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I2P: End-to-End encrypted and anonymous Internet https://i2pd.website/
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i2pd/Signature.cpp

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#include <memory>
#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 Bi16 table
Bi16Carry = { Bx, By }; // B
for (int i = 0; i < 64; i++)
{
Bi16[i][0] = Bi16Carry;
for (int j = 1; j < 8; j++)
Bi16[i][j] = Sum (Bi16[i][j-1], Bi16[i][0], ctx); // (16+j+1)^i*B
Bi16Carry = Bi16[i][7];
for (int j = 8; j < 16; j++)
Bi16Carry = Sum (Bi16Carry, Bi16[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)),
Bi16Carry (other.Bi16Carry)
{
for (int i = 0; i < 64; i++)
for (int j = 0; j < 8; j++)
Bi16[i][j] = other.Bi16[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<EDDSA25519_PRIVATE_KEY_LENGTH> (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 ();
BIGNUM * z1 = p1.z, * t1 = p1.t;
if (!z1) { z1 = BN_new (); BN_one (z1); }
if (!t1) { t1 = BN_new (); BN_mul (t1, p1.x, p1.y, ctx); }
BIGNUM * z2 = p2.z, * t2 = p2.t;
if (!z2) { z2 = BN_new (); BN_one (z2); }
if (!t2) { t2 = BN_new (); BN_mul (t2, p2.x, p2.y, ctx); }
BN_mul (x3, p1.x, p2.x, ctx); // A = x1*x2
BN_mul (y3, p1.y, p2.y, ctx); // B = y1*y2
BN_mul (t3, t1, t2, ctx);
BN_mul (t3, t3, d, ctx); // C = d*t1*t2
BN_mul (z3, z1, z2, ctx); // D = z1*z2
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
if (!p1.z) BN_free (z1);
if (!p1.t) BN_free (t1);
if (!p2.z) BN_free (z2);
if (!p2.t) BN_free (t2);
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 ();
BIGNUM * z = p.z, * t = p.t;
if (!z) { z = BN_new (); BN_one (z); }
if (!t) { t = BN_new (); BN_mul (t, p.x, p.y, ctx); }
BN_sqr (x2, p.x, ctx); // x2 = A = x^2
BN_sqr (y2, p.y, ctx); // y2 = B = y^2
BN_sqr (t2, t, ctx);
BN_mul (t2, t2, d, ctx); // t2 = C = d*t^2
BN_sqr (z2, z, ctx); // z2 = D = z^2
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
if (!p.z) BN_free (z);
if (!p.t) BN_free (t);
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] & 0x0F; // 4 low bits
if (carry) { x++; if (x <= 15) carry = false; else x = 0; };
if (x > 0)
{
if (x <= 8)
res = Sum (res, Bi16[i*2][x-1], ctx);
else
{
res = Sum (res, -Bi16[i*2][15-x], ctx); // -Bi[16-x]
carry = true;
}
}
x = e[i] >> 4; // 4 high bits
if (carry) { x++; if (x <= 15) carry = false; else x = 0; };
if (x > 0)
{
if (x <= 8)
res = Sum (res, Bi16[i*2+1][x-1], ctx);
else
{
res = Sum (res, -Bi16[i*2+1][15-x], ctx); // -Bi[16-x]
carry = true;
}
}
}
if (carry) res = Sum (res, Bi16Carry, 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
EDDSAPoint p {x, y};
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<int len>
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 Bi16[64][8]; // per 4-bits, Bi16[i][j] = (16+j+1)^i*B, we don't store zeroes
// if j > 8 we use 16 - j and carry 1 to next 4-bits
// Bi16[0][0] = B, base point
EDDSAPoint Bi16Carry; // Bi16[64][0]
};
static std::unique_ptr<Ed25519> g_Ed25519;
std::unique_ptr<Ed25519>& GetEd25519 ()
{
if (!g_Ed25519)
g_Ed25519.reset (new Ed25519());
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);
}
}
}