1] >> 25); out[1] &= reduce_mask_25; - out[2] = a[2] + b[2] + c; c = (out[2] >> 26); out[2] &= reduce_mask_26; - out[3] = a[3] + b[3] + c; c = (out[3] >> 25); out[3] &= reduce_mask_25; - out[4] = a[4] + b[4] + c; c = (out[4] >> 26); out[4] &= reduce_mask_26; - out[5] = a[5] + b[5] + c; c = (out[5] >> 25); out[5] &= reduce_mask_25; - out[6] = a[6] + b[6] + c; c = (out[6] >> 26); out[6] &= reduce_mask_26; - out[7] = a[7] + b[7] + c; c = (out[7] >> 25); out[7] &= reduce_mask_25; - out[8] = a[8] + b[8] + c; c = (out[8] >> 26); out[8] &= reduce_mask_26; - out[9] = a[9] + b[9] + c; c = (out[9] >> 25); out[9] &= reduce_mask_25; - out[0] += 19 * c; -} - -inline void -curve25519_add_reduce(bignum25519 out, const bignum25519 a, const bignum25519 b) { - word32 c; - out[0] = a[0] + b[0] ; c = (out[0] >> 26); out[0] &= reduce_mask_26; - out[1] = a[1] + b[1] + c; c = (out[1] >> 25); out[1] &= reduce_mask_25; - out[2] = a[2] + b[2] + c; c = (out[2] >> 26); out[2] &= reduce_mask_26; - out[3] = a[3] + b[3] + c; c = (out[3] >> 25); out[3] &= reduce_mask_25; - out[4] = a[4] + b[4] + c; c = (out[4] >> 26); out[4] &= reduce_mask_26; - out[5] = a[5] + b[5] + c; c = (out[5] >> 25); out[5] &= reduce_mask_25; - out[6] = a[6] + b[6] + c; c = (out[6] >> 26); out[6] &= reduce_mask_26; - out[7] = a[7] + b[7] + c; c = (out[7] >> 25); out[7] &= reduce_mask_25; - out[8] = a[8] + b[8] + c; c = (out[8] >> 26); out[8] &= reduce_mask_26; - out[9] = a[9] + b[9] + c; c = (out[9] >> 25); out[9] &= reduce_mask_25; - out[0] += 19 * c; -} - -/* multiples of p */ -const word32 twoP0 = 0x07ffffda; -const word32 twoP13579 = 0x03fffffe; -const word32 twoP2468 = 0x07fffffe; -const word32 fourP0 = 0x0fffffb4; -const word32 fourP13579 = 0x07fffffc; -const word32 fourP2468 = 0x0ffffffc; - /* out = a - b */ inline void curve25519_sub(bignum25519 out, const bignum25519 a, const bignum25519 b) { word32 c; - out[0] = twoP0 + a[0] - b[0] ; c = (out[0] >> 26); out[0] &= reduce_mask_26; - out[1] = twoP13579 + a[1] - b[1] + c; c = (out[1] >> 25); out[1] &= reduce_mask_25; - out[2] = twoP2468 + a[2] - b[2] + c; c = (out[2] >> 26); out[2] &= reduce_mask_26; - out[3] = twoP13579 + a[3] - b[3] + c; c = (out[3] >> 25); out[3] &= reduce_mask_25; - out[4] = twoP2468 + a[4] - b[4] + c; - out[5] = twoP13579 + a[5] - b[5] ; - out[6] = twoP2468 + a[6] - b[6] ; - out[7] = twoP13579 + a[7] - b[7] ; - out[8] = twoP2468 + a[8] - b[8] ; - out[9] = twoP13579 + a[9] - b[9] ; -} - -/* out = a - b, where a is the result of a basic op (add,sub) */ -inline void -curve25519_sub_after_basic(bignum25519 out, const bignum25519 a, const bignum25519 b) { - word32 c; - out[0] = fourP0 + a[0] - b[0] ; c = (out[0] >> 26); out[0] &= reduce_mask_26; - out[1] = fourP13579 + a[1] - b[1] + c; c = (out[1] >> 