diff --git a/vendor/ristretto.sage b/vendor/ristretto.sage index f40bebd..04cf4f9 100644 --- a/vendor/ristretto.sage +++ b/vendor/ristretto.sage @@ -49,7 +49,11 @@ def isqrt(x,exn=InvalidEncodingException("Not on curve")): """Return 1/sqrt(x)""" if x==0: return 0 if not is_square(x): raise exn - return 1/sqrt(x) + s = sqrt(x) + #if negative(s): s=-s + return 1/s + +def inv0(x): return 1/x if x != 0 else 0 def isqrt_i(x): """Return 1/sqrt(x) or 1/sqrt(zeta * x)""" @@ -117,16 +121,20 @@ class QuotientEdwardsPoint(object): else: return self.__class__(-self.x, -self.y) + def doubleAndEncodeSpec(self): + return (self+self).encode() # Utility functions @classmethod - def bytesToGf(cls,bytes,mustBeProper=True,mustBePositive=False): + def bytesToGf(cls,bytes,mustBeProper=True,mustBePositive=False,maskHiBits=False): """Convert little-endian bytes to field element, sanity check length""" if len(bytes) != cls.encLen: raise InvalidEncodingException("wrong length %d" % len(bytes)) s = dec_le(bytes) - if mustBeProper and s >= cls.F.modulus(): + if mustBeProper and s >= cls.F.order(): raise InvalidEncodingException("%d out of range!" % s) + bitlen = int(ceil(log(cls.F.order())/log(2))) + if maskHiBits: s &= 2^bitlen-1 s = cls.F(s) if mustBePositive and negative(s): raise InvalidEncodingException("%d is negative!" % s) @@ -197,7 +205,42 @@ class RistrettoPoint(QuotientEdwardsPoint): if negative(isr^2*num*y*t): y = -y s = isr*y*(z-y) + return self.gfToBytes(s,mustBePositive=True) + + @optimized_version_of("doubleAndEncodeSpec") + def doubleAndEncode(self): + X,Y,Z,T = self.xyzt() + a,d,mneg = self.a,self.d,self.mneg + + if self.cofactor==8: + e = 2*X*Y + f = Z^2+d*T^2 + g = Y^2-a*X^2 + h = Z^2-d*T^2 + + inv1 = 1/(e*f*g*h) + z_inv = inv1*e*g # 1 / (f*h) + t_inv = inv1*f*h + if negative(e*g*z_inv): + if a==-1: sqrta = self.i + else: sqrta = -1 + e,f,g,h = g,h,-e,f*sqrta + factor = self.i + else: + factor = self.magic + + if negative(h*e*z_inv): g=-g + s = (h-g)*factor*g*t_inv + + else: + foo = Y^2+a*X^2 + bar = X*Y + den = 1/(foo*bar) + if negative(2*bar^2*den): tmp = a*X^2 + else: tmp = Y^2 + s = self.magic*(Z^2-tmp)*foo*den + return self.gfToBytes(s,mustBePositive=True) @classmethod @@ -238,8 +281,9 @@ class RistrettoPoint(QuotientEdwardsPoint): @classmethod def elligatorSpec(cls,r0): a,d = cls.a,cls.d - r = cls.qnr * cls.bytesToGf(r0)^2 + r = cls.qnr * cls.bytesToGf(r0,mustBeProper=False,maskHiBits=True)^2 den = (d*r-a)*(a*r-d) + if den == 0: return cls() n1 = cls.a*(r+1)*(a+d)*(d-a)/den n2 = r*n1 if is_square(n1): @@ -253,7 +297,7 @@ class RistrettoPoint(QuotientEdwardsPoint): @optimized_version_of("elligatorSpec") def elligator(cls,r0): a,d = cls.a,cls.d - r0 = cls.bytesToGf(r0) + r0 = cls.bytesToGf(r0,mustBeProper=False,maskHiBits=True) r = cls.qnr * r0^2 den = (d*r-a)*(a*r-d) num = cls.a*(r+1)*(a+d)*(d-a) @@ -278,15 +322,11 @@ class Decaf_1_1_Point(QuotientEdwardsPoint): if self.cofactor==8 and negative(x*y*self.isoMagic): x,y = self.torque() - - isr2 = isqrt(a*(y^2-1)) * sqrt(a*d-1) - + sr = xsqrt(1-a*x^2) - assert sr in [isr2*x*y,-isr2*x*y] - - altx = 1/isr2*self.isoMagic - if negative(altx): s = (1+x*y*isr2)/(a*x) - else: s = (1-x*y*isr2)/(a*x) + altx = x*y*self.isoMagic / sr + if negative(altx): s = (1+sr)/x + else: s = (1-sr)/x return self.gfToBytes(s,mustBePositive=True) @@ -297,52 +337,141 @@ class Decaf_1_1_Point(QuotientEdwardsPoint): s = cls.bytesToGf(s,mustBePositive=True) if s==0: return cls() - isr = isqrt(s^4 + 2*(a-2*d)*s^2 + 1) - altx = 2*s*isr*cls.isoMagic - if negative(altx): isr = -isr + t = xsqrt(s^4 + 2*(a-2*d)*s^2 + 1) + altx = 2*s*cls.isoMagic/t + if negative(altx): t = -t x = 2*s / (1+a*s^2) - y = (1-a*s^2) * isr + y = (1-a*s^2) / t if cls.cofactor==8 and (negative(x*y*cls.isoMagic) or y==0): raise InvalidEncodingException("x*y is invalid: %d, %d" % (x,y)) return cls(x,y) - @optimized_version_of("encodeSpec") - def encode(self): - """Encode, optimized version""" + def toJacobiQuartic(self,toggle_rotation=False,toggle_altx=False,toggle_s=False): + "Return s,t on jacobi curve" a,d = self.a,self.d x,y,z,t = self.xyzt() if self.cofactor == 8: # Cofactor 8 version + # Simulate IMAGINE_TWIST because that's how libdecaf does it + x = self.i*x + t = self.i*t + a = -a + d = -d + + # OK, the actual libdecaf code should be here num = (z+y)*(z-y) den = x*y - tmp = isqrt(num*(a-d)*den^2) - - if negative(tmp^2*den*num*(a-d)*t^2*self.isoMagic): - den,num = num,den - tmp *= sqrt(a-d) # witness that cofactor is 8 - yisr = x*sqrt(a) - toggle = (a==1) - else: - yisr = y*(a*d-1) - toggle = False + isr = isqrt(num*(a-d)*den^2) + + iden = isr * den * self.isoMagic # 1/sqrt((z+y)(z-y)) = 1/sqrt(1-Y^2) / z + inum = isr * num # sqrt(1-Y^2) * z / xysqrt(a-d) ~ 1/sqrt(1-ax^2)/z - tiisr = tmp*num - altx = tiisr*t*self.isoMagic - if negative(altx) != toggle: tiisr =- tiisr - s = tmp*den*yisr*(tiisr*z - 1) + if negative(iden*inum*self.i*t^2*(d-a)) != toggle_rotation: + iden,inum = inum,iden + fac = x*sqrt(a) + toggle=(a==-1) + else: + fac = y + toggle=False + + imi = self.isoMagic * self.i + altx = inum*t*imi + neg_altx = negative(altx) != toggle_altx + if neg_altx != toggle: inum =- inum + + tmp = fac*(inum*z + 1) + s = iden*tmp*imi + + negm1 = (negative(s) != toggle_s) != neg_altx + if negm1: m1 = a*fac + z + else: m1 = a*fac - z + + swap = toggle_s else: # Much simpler cofactor 4 version num = (x+t)*(x-t) isr = isqrt(num*(a-d)*x^2) - ratio = isr*num - if negative(ratio*self.isoMagic): ratio=-ratio - s = (a-d)*isr*x*(ratio*z - t) + ratio = isr*num + altx = ratio*self.isoMagic + + neg_altx = negative(altx) != toggle_altx + if neg_altx: ratio =- ratio + + tmp = ratio*z - t + s = (a-d)*isr*x*tmp + + negx = (negative(s) != toggle_s) != neg_altx + if negx: m1 = -a*t + x + else: m1 = -a*t - x + + swap = toggle_s + + if negative(s): s = -s - return self.gfToBytes(s,mustBePositive=True) + return s,m1,a*tmp,swap + + def invertElligator(self,toggle_r=False,*args,**kwargs): + "Produce preimage of self under elligator, or None" + a,d = self.a,self.d + + rets = [] + + tr = [False,True] if self.cofactor == 8 else [False] + for toggle_rotation in tr: + for toggle_altx in [False,True]: + for toggle_s in [False,True]: + for toggle_r in [False,True]: + s,m1,m12,swap = self.toJacobiQuartic(toggle_rotation,toggle_altx,toggle_s) + + #print + #print toggle_rotation,toggle_altx,toggle_s + #print m1 + #print m12 + + + if self == self.