# ed25519ll / ed25519ll / djbec.py

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241``` ```# Ed25519 digital signatures # Based on http://ed25519.cr.yp.to/python/ed25519.py # See also http://ed25519.cr.yp.to/software.html # Adapted by Ron Garret # Sped up considerably using coordinate transforms found on: # http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html # Specifically add-2008-hwcd-4 and dbl-2008-hwcd import hashlib b = 256 q = 2**255 - 19 l = 2**252 + 27742317777372353535851937790883648493 def H(m): return hashlib.sha512(m).digest() def expmod(b,e,m): if e == 0: return 1 t = expmod(b,e/2,m)**2 % m if e & 1: t = (t*b) % m return t # Can probably get some extra speedup here by replacing this with # an extended-euclidean, but performance seems OK without that def inv(x): return expmod(x,q-2,q) d = -121665 * inv(121666) I = expmod(2,(q-1)/4,q) def xrecover(y): xx = (y*y-1) * inv(d*y*y+1) x = expmod(xx,(q+3)/8,q) if (x*x - xx) % q != 0: x = (x*I) % q if x % 2 != 0: x = q-x return x By = 4 * inv(5) Bx = xrecover(By) B = [Bx % q,By % q] def edwards(P,Q): x1 = P[0] y1 = P[1] x2 = Q[0] y2 = Q[1] x3 = (x1*y2+x2*y1) * inv(1+d*x1*x2*y1*y2) y3 = (y1*y2+x1*x2) * inv(1-d*x1*x2*y1*y2) return (x3 % q,y3 % q) #def scalarmult(P,e): # if e == 0: return [0,1] # Q = scalarmult(P,e/2) # Q = edwards(Q,Q) # if e & 1: Q = edwards(Q,P) # return Q # Faster (!) version based on: # http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html def xpt_add(pt1, pt2): (X1, Y1, Z1, T1) = pt1 (X2, Y2, Z2, T2) = pt2 A = ((Y1-X1)*(Y2+X2)) % q B = ((Y1+X1)*(Y2-X2)) % q C = (Z1*2*T2) % q D = (T1*2*Z2) % q E = (D+C) % q F = (B-A) % q G = (B+A) % q H = (D-C) % q X3 = (E*F) % q Y3 = (G*H) % q Z3 = (F*G) % q T3 = (E*H) % q return (X3, Y3, Z3, T3) def xpt_double (pt): (X1, Y1, Z1, _) = pt A = (X1*X1) B = (Y1*Y1) C = (2*Z1*Z1) D = (-A) % q J = (X1+Y1) % q E = (J*J-A-B) % q G = (D+B) % q F = (G-C) % q H = (D-B) % q X3 = (E*F) % q Y3 = (G*H) % q Z3 = (F*G) % q T3 = (E*H) % q return (X3, Y3, Z3, T3) def pt_xform (pt): (x, y) = pt return (x, y, 1, (x*y)%q) def pt_unxform (pt): (x, y, z, _) = pt return ((x*inv(z))%q, (y*inv(z))%q) def xpt_mult (pt, n): if n==0: return pt_xform((0,1)) _ = xpt_double(xpt_mult(pt, n>>1)) return xpt_add(_, pt) if n&1 else _ def scalarmult(pt, e): return pt_unxform(xpt_mult(pt_xform(pt), e)) def encodeint(y): bits = [(y >> i) & 1 for i in range(b)] return ''.join([chr(sum([bits[i * 8 + j] << j for j in range(8)])) for i in range(b/8)]) def encodepoint(P): x = P[0] y = P[1] bits = [(y >> i) & 1 for i in range(b - 1)] + [x & 1] return ''.join([chr(sum([bits[i * 8 + j] << j for j in range(8)])) for i in range(b/8)]) def bit(h,i): return (ord(h[i/8]) >> (i%8)) & 1 def publickey(sk): h = H(sk) a = 2**(b-2) + sum(2**i * bit(h,i) for i in range(3,b-2)) A = scalarmult(B,a) return encodepoint(A) def Hint(m): h = H(m) return sum(2**i * bit(h,i) for i in range(2*b)) def signature(m,sk,pk): h = H(sk) a = 2**(b-2) + sum(2**i * bit(h,i) for i in range(3,b-2)) r = Hint(''.join([h[i] for i in range(b/8,b/4)]) + m) R = scalarmult(B,r) S = (r + Hint(encodepoint(R) + pk + m) * a) % l return encodepoint(R) + encodeint(S) def isoncurve(P): x = P[0] y = P[1] return (-x*x + y*y - 1 - d*x*x*y*y) % q == 0 def decodeint(s): return sum(2**i * bit(s,i) for i in range(0,b)) def decodepoint(s): y = sum(2**i * bit(s,i) for i in range(0,b-1)) x = xrecover(y) if x & 1 != bit(s,b-1): x = q-x P = [x,y] if not isoncurve(P): raise Exception("decoding point that is not on curve") return P def checkvalid(s,m,pk): if len(s) != b/4: raise Exception("signature length is wrong") if len(pk) != b/8: raise Exception("public-key length is wrong") R = decodepoint(s[0:b/8]) A = decodepoint(pk) S = decodeint(s[b/8:b/4]) h = Hint(encodepoint(R) + pk + m) v1 = scalarmult(B,S) # v2 = edwards(R,scalarmult(A,h)) v2 = pt_unxform(xpt_add(pt_xform(R), pt_xform(scalarmult(A, h)))) return v1==v2 ########################################################## # # Curve25519 reference implementation by Matthew Dempsky, from: # http://cr.yp.to/highspeed/naclcrypto-20090310.pdf P = 2 ** 255 - 19 A = 486662 def expmod(b, e, m): if e == 0: return 1 t = expmod(b, e / 2, m) ** 2 % m if e & 1: t = (t * b) % m return t def inv(x): return expmod(x, P - 2, P) def add((xn,zn), (xm,zm), (xd,zd)): x = 4 * (xm * xn - zm * zn) ** 2 * zd z = 4 * (xm * zn - zm * xn) ** 2 * xd return (x % P, z % P) def double((xn,zn)): x = (xn ** 2 - zn ** 2) ** 2 z = 4 * xn * zn * (xn ** 2 + A * xn * zn + zn ** 2) return (x % P, z % P) def curve25519(n, base=9): one = (base,1) two = double(one) # f(m) evaluates to a tuple # containing the mth multiple and the # (m+1)th multiple of base. def f(m): if m == 1: return (one, two) (pm, pm1) = f(m / 2) if (m & 1): return (add(pm, pm1, one), double(pm1)) return (double(pm), add(pm, pm1, one)) ((x,z), _) = f(n) return (x * inv(z)) % P import random def genkey(n=0): n = n or random.randint(0,P) n &= ~7 n &= ~(128 << 8 * 31) n |= 64 << 8 * 31 return n def str2int(s): return sum(ord(s[i]) << (8 * i) for i in range(32)) def int2str(n): return ''.join([chr((n >> (8 * i)) & 255) for i in range(32)]) ################################################# def dsa_test(): msg = str(random.randint(q,q+q)) sk = str(random.randint(q,q+q)) pk = publickey(sk) sig = signature(msg, sk, pk) return checkvalid(sig, msg, pk) def dh_test(): sk1 = genkey() sk2 = genkey() return curve25519(sk1, curve25519(sk2)) == curve25519(sk2, curve25519(sk1)) ```