#include #include "Log.h" #include "Signature.h" namespace i2p { namespace crypto { class Ed25519 { public: Ed25519 () { BN_CTX * ctx = BN_CTX_new (); BIGNUM * tmp = BN_new (); q = BN_new (); // 2^255-19 BN_set_bit (q, 255); // 2^255 BN_sub_word (q, 19); l = BN_new (); // 2^252 + 27742317777372353535851937790883648493 BN_set_bit (l, 252); two_252_2 = BN_dup (l); BN_dec2bn (&tmp, "27742317777372353535851937790883648493"); BN_add (l, l, tmp); BN_sub_word (two_252_2, 2); // 2^252 - 2 // -121665*inv(121666) d = BN_new (); BN_set_word (tmp, 121666); BN_mod_inverse (tmp, tmp, q, ctx); BN_set_word (d, 121665); BN_set_negative (d, 1); BN_mul (d, d, tmp, ctx); // 2^((q-1)/4) I = BN_new (); BN_free (tmp); tmp = BN_dup (q); BN_sub_word (tmp, 1); BN_div_word (tmp, 4); BN_set_word (I, 2); BN_mod_exp (I, I, tmp, q, ctx); BN_free (tmp); // 4*inv(5) BIGNUM * By = BN_new (); BN_set_word (By, 5); BN_mod_inverse (By, By, q, ctx); BN_mul_word (By, 4); BIGNUM * Bx = RecoverX (By, ctx); BN_mod (Bx, Bx, q, ctx); // % q BN_mod (By, By, q, ctx); // % q // precalculate Bi256 table Bi256Carry = { Bx, By }; // B for (int i = 0; i < 32; i++) { Bi256[i][0] = Bi256Carry; // first point for (int j = 1; j < 128; j++) Bi256[i][j] = Sum (Bi256[i][j-1], Bi256[i][0], ctx); // (256+j+1)^i*B Bi256Carry = Bi256[i][127]; for (int j = 0; j < 128; j++) // add first point 128 more times Bi256Carry = Sum (Bi256Carry, Bi256[i][0], ctx); } BN_CTX_free (ctx); } Ed25519 (const Ed25519& other): q (BN_dup (other.q)), l (BN_dup (other.l)), d (BN_dup (other.d)), I (BN_dup (other.I)), two_252_2 (BN_dup (other.two_252_2)), Bi256Carry (other.Bi256Carry) { for (int i = 0; i < 32; i++) for (int j = 0; j < 128; j++) Bi256[i][j] = other.Bi256[i][j]; } ~Ed25519 () { BN_free (q); BN_free (l); BN_free (d); BN_free (I); BN_free (two_252_2); } EDDSAPoint GeneratePublicKey (const uint8_t * expandedPrivateKey, BN_CTX * ctx) const { return MulB (expandedPrivateKey, ctx); // left half of expanded key, considered as Little Endian } EDDSAPoint DecodePublicKey (const uint8_t * buf, BN_CTX * ctx) const { return DecodePoint (buf, ctx); } void EncodePublicKey (const EDDSAPoint& publicKey, uint8_t * buf, BN_CTX * ctx) const { EncodePoint (Normalize (publicKey, ctx), buf); } bool Verify (const EDDSAPoint& publicKey, const uint8_t * digest, const uint8_t * signature) const { BN_CTX * ctx = BN_CTX_new (); BIGNUM * h = DecodeBN<64> (digest); // signature 0..31 - R, 32..63 - S // B*S = R + PK*h => R = B*S - PK*h // we don't decode R, but encode (B*S - PK*h) auto Bs = MulB (signature + EDDSA25519_SIGNATURE_LENGTH/2, ctx); // B*S; BN_mod (h, h, l, ctx); // public key is multiple of B, but B%l = 0 auto PKh = Mul (publicKey, h, ctx); // PK*h uint8_t diff[32]; EncodePoint (Normalize (Sum (Bs, -PKh, ctx), ctx), diff); // Bs - PKh encoded bool passed = !memcmp (signature, diff, 32); // R BN_free (h); BN_CTX_free (ctx); if (!passed) LogPrint (eLogError, "25519 signature verification failed"); return passed; } void Sign (const uint8_t * expandedPrivateKey, const uint8_t * publicKeyEncoded, const uint8_t * buf, size_t len, uint8_t * signature) const { BN_CTX * bnCtx = BN_CTX_new (); // calculate r SHA512_CTX ctx; SHA512_Init (&ctx); SHA512_Update (&ctx, expandedPrivateKey + EDDSA25519_PRIVATE_KEY_LENGTH, EDDSA25519_PRIVATE_KEY_LENGTH); // right half of expanded key SHA512_Update (&ctx, buf, len); // data uint8_t digest[64]; SHA512_Final (digest, &ctx); BIGNUM * r = DecodeBN<32> (digest); // DecodeBN<64> (digest); // for test vectors // calculate R uint8_t R[EDDSA25519_SIGNATURE_LENGTH/2]; // we must use separate buffer because signature might be inside buf EncodePoint (Normalize (MulB (digest, bnCtx), bnCtx), R); // EncodePoint (Mul (B, r, bnCtx), R); // for test vectors // calculate S SHA512_Init (&ctx); SHA512_Update (&ctx, R, EDDSA25519_SIGNATURE_LENGTH/2); // R SHA512_Update (&ctx, publicKeyEncoded, EDDSA25519_PUBLIC_KEY_LENGTH); // public key SHA512_Update (&ctx, buf, len); // data SHA512_Final (digest, &ctx); BIGNUM * h = DecodeBN<64> (digest); // S = (r + h*a) % l BIGNUM * a = DecodeBN (expandedPrivateKey); // left half of expanded key BN_mod_mul (h, h, a, l, bnCtx); // %l BN_mod_add (h, h, r, l, bnCtx); // %l memcpy (signature, R, EDDSA25519_SIGNATURE_LENGTH/2); EncodeBN (h, signature + EDDSA25519_SIGNATURE_LENGTH/2, EDDSA25519_SIGNATURE_LENGTH/2); // S BN_free (r); BN_free (h); BN_free (a); BN_CTX_free (bnCtx); } private: EDDSAPoint Sum (const EDDSAPoint& p1, const EDDSAPoint& p2, BN_CTX * ctx) const { // x3 = (x1*y2+y1*x2)*(z1*z2-d*t1*t2) // y3 = (y1*y2+x1*x2)*(z1*z2+d*t1*t2) // z3 = (z1*z2-d*t1*t2)*(z1*z2+d*t1*t2) // t3 = (y1*y2+x1*x2)*(x1*y2+y1*x2) BIGNUM * x3 = BN_new (), * y3 = BN_new (), * z3 = BN_new (), * t3 = BN_new (); BN_mul (x3, p1.x, p2.x, ctx); // A = x1*x2 BN_mul (y3, p1.y, p2.y, ctx); // B = y1*y2 BIGNUM * t1 = p1.t, * t2 = p2.t; if (!t1) { t1 = BN_new (); BN_mul (t1, p1.x, p1.y, ctx); } if (!t2) { t2 = BN_new (); BN_mul (t2, p2.x, p2.y, ctx); } BN_mul (t3, t1, t2, ctx); BN_mul (t3, t3, d, ctx); // C = d*t1*t2 if (!p1.t) BN_free (t1); if (!p2.t) BN_free (t2); if (p1.z) { if (p2.z) BN_mul (z3, p1.z, p2.z, ctx); // D = z1*z2 else BN_copy (z3, p1.z); // D = z1 } else { if (p2.z) BN_copy (z3, p2.z); // D = z2 else BN_one (z3); // D = 1 } BIGNUM * E = BN_new (), * F = BN_new (), * G = BN_new (), * H = BN_new (); BN_add (E, p1.x, p1.y); BN_add (F, p2.x, p2.y); BN_mul (E, E, F, ctx); // (x1 + y1)*(x2 + y2) BN_sub (E, E, x3); BN_sub (E, E, y3); // E = (x1 + y1)*(x2 + y2) - A - B BN_sub (F, z3, t3); // F = D - C BN_add (G, z3, t3); // G = D + C BN_add (H, y3, x3); // H = B + A BN_mod_mul (x3, E, F, q, ctx); // x3 = E*F BN_mod_mul (y3, G, H, q, ctx); // y3 = G*H BN_mod_mul (z3, F, G, q, ctx); // z3 = F*G BN_mod_mul (t3, E, H, q, ctx); // t3 = E*H BN_free (E); BN_free (F); BN_free (G); BN_free (H); return EDDSAPoint {x3, y3, z3, t3}; } EDDSAPoint Double (const EDDSAPoint& p, BN_CTX * ctx) const { BIGNUM * x2 = BN_new (), * y2 = BN_new (), * z2 = BN_new (), * t2 = BN_new (); BN_sqr (x2, p.x, ctx); // x2 = A = x^2 BN_sqr (y2, p.y, ctx); // y2 = B = y^2 if (p.t) BN_sqr (t2, p.t, ctx); // t2 = t^2 else { BN_mul (t2, p.x, p.y, ctx); // t = x*y BN_sqr (t2, t2, ctx); // t2 = t^2 } BN_mul (t2, t2, d, ctx); // t2 = C = d*t^2 if (p.z) BN_sqr (z2, p.z, ctx); // z2 = D = z^2 else BN_one (z2); // z2 = 1 BIGNUM * E = BN_new (), * F = BN_new (), * G = BN_new (), * H = BN_new (); // E = (x+y)*(x+y)-A-B = x^2+y^2+2xy-A-B = 2xy BN_mul (E, p.x, p.y, ctx); BN_lshift1 (E, E); // E =2*x*y BN_sub (F, z2, t2); // F = D - C BN_add (G, z2, t2); // G = D + C BN_add (H, y2, x2); // H = B + A BN_mod_mul (x2, E, F, q, ctx); // x2 = E*F BN_mod_mul (y2, G, H, q, ctx); // y2 = G*H BN_mod_mul (z2, F, G, q, ctx); // z2 = F*G BN_mod_mul (t2, E, H, q, ctx); // t2 = E*H BN_free (E); BN_free (F); BN_free (G); BN_free (H); return EDDSAPoint {x2, y2, z2, t2}; } EDDSAPoint Mul (const EDDSAPoint& p, const BIGNUM * e, BN_CTX * ctx) const { BIGNUM * zero = BN_new (), * one = BN_new (); BN_zero (zero); BN_one (one); EDDSAPoint res {zero, one}; if (!BN_is_zero (e)) { int bitCount = BN_num_bits (e); for (int i = bitCount - 1; i >= 0; i--) { res = Double (res, ctx); if (BN_is_bit_set (e, i)) res = Sum (res, p, ctx); } } return res; } EDDSAPoint MulB (const uint8_t * e, BN_CTX * ctx) const // B*e, e is 32 bytes Little Endian { BIGNUM * zero = BN_new (), * one = BN_new (); BN_zero (zero); BN_one (one); EDDSAPoint res {zero, one}; bool carry = false; for (int i = 0; i < 32; i++) { uint8_t x = e[i]; if (carry) { if (x < 255) { x++; carry = false; } else x = 0; } if (x > 0) { if (x <= 128) res = Sum (res, Bi256[i][x-1], ctx); else { res = Sum (res, -Bi256[i][255-x], ctx); // -Bi[256-x] carry = true; } } } if (carry) res = Sum (res, Bi256Carry, ctx); return res; } EDDSAPoint Normalize (const EDDSAPoint& p, BN_CTX * ctx) const { if (p.z) { BIGNUM * x = BN_new (), * y = BN_new (); BN_mod_inverse (y, p.z, q, ctx); BN_mod_mul (x, p.x, y, q, ctx); // x = x/z BN_mod_mul (y, p.y, y, q, ctx); // y = y/z return EDDSAPoint{x, y}; } else return EDDSAPoint{BN_dup (p.x), BN_dup (p.y)}; } bool IsOnCurve (const EDDSAPoint& p, BN_CTX * ctx) const { BIGNUM * x2 = BN_new (); BN_sqr (x2, p.x, ctx); // x^2 BIGNUM * y2 = BN_new (); BN_sqr (y2, p.y, ctx); // y^2 // y^2 - x^2 - 1 - d*x^2*y^2 BIGNUM * tmp = BN_new (); BN_mul (tmp, d, x2, ctx); BN_mul (tmp, tmp, y2, ctx); BN_sub (tmp, y2, tmp); BN_sub (tmp, tmp, x2); BN_sub_word (tmp, 1); BN_mod (tmp, tmp, q, ctx); // % q bool ret = BN_is_zero (tmp); BN_free (x2); BN_free (y2); BN_free (tmp); return ret; } BIGNUM * RecoverX (const BIGNUM * y, BN_CTX * ctx) const { BIGNUM * y2 = BN_new (); BN_sqr (y2, y, ctx); // y^2 // xx = (y^2 -1)*inv(d*y^2 +1) BIGNUM * xx = BN_new (); BN_mul (xx, d, y2, ctx); BN_add_word (xx, 1); BN_mod_inverse (xx, xx, q, ctx); BN_sub_word (y2, 1); BN_mul (xx, y2, xx, ctx); // x = srqt(xx) = xx^(2^252-2) BIGNUM * x = BN_new (); BN_mod_exp (x, xx, two_252_2, q, ctx); // check (x^2 -xx) % q BN_sqr (y2, x, ctx); BN_mod_sub (y2, y2, xx, q, ctx); if (!BN_is_zero (y2)) BN_mod_mul (x, x, I, q, ctx); if (BN_is_odd (x)) BN_sub (x, q, x); BN_free (y2); BN_free (xx); return x; } EDDSAPoint DecodePoint (const uint8_t * buf, BN_CTX * ctx) const { // buf is 32 bytes Little Endian, convert it to Big Endian uint8_t buf1[EDDSA25519_PUBLIC_KEY_LENGTH]; for (size_t i = 0; i < EDDSA25519_PUBLIC_KEY_LENGTH/2; i++) // invert bytes { buf1[i] = buf[EDDSA25519_PUBLIC_KEY_LENGTH -1 - i]; buf1[EDDSA25519_PUBLIC_KEY_LENGTH -1 - i] = buf[i]; } bool isHighestBitSet = buf1[0] & 0x80; if (isHighestBitSet) buf1[0] &= 0x7f; // clear highest bit