__class__(): + if self.cofactor == 4: + # Hacks for identity! + if toggle_altx: m12 = 1 + elif toggle_s: m1 = 1 + elif toggle_r: continue + ## BOTH??? + + else: + m12 = 1 + imi = self.isoMagic * self.i + if toggle_rotation: + if toggle_altx: m1 = -imi + else: m1 = +imi + else: + if toggle_altx: m1 = 0 + else: m1 = a-d + + rnum = (d*a*m12-m1) + rden = ((d*a-1)*m12+m1) + if swap: rnum,rden = rden,rnum + + ok,sr = isqrt_i(rnum*rden*self.qnr) + if not ok: continue + sr *= rnum + #print "Works! %d %x" % (swap,sr) + + if negative(sr) != toggle_r: sr = -sr + ret = self.gfToBytes(sr) + if self.elligator(ret) != self and self.elligator(ret) != -self: + print "WRONG!",[toggle_rotation,toggle_altx,toggle_s] + if self.elligator(ret) == -self and self != -self: print "Negated!",[toggle_rotation,toggle_altx,toggle_s] + rets.append(bytes(ret)) + return rets + + @optimized_version_of("encodeSpec") + def encode(self): + """Encode, optimized version""" + return self.gfToBytes(self.toJacobiQuartic()[0]) @classmethod @optimized_version_of("decodeSpec") @@ -351,7 +480,7 @@ class Decaf_1_1_Point(QuotientEdwardsPoint): a,d = cls.a,cls.d s = cls.bytesToGf(s,mustBePositive=True) - if s==0: return cls() + #if s==0: return cls() s2 = s^2 den = 1+a*s2 num = den^2 - 4*d*s2 @@ -374,13 +503,63 @@ class Decaf_1_1_Point(QuotientEdwardsPoint): x = 2*s / (1+a*s^2) y = (1-a*s^2) / t return cls(x,sgn*y) + + @optimized_version_of("doubleAndEncodeSpec") + def doubleAndEncode(self): + X,Y,Z,T = self.xyzt() + a,d = self.a,self.d + + if self.cofactor == 8: + # Cofactor 8 version + # Simulate IMAGINE_TWIST because that's how libdecaf does it + X = self.i*X + T = self.i*T + a = -a + d = -d + # TODO: This is only being called for a=-1, so could + # be wrong for a=1 + + e = 2*X*Y + f = Y^2+a*X^2 + g = Y^2-a*X^2 + h = Z^2-d*T^2 + + eim = e*self.isoMagic + inv = 1/(eim*g*f*h) + fh_inv = eim*g*inv*self.i + + if negative(eim*g*fh_inv): + idf = g*self.isoMagic*self.i + bar = f + foo = g + test = eim*f + else: + idf = eim + bar = h + foo = -eim + test = g*h + + if negative(test*fh_inv): bar =- bar + s = idf*(foo+bar)*inv*f*h + + else: + xy = X*Y + h = Z^2-d*T^2 + inv = 1/(xy*h) + if negative(inv*2*xy^2*self.isoMagic): tmp = Y + else: tmp = X + s = tmp^2*h*inv # = X/Y or Y/X, interestingly + + return self.gfToBytes(s,mustBePositive=True) @classmethod - def elligatorSpec(cls,r0): + def elligatorSpec(cls,r0,fromR=False): a,d = cls.a,cls.d - r = cls.qnr * cls.bytesToGf(r0)^2 + if fromR: r = r0 + else: r = cls.qnr * cls.bytesToGf(r0,mustBeProper=False,maskHiBits=True)^2 den = (d*r-(d-a))*((d-a)*r-d) + if den == 0: return cls() n1 = (r+1)*(a-2*d)/den n2 = r*n1 if is_square(n1): @@ -394,7 +573,7 @@ class