25); out[1] &= reduce_mask_25; - out[2] = fourP2468 + a[2] - b[2] + c; c = (out[2] >> 26); out[2] &= reduce_mask_26; - out[3] = fourP13579 + a[3] - b[3] + c; c = (out[3] >> 25); out[3] &= reduce_mask_25; - out[4] = fourP2468 + a[4] - b[4] + c; c = (out[4] >> 26); out[4] &= reduce_mask_26; - out[5] = fourP13579 + a[5] - b[5] + c; c = (out[5] >> 25); out[5] &= reduce_mask_25; - out[6] = fourP2468 + a[6] - b[6] + c; c = (out[6] >> 26); out[6] &= reduce_mask_26; - out[7] = fourP13579 + a[7] - b[7] + c; c = (out[7] >> 25); out[7] &= reduce_mask_25; - out[8] = fourP2468 + a[8] - b[8] + c; c = (out[8] >> 26); out[8] &= reduce_mask_26; - out[9] = fourP13579 + a[9] - b[9] + c; c = (out[9] >> 25); out[9] &= reduce_mask_25; + out[0] = 0x7ffffda + a[0] - b[0] ; c = (out[0] >> 26); out[0] &= reduce_mask_26; + out[1] = 0x3fffffe + a[1] - b[1] + c; c = (out[1] >> 25); out[1] &= reduce_mask_25; + out[2] = 0x7fffffe + a[2] - b[2] + c; c = (out[2] >> 26); out[2] &= reduce_mask_26; + out[3] = 0x3fffffe + a[3] - b[3] + c; c = (out[3] >> 25); out[3] &= reduce_mask_25; + out[4] = 0x7fffffe + a[4] - b[4] + c; c = (out[4] >> 26); out[4] &= reduce_mask_26; + out[5] = 0x3fffffe + a[5] - b[5] + c; c = (out[5] >> 25); out[5] &= reduce_mask_25; + out[6] = 0x7fffffe + a[6] - b[6] + c; c = (out[6] >> 26); out[6] &= reduce_mask_26; + out[7] = 0x3fffffe + a[7] - b[7] + c; c = (out[7] >> 25); out[7] &= reduce_mask_25; + out[8] = 0x7fffffe + a[8] - b[8] + c; c = (out[8] >> 26); out[8] &= reduce_mask_26; + out[9] = 0x3fffffe + a[9] - b[9] + c; c = (out[9] >> 25); out[9] &= reduce_mask_25; out[0] += 19 * c; } +/* out = in * scalar */ inline void -curve25519_sub_reduce(bignum25519 out, const bignum25519 a, const bignum25519 b) { +curve25519_scalar_product(bignum25519 out, const bignum25519 in, const word32 scalar) { + word64 a; word32 c; - out[0] = fourP0 + a[0] - b[0] ; c = (out[0] >> 26); out[0] &= reduce_mask_26; - out[1] = fourP13579 + a[1] - b[1] + c; c = (out[1] >> 25); out[1] &= reduce_mask_25; - out[2] = fourP2468 + a[2] - b[2] + c; c = (out[2] >> 26); out[2] &= reduce_mask_26; - out[3] = fourP13579 + a[3] - b[3] + c; c = (out[3] >> 25); out[3] &= reduce_mask_25; - out[4] = fourP2468 + a[4] - b[4] + c; c = (out[4] >> 26); out[4] &= reduce_mask_26; - out[5] = fourP13579 + a[5] - b[5] + c; c = (out[5] >> 25); out[5] &= reduce_mask_25; - out[6] = fourP2468 + a[6] - b[6] + c; c = (out[6] >> 26); out[6] &= reduce_mask_26; - out[7] = fourP13579 + a[7] - b[7] + c; c = (out[7] >> 25); out[7] &= reduce_mask_25; - out[8] = fourP2468 + a[8] - b[8] + c; c = (out[8] >> 26); out[8] &= reduce_mask_26; - out[9] = fourP13579 + a[9] - b[9] + c; c = (out[9] >> 25); out[9] &= reduce_mask_25; - out[0] += 19 * c; -} - -/* out = -a */ -inline void -curve25519_neg(bignum25519 out, const bignum25519 a) { - word32 c; - out[0] = twoP0 - a[0] ; c = (out[0] >> 26); out[0] &= reduce_mask_26; - out[1] = twoP13579 - a[1] + c; c = (out[1] >> 25); out[1] &= reduce_mask_25; - out[2] = twoP2468 - a[2] + c; c = (out[2] >> 26); out[2] &= reduce_mask_26; - out[3] = twoP13579 - a[3] + c; c = (out[3] >> 25); out[3] &= reduce_mask_25; - out[4] = twoP2468 - a[4] + c; c = (out[4] >> 26); out[4] &= reduce_mask_26; - out[5] = twoP13579 - a[5] + c; c = (out[5] >> 25); out[5] &= reduce_mask_25; - out[6] = twoP2468 - a[6] + c; c = (out[6] >> 26); out[6] &= reduce_mask_26; - out[7] = twoP13579 - a[7] + c; c = (out[7] >> 25); out[7] &= reduce_mask_25; - out[8] = twoP2468 - a[8] + c; c = (out[8] >> 26); out[8] &= reduce_mask_26; - out[9] = twoP13579 - a[9] + c; c = (out[9] >> 25); out[9] &= reduce_mask_25; - out[0] += 19 * c; + a = mul32x32_64(in[0], scalar); out[0] = (word32)a & reduce_mask_26; c = (word32)(a >> 26); + a = mul32x32_64(in[1], scalar) + c; out[1] = (word32)a & reduce_mask_25; c = (word32)(a >> 25); + a = mul32x32_64(in[2], scalar) + c; out[2] = (word32)a & reduce_mask_26; c = (word32)(a >> 26); + a = mul32x32_64(in[3], scalar) + c; out[3] = (word32)a & reduce_mask_25; c = (word32)(a >> 25); + a = mul32x32_64(in[4], scalar) + c; out[4] = (word32)a & reduce_mask_26; c = (word32)(a >> 26); + a = mul32x32_64(in[5], scalar) + c; out[5] = (word32)a & reduce_mask_25; c = (word32)(a >> 25); + a = mul32x32_64(in[6], scalar) + c; out[6] = (word32)a & reduce_mask_26; c = (word32)(a >> 26); + a = mul32x32_64(in[7], scalar) + c; out[7] = (word32)a & reduce_mask_25; c = (word32)(a >> 25); + a = mul32x32_64(in[8], scalar) + c; out[8] = (word32)a & reduce_mask_26; c = (word32)(a >> 26); + a = mul32x32_64(in[9], scalar) + c; out[9] = (word32)a & reduce_mask_25; c = (word32)(a >> 25); + out[0] += c * 19; } /* out = a * b */ -void +inline void curve25519_mul(bignum25519 out, const bignum25519 a, const bignum25519 b) { word32 r0,r1,r2,r3,r4,r5,r6,r7,r8,r9; word32 s0,s1,s2,s3,s4,s5,s6,s7,s8,s9; @@ -238,8 +167,8 @@ curve25519_mul(bignum25519 out, const bignum25519 a, const bignum25519 b) { out[5] = r5; out[6] = r6; out[7] = r7; out[8] = r8; out[9] = r9; } -/* out = in*in */ -void +/* out = in * in */ +inline void curve25519_square(bignum25519 out, const bignum25519 in) { word32 r0,r1,r2,r3,r4,r5,r6,r7,r8,r9; word32 d6,d7,d8,d9; @@ -294,8 +223,7 @@ curve25519_square(bignum25519 out, const bignum25519 in) { out[5] = r5; out[6] = r6; out[7] = r7; out[8] = r8; out[9] = r9; } - -/* out = in ^ (2 * count) */ +/* out = in^(2 * count) */ void curve25519_square_times(bignum25519 out, const bignum25519 in, int count) { word32 r0,r1,r2,r3,r4,r5,r6,r7,r8,r9; @@ -303,7 +231,7 @@ curve25519_square_times(bignum25519 