BIGNUM * y = BN_new (); BN_bin2bn (buf1, EDDSA25519_PUBLIC_KEY_LENGTH, y); auto x = RecoverX (y, ctx); if (BN_is_bit_set (x, 0) != isHighestBitSet) BN_sub (x, q, x); // x = q - x BIGNUM * z = BN_new (), * t = BN_new (); BN_one (z); BN_mod_mul (t, x, y, q, ctx); // pre-calculate t EDDSAPoint p {x, y, z, t}; if (!IsOnCurve (p, ctx)) LogPrint (eLogError, "Decoded point is not on 25519"); return p; } void EncodePoint (const EDDSAPoint& p, uint8_t * buf) const { EncodeBN (p.y, buf,EDDSA25519_PUBLIC_KEY_LENGTH); if (BN_is_bit_set (p.x, 0)) // highest bit buf[EDDSA25519_PUBLIC_KEY_LENGTH - 1] |= 0x80; // set highest bit } template BIGNUM * DecodeBN (const uint8_t * buf) const { // buf is Little Endian convert it to Big Endian uint8_t buf1[len]; for (size_t i = 0; i < len/2; i++) // invert bytes { buf1[i] = buf[len -1 - i]; buf1[len -1 - i] = buf[i]; } BIGNUM * res = BN_new (); BN_bin2bn (buf1, len, res); return res; } void EncodeBN (const BIGNUM * bn, uint8_t * buf, size_t len) const { bn2buf (bn, buf, len); // To Little Endian for (size_t i = 0; i < len/2; i++) // invert bytes { uint8_t tmp = buf[i]; buf[i] = buf[len -1 - i]; buf[len -1 - i] = tmp; } } private: BIGNUM * q, * l, * d, * I; // transient values BIGNUM * two_252_2; // 2^252-2 EDDSAPoint Bi256[32][128]; // per byte, Bi256[i][j] = (256+j+1)^i*B, we don't store zeroes // if j > 128 we use 256 - j and carry 1 to next byte // Bi256[0][0] = B, base point EDDSAPoint Bi256Carry; // Bi256[32][0] }; static std::unique_ptr g_Ed25519; std::unique_ptr& GetEd25519 () { if (!g_Ed25519) { auto c = new Ed25519(); if (!g_Ed25519) // make sure it was not created already g_Ed25519.reset (c); else delete c; } return g_Ed25519; } EDDSA25519Verifier::EDDSA25519Verifier (const uint8_t * signingKey) { memcpy (m_PublicKeyEncoded, signingKey, EDDSA25519_PUBLIC_KEY_LENGTH); BN_CTX * ctx = BN_CTX_new (); m_PublicKey = GetEd25519 ()->DecodePublicKey (m_PublicKeyEncoded, ctx); BN_CTX_free (ctx); } bool EDDSA25519Verifier::Verify (const uint8_t * buf, size_t len, const uint8_t * signature) const { uint8_t digest[64]; SHA512_CTX ctx; SHA512_Init (&ctx); SHA512_Update (&ctx, signature, EDDSA25519_SIGNATURE_LENGTH/2); // R SHA512_Update (&ctx, m_PublicKeyEncoded, EDDSA25519_PUBLIC_KEY_LENGTH); // public key SHA512_Update (&ctx, buf, len); // data SHA512_Final (digest, &ctx); return GetEd25519 ()->Verify (m_PublicKey, digest, signature); } EDDSA25519Signer::EDDSA25519Signer (const uint8_t * signingPrivateKey) { // expand key SHA512 (signingPrivateKey, EDDSA25519_PRIVATE_KEY_LENGTH, m_ExpandedPrivateKey); m_ExpandedPrivateKey[0] &= 0xF8; // drop last 3 bits m_ExpandedPrivateKey[EDDSA25519_PRIVATE_KEY_LENGTH - 1] &= 0x1F; // drop first 3 bits m_ExpandedPrivateKey[EDDSA25519_PRIVATE_KEY_LENGTH - 1] |= 0x40; // set second bit // generate and encode public key BN_CTX * ctx = BN_CTX_new (); auto publicKey = GetEd25519 ()->GeneratePublicKey (m_ExpandedPrivateKey, ctx); GetEd25519 ()->EncodePublicKey (publicKey, m_PublicKeyEncoded, ctx); BN_CTX_free (ctx); } void EDDSA25519Signer::Sign (const uint8_t * buf, int len, uint8_t * signature) const { GetEd25519 ()->Sign (m_ExpandedPrivateKey, m_PublicKeyEncoded, buf, len, signature); } } }