Decaf_1_1_Point(QuotientEdwardsPoint): @optimized_version_of("elligatorSpec") def elligator(cls,r0): a,d = cls.a,cls.d - r0 = cls.bytesToGf(r0) + r0 = cls.bytesToGf(r0,mustBeProper=False,maskHiBits=True) r = cls.qnr * r0^2 den = (d*r-(d-a))*((d-a)*r-d) num = (r+1)*(a-2*d) @@ -408,6 +587,40 @@ class Decaf_1_1_Point(QuotientEdwardsPoint): if negative(s) == iss: s = -s return cls.fromJacobiQuartic(s,t) + def elligatorInverseBruteForce(self): + """Invert Elligator using SAGE's polynomial solver""" + a,d = self.a,self.d + R. = self.F[] + r = self.qnr * r0^2 + den = (d*r-(d-a))*((d-a)*r-d) + n1 = (r+1)*(a-2*d)/den + n2 = r*n1 + ret = set() + for s2,t in [(n1, -(r-1)*(a-2*d)^2 / den - 1), + (n2,r*(r-1)*(a-2*d)^2 / den - 1)]: + x2 = 4*s2/(1+a*s2)^2 + y = (1-a*s2) / t + + selfT = self + for i in xrange(self.cofactor/2): + xT,yT = selfT + polyX = xT^2-x2 + polyY = yT-y + sx = set(r for r,_ in polyX.numerator().roots()) + sy = set(r for r,_ in polyY.numerator().roots()) + ret = ret.union(sx.intersection(sy)) + + selfT = selfT.torque() + + ret = [self.gfToBytes(r) for r in ret] + + for r in ret: + assert self.elligator(r) in [self,-self] + + ret = [r for r in ret if self.elligator(r) == self] + + return ret + class Ed25519Point(RistrettoPoint): F = GF(2^255-19) d = F(-121665/121666) @@ -455,7 +668,7 @@ class IsoEd448Point(RistrettoPoint): @classmethod def base(cls): return cls( # RFC has it wrong - -345397493039729516374008604150537410266655260075183290216406970281645695073672344430481787759340633221708391583424041788924124567700732, + 345397493039729516374008604150537410266655260075183290216406970281645695073672344430481787759340633221708391583424041788924124567700732, -363419362147803445274661903944002267176820680343659030140745099590306164083365386343198191849338272965044442230921818680526749009182718 ) @@ -464,7 +677,6 @@ class TwistedEd448GoldilocksPoint(Decaf_1_1_Point): d = F(-39082) a = F(-1) qnr = -1 - magic = isqrt(a*d-1) cofactor = 4 encLen = 56 isoMagic = IsoEd448Point.magic @@ -478,14 +690,13 @@ class Ed448GoldilocksPoint(Decaf_1_1_Point): d = F(-39081) a = F(1) qnr = -1 - magic = isqrt(a*d-1) cofactor = 4 encLen = 56 isoMagic = IsoEd448Point.magic @classmethod def base(cls): - return -2*cls( # FIXME: make not negative + return 2*cls( 224580040295924300187604334099896036246789641632564134246125461686950415467406032909029192869357953282578032075146446173674602635247710, 298819210078481492676017930443930673437544040154080242095928241372331506189835876003536878655418784733982303233503462500531545062832660 ) @@ -532,19 +743,29 @@ def test(cls,n): P = cls.base() - print "base", list(P.encode()) - for i in xrange(16): - Q = P*i - print i, list(Q.encode()) - Q = cls() for i in xrange(n): - #print i, binascii.hexlify(Q.encode()) - QQ = cls.decode(Q.encode()) + #print binascii.hexlify(Q.encode()) + QE = Q.encode() + QQ = cls.decode(QE) if QQ != Q: raise TestFailedException("Round trip %s != %s" % (str(QQ),str(Q))) + + # Testing s -> 1/s: encodes -point on cofactor + s = cls.bytesToGf(QE) + if s != 0: + ss = cls.gfToBytes(1/s,mustBePositive=True) + try: + QN = cls.decode(ss) + if cls.cofactor == 8: + raise TestFailedException("1/s shouldnt work for cofactor 8") + if QN != -Q: + raise TestFailedException("s -> 1/s should negate point for cofactor 4") + except InvalidEncodingException as e: + # Should be raised iff cofactor==8 + if cls.cofactor == 4: + raise TestFailedException("s -> 1/s should work for cofactor 4") QT = Q - QE = Q.encode() for h in xrange(cls.cofactor): QT = QT.torque() if QT.encode() != QE: @@ -559,27 +780,26 @@ def test(cls,n): Q2 = Q0*(r+1) if Q1 + Q0 != Q2: raise TestFailedException("Scalarmul doesn't work") Q = Q1 - -test(Ed25519Point,100) -#test(NegEd25519Point,100) -#test(IsoEd25519Point,100) -#test(IsoEd448Point,100) -#test(TwistedEd448GoldilocksPoint,100) -#test(Ed448GoldilocksPoint,100) - def testElligator(cls,n): print "Testing elligator on %s" % cls.__name__ for i in xrange(n): r = randombytes(cls.encLen) - Q = cls.elligator(r) - print list(r), list(Q.encode()) - -testElligator(Ed25519Point,100) -#testElligator(NegEd25519Point,100) -#testElligator(IsoEd448Point,100) -#testElligator(Ed448GoldilocksPoint,100) -#testElligator(TwistedEd448GoldilocksPoint,100) + P = cls.elligator(r) + if hasattr(P,"invertElligator"): + iv = P.invertElligator() + modr = bytes(cls.gfToBytes(cls.bytesToGf(r,mustBeProper=False,maskHiBits=True))) + iv2 = P.torque().invertElligator() + if modr not in iv: print "Failed to invert Elligator!" + if len(iv) != len(set(iv)): + print "Elligator inverses not unique!", len(set(iv)), len(iv) + if iv != iv2: + print "Elligator is untorqueable!" + #print [binascii.hexlify(j) for j in iv] + #print [binascii.hexlify(j) for j in iv2] + #break + else: + pass # TODO def gangtest(classes,n): print "Gang test",[cls.__name__ for cls in classes] @@ -607,5 +827,31 @@ def gangtest(classes,n): for c,ret in zip(classes,rets): print c,binascii.hexlify(ret) print -gangtest([IsoEd448Point,TwistedEd448GoldilocksPoint,Ed448GoldilocksPoint],100) -gangtest([Ed25519Point,IsoEd25519Point],100) + +def testDoubleAndEncode(cls,n): + print "Testing doubleAndEncode on %s" % cls.__name__ + for i in xrange(n): + r1 = randombytes(cls.encLen) + r2 = randombytes(cls.encLen) + u = cls.elligator(r1) + cls.elligator(r2) + u.doubleAndEncode() + +testDoubleAndEncode(Ed25519Point,100) +testDoubleAndEncode(NegEd25519Point,100) +testDoubleAndEncode(IsoEd25519Point,100) +testDoubleAndEncode(IsoEd448Point,100) +testDoubleAndEncode(TwistedEd448GoldilocksPoint,100) +#test(Ed25519Point,100) +#test(NegEd25519Point,100) +#test(IsoEd25519Point,100) +#test(IsoEd448Point,100) +#test(TwistedEd448GoldilocksPoint,100) +#test(Ed448GoldilocksPoint,100) +#testElligator(Ed25519Point,100) +#testElligator(NegEd25519Point,100) +#testElligator(IsoEd25519Point,100) +#testElligator(IsoEd448Point,100) +#testElligator(Ed448GoldilocksPoint,100) +#testElligator(TwistedEd448GoldilocksPoint,100) +#gangtest([IsoEd448Point,TwistedEd448GoldilocksPoint,Ed448GoldilocksPoint],100) +#gangtest([Ed25519Point,IsoEd25519Point],100)