out, const bignum25519 in, int count) { word64 m0,m1,m2,m3,m4,m5,m6,m7,m8,m9,c; word32 p; - r0 = in[0]; r1 = in[1]; r2 = in[2]; r3 = in[3]; r4 = in[4]; + r0 = in[0]; r1 = in[1]; r2 = in[2]; r3 = in[3]; r4 = in[4]; r5 = in[5]; r6 = in[6]; r7 = in[7]; r8 = in[8]; r9 = in[9]; do { @@ -356,26 +284,54 @@ curve25519_square_times(bignum25519 out, const bignum25519 in, int count) { /* Take a little-endian, 32-byte number and expand it into polynomial form */ void curve25519_expand(bignum25519 out, const byte in[32]) { + word32 x0,x1,x2,x3,x4,x5,x6,x7; GetBlock block(in); block(x0)(x1)(x2)(x3)(x4)(x5)(x6)(x7); - out[0] = ( x0 ) & 0x3ffffff; - out[1] = ((((word64)x1 << 32) | x0) >> 26) & 0x1ffffff; - out[2] = ((((word64)x2 << 32) | x1) >> 19) & 0x3ffffff; - out[3] = ((((word64)x3 << 32) | x2) >> 13) & 0x1ffffff; - out[4] = (( x3) >> 6) & 0x3ffffff; - out[5] = ( x4 ) & 0x1ffffff; - out[6] = ((((word64)x5 << 32) | x4) >> 25) & 0x3ffffff; - out[7] = ((((word64)x6 << 32) | x5) >> 19) & 0x1ffffff; - out[8] = ((((word64)x7 << 32) | x6) >> 12) & 0x3ffffff; - out[9] = (( x7) >> 6) & 0x1ffffff; +#if 0 +#if defined(CRYPTOPP_LITTLE_ENDIAN) + x0 = *(word32 *)(in + 0); + x1 = *(word32 *)(in + 4); + x2 = *(word32 *)(in + 8); + x3 = *(word32 *)(in + 12); + x4 = *(word32 *)(in + 16); + x5 = *(word32 *)(in + 20); + x6 = *(word32 *)(in + 24); + x7 = *(word32 *)(in + 28); +#else + #define F(s) \ + ((((word32)in[s + 0]) ) | \ + (((word32)in[s + 1]) << 8) | \ + (((word32)in[s + 2]) << 16) | \ + (((word32)in[s + 3]) << 24)) + x0 = F(0); + x1 = F(4); + x2 = F(8); + x3 = F(12); + x4 = F(16); + x5 = F(20); + x6 = F(24); + x7 = F(28); + #undef F +#endif +#endif + + out[0] = ( x0 ) & reduce_mask_26; + out[1] = ((((word64)x1 << 32) | x0) >> 26) & reduce_mask_25; + out[2] = ((((word64)x2 << 32) | x1) >> 19) & reduce_mask_26; + out[3] = ((((word64)x3 << 32) | x2) >> 13) & reduce_mask_25; + out[4] = (( x3) >> 6) & reduce_mask_26; + out[5] = ( x4 ) & reduce_mask_25; + out[6] = ((((word64)x5 << 32) | x4) >> 25) & reduce_mask_26; + out[7] = ((((word64)x6 << 32) | x5) >> 19) & reduce_mask_25; + out[8] = ((((word64)x7 << 32) | x6) >> 12) & reduce_mask_26; + out[9] = (( x7) >> 6) & reduce_mask_25; /* ignore the top bit */ + // out[9] = (( x7) >> 6) & reduce_mask_26; } -/* Take a fully reduced polynomial form number and contract it into a - * little-endian, 32-byte array - */ +/* Take a fully reduced polynomial form number and contract it into a little-endian, 32-byte array */ void curve25519_contract(byte out[32], const bignum25519 in) { bignum25519 f; @@ -409,16 +365,16 @@ curve25519_contract(byte out[32], const bignum25519 in) { carry_pass_full() /* now between 19 and 2^255-1 in both cases, and offset by 19. */ - f[0] += (reduce_mask_26 + 1) - 19; - f[1] += (reduce_mask_25 + 1) - 1; - f[2] += (reduce_mask_26 + 1) - 1; - f[3] += (reduce_mask_25 + 1) - 1; - f[4] += (reduce_mask_26 + 1) - 1; - f[5] += (reduce_mask_25 + 1) - 1; - f[6] += (reduce_mask_26 + 1) - 1; - f[7] += (reduce_mask_25 + 1) - 1; - f[8] += (reduce_mask_26 + 1) - 1; - f[9] += (reduce_mask_25 + 1) - 1; + f[0] += (1 << 26) - 19; + f[1] += (1 << 25) - 1; + f[2] += (1 << 26) - 1; + f[3] += (1 << 25) - 1; + f[4] += (1 << 26) - 1; + f[5] += (1 << 25) - 1; + f[6] += (1 << 26) - 1; + f[7] += (1 << 25) - 1; + f[8] += (1 << 26) - 1; + f[9] += (1 << 25) - 1; /* now between 2^255 and 2^256-20, and offset by 2^255. */ carry_pass_final() @@ -437,7 +393,7 @@ curve25519_contract(byte out[32], const bignum25519 in) { f[9] <<= 6; #define F(i, s) \ - out[s+0] |= (byte)(f[i] & 0xff); \ + out[s+0] |= (byte)( f[i] & 0xff); \ out[s+1] = (byte)((f[i] >> 8) & 0xff); \ out[s+2] = (byte)((f[i] >> 16) & 0xff); \ out[s+3] = (byte)((f[i] >> 24) & 0xff); @@ -457,6 +413,66 @@ curve25519_contract(byte out[32], const bignum25519 in) { #undef F } +inline void +curve25519_swap_conditional(bignum25519 x, bignum25519 qpx, word32 iswap) { + const word32 swap = (word32)(-(sword32)iswap); + word32 x0,x1,x2,x3,x4,x5,x6,x7,x8,x9; + + x0 = swap & (x[0] ^ qpx[0]); x[0] ^= x0; qpx[0] ^= x0; + x1 = swap & (x[1] ^ qpx[1]); x[1] ^= x1; qpx[1] ^= x1; + x2 = swap & (x[2] ^ qpx[2]); x[2] ^= x2; qpx[2] ^= x2; + x3 = swap & (x[3] ^ qpx[3]); x[3] ^= x3; qpx[3] ^= x3; + x4 = swap & (x[4] ^ qpx[4]); x[4] ^= x4; qpx[4] ^= x4; + x5 = swap & (x[5] ^ qpx[5]); x[5] ^= x5; qpx[5] ^= x5; + x6 = swap & (x[6] ^ qpx[6]); x[6] ^= x6; qpx[6] ^= x6; + x7 = swap & (x[7] ^ qpx[7]); x[7] ^= x7; qpx[7] ^= x7; + x8 = swap & (x[8] ^ qpx[8]); x[8] ^= x8; qpx[8] ^= x8; + x9 = swap & (x[9] ^ qpx[9]); x[9] ^= x9; qpx[9] ^= x9; +} + +/* + * In: b = 2^5 - 2^0 + * Out: b = 2^250 - 2^0 + */ +void +curve25519_pow_two5mtwo0_two250mtwo0(bignum25519 b) { + bignum25519 ALIGN(16) t0,c; + + /* 2^5 - 2^0 */ /* b */ + /* 2^10 - 2^5 */ curve25519_square_times(t0, b, 5); + /* 2^10 - 2^0 */ curve25519_mul(b, t0, b); + /* 2^20 - 2^10 */ curve25519_square_times(t0, b, 10); + /* 2^20 - 2^0 */ curve25519_mul(c, t0, b); + /* 2^40 - 2^20 */ curve25519_square_times(t0, c, 20); + /* 2^40 - 2^0 */ curve25519_mul(t0, t0, c); + /* 2^50 - 2^10 */ curve25519_square_times(t0, t0, 10); + /* 2^50 - 2^0 */ curve25519_mul(b, t0, b); + /* 2^100 - 2^50 */ curve25519_square_times(t0, b, 50); + /* 2^100 - 2^0 */ curve25519_mul(c, t0, b); + /* 2^200 - 2^100 */ curve25519_square_times(t0, c, 100); + /* 2^200 - 2^0 */ curve25519_mul(t0, t0, c); + /* 2^250 - 2^50 */ curve25519_square_times(t0, t0, 50); + /* 2^250 - 2^0 */ curve25519_mul(b, t0, b); +} + +/* + * z^(p - 2) = z(2^255 - 21) + */ +void +curve25519_recip(bignum25519 out, const bignum25519 z) { + ALIGN(16) bignum25519 a, t0, b; + + /* 2 */ curve25519_square(a, z); /* a = 2 */ + /* 8 */ curve25519_square_times(t0, a, 2); + /* 9 */ curve25519_mul(b, t0, z); /* b = 9 */ + /* 11 */ curve25519_mul(a, b, a); /* a = 11 */ + /* 22 */ curve25519_square(t0, a); + /* 2^5 - 2^0 = 31 */ curve25519_mul(b, t0, b); + /* 2^250 - 2^0 */ curve25519_pow_two5mtwo0_two250mtwo0(b); + /* 2^255 - 2^5 */ curve25519_square_times(b, b, 5); + /* 2^255 - 21 */ curve25519_mul(out, b, a); +} + ANONYMOUS_NAMESPACE_END NAMESPACE_BEGIN(CryptoPP) @@ -464,12 +480,68 @@ NAMESPACE_BEGIN(Donna) int curve25519_CXX(byte sharedKey[32], const byte secretKey[32], const byte othersKey[32]) { - bignum25519 out, r, s; - curve25519_expand(r, secretKey); - curve25519_expand(s, othersKey); + FixedSizeSecBlock e; + for (size_t i = 0;i < 32;++i) + e[i] = secretKey[i]; - curve25519_mul(out, r, s); - curve25519_contract(sharedKey, out); + e[ 0] &= 0xf8; + e[31] &= 0x7f; + e[31] |= 0x40; + + bignum25519 nqpqx = {1}, nqpqz = {0}, nqz = {1}, nqx; + bignum25519 q, qx, qpqx, qqx, zzz, zmone; + size_t bit, lastbit; + + curve25519_expand(q, othersKey); + curve25519_copy(nqx, q); + + /* bit 255 is always 0, and bit 254 is always 1, so skip bit 255 and + start pre-swapped on bit 254 */ + lastbit = 1; + + /* we are doing bits 254..3 in the loop, but are swapping in bits 253..2 */ + for (int i = 253; i >= 2; i--) { + curve25519_add(qx, nqx, nqz); + curve25519_sub(nqz, nqx, nqz); + curve25519_add(qpqx, nqpqx, nqpqz); + curve25519_sub(nqpqz, nqpqx, nqpqz); + curve25519_mul(nqpqx, qpqx, nqz); + curve25519_mul(nqpqz, qx, nqpqz); + curve25519_add(qqx, nqpqx, nqpqz); + curve25519_sub(nqpqz, nqpqx, nqpqz); + curve25519_square(nqpqz, nqpqz); + curve25519_square(nqpqx, qqx); + curve25519_mul(nqpqz, nqpqz, q); + curve25519_square(qx, qx); + curve25519_square(nqz, nqz); + curve25519_mul(nqx, qx, nqz); + curve25519_sub(nqz, qx, nqz); + curve25519_scalar_product(zzz, nqz, 121665); + curve25519_add(zzz, zzz, qx); + curve25519_mul(nqz, nqz, zzz); + + bit = (e[i/8] >> (i & 7)) & 1; + curve25519_swap_conditional(nqx, nqpqx, bit ^ lastbit); + curve25519_swap_conditional(nqz, nqpqz, bit ^ lastbit); + lastbit = bit; + } + + /* the final 3 bits are always zero, so we only need to double */ + for (int i = 0; i < 3; i++) { + curve25519_add(qx, nqx, nqz); + curve25519_sub(nqz, nqx, nqz); + curve25519_square(qx, qx); + curve25519_square(nqz, nqz); + curve25519_mul(nqx, qx, nqz); + curve25519_sub(nqz, qx, nqz); + curve25519_scalar_product(zzz, nqz, 121665); + curve25519_add(zzz, zzz, qx); + curve25519_mul(nqz, nqz, zzz); + } + + curve25519_recip(zmone, nqz); + curve25519_mul(nqz, nqx, zmone); + curve25519_contract(sharedKey, nqz); return 0; }