diff --git a/aux/ristretto.sage b/aux/ristretto.sage
index 24ac5e1..68cfaa4 100644
--- a/aux/ristretto.sage
+++ b/aux/ristretto.sage
@@ -50,7 +50,7 @@ def isqrt(x,exn=InvalidEncodingException("Not on curve")):
 
 def isqrt_i(x):
     """Return 1/sqrt(x) or 1/sqrt(zeta * x)"""
-    if x==0: return 0
+    if x==0: return True,0
     gen = x.parent(-1)
     while is_square(gen): gen = sqrt(gen)
     if is_square(x): return True,1/sqrt(x)
@@ -225,7 +225,6 @@ class RistrettoPoint(QuotientEdwardsPoint):
         a,d = cls.a,cls.d
         assert s^4 - 2*cls.a*(1-2*d/(d-a))*s^2 + 1 == t^2
         x = 2*s*cls.magic / t
-        if negative(x): x = -x # TODO: doesn't work without resolving x
         y = (1+a*s^2) / (1-a*s^2)
         return cls(sgn*x,y)
             
@@ -237,11 +236,11 @@ class RistrettoPoint(QuotientEdwardsPoint):
         n1 = cls.a*(r+1)*(a+d)*(d-a)/den
         n2 = r*n1
         if is_square(n1):
-            sgn,s,t = 1,xsqrt(n1), -(r-1)*(a+d)^2 / den - 1
+            sgn,s,t =  1, xsqrt(n1), -(r-1)*(a+d)^2 / den - 1
         else:
-            sgn,s,t = -1,xsqrt(n2), r*(r-1)*(a+d)^2 / den - 1
+            sgn,s,t = -1,-xsqrt(n2), r*(r-1)*(a+d)^2 / den - 1
         
-        return cls.fromJacobiQuartic(s,t,sgn)
+        return cls.fromJacobiQuartic(s,t)
             
     @classmethod
     @optimized_version_of("elligatorSpec")
@@ -257,8 +256,9 @@ class RistrettoPoint(QuotientEdwardsPoint):
         else:   sgn,twiddle = -1,r0*cls.qnr
         isri *= twiddle
         s = isri*num
-        t = isri*s*(r-1)*(d+a)^2 + sgn
-        return cls.fromJacobiQuartic(s,t,sgn)
+        t = -sgn*isri*s*(r-1)*(d+a)^2 - 1
+        if negative(s) == iss: s = -s
+        return cls.fromJacobiQuartic(s,t)
 
 
 class Decaf_1_1_Point(QuotientEdwardsPoint):
@@ -364,8 +364,6 @@ class Decaf_1_1_Point(QuotientEdwardsPoint):
         """Convert point from its Jacobi Quartic representation"""
         a,d = cls.a,cls.d
         if s==0: return cls()
-        altx = 2*s*cls.isoMagic / t
-        if negative(altx): t = -t
         x = 2*s / (1+a*s^2)
         y = (1-a*s^2) / t
         return cls(x,sgn*y)
@@ -379,16 +377,29 @@ class Decaf_1_1_Point(QuotientEdwardsPoint):
         n1 = (r+1)*(a-2*d)/den
         n2 = r*n1
         if is_square(n1):
-            sgn,s,t = 1,xsqrt(n1), -(r-1)*(a-2*d)^2 / den - 1
+            sgn,s,t = 1,   xsqrt(n1),  -(r-1)*(a-2*d)^2 / den - 1
         else:
-            sgn,s,t = -1,xsqrt(n2), r*(r-1)*(a-2*d)^2 / den - 1
+            sgn,s,t = -1, -xsqrt(n2), r*(r-1)*(a-2*d)^2 / den - 1
         
-        return cls.fromJacobiQuartic(s,t,sgn)
+        return cls.fromJacobiQuartic(s,t)
             
     @classmethod
     @optimized_version_of("elligatorSpec")
     def elligator(cls,r0):
-        return cls.elligatorSpec(r0)
+        a,d = cls.a,cls.d
+        r0 = cls.bytesToGf(r0)
+        r = cls.qnr * r0^2
+        den = (d*r-(d-a))*((d-a)*r-d)
+        num = (r+1)*(a-2*d)
+        
+        iss,isri = isqrt_i(num*den)
+        if iss: sgn,twiddle =  1,1
+        else:   sgn,twiddle = -1,r0*cls.qnr
+        isri *= twiddle
+        s = isri*num
+        t = -sgn*isri*s*(r-1)*(a-2*d)^2 - 1
+        if negative(s) == iss: s = -s
+        return cls.fromJacobiQuartic(s,t)
             
 class Ed25519Point(RistrettoPoint):
     F = GF(2^255-19)
@@ -477,7 +488,26 @@ class TestFailedException(Exception): pass
 
 def test(cls,n):
     print "Testing curve %s" % cls.__name__
-    # TODO: test corner cases like 0,1,i
+    
+    specials = [1]
+    ii = cls.F(-1)
+    while is_square(ii):
+        specials.append(ii)
+        ii = sqrt(ii)
+    specials.append(ii)
+    for i in specials:
+        if negative(cls.F(i)): i = -i
+        i = enc_le(i,cls.encLen)
+        try:
+            Q = cls.decode(i)
+            QE = Q.encode()
+            if QE != i:
+                raise TestFailedException("Round trip special %s != %s" %
+                     (binascii.hexlify(QE),binascii.hexlify(i)))
+        except NotOnCurveException: pass
+        except InvalidEncodingException: pass
+        
+    
     P = cls.base()
     Q = cls()
     for i in xrange(n):
@@ -520,6 +550,13 @@ testElligator(TwistedEd448GoldilocksPoint,100)
 
 def gangtest(classes,n):
     print "Gang test",[cls.__name__ for cls in classes]
+    specials = [1]
+    ii = classes[0].F(-1)
+    while is_square(ii):
+        specials.append(ii)
+        ii = sqrt(ii)
+    specials.append(ii)
+    
     for i in xrange(n):
         rets = [bytes((cls.base()*i).encode()) for cls in classes]
         if len(set(rets)) != 1:
@@ -528,7 +565,9 @@ def gangtest(classes,n):
                 print c,binascii.hexlify(ret)
             print
         
-        r0 = randombytes(classes[0].encLen)
+        if i < len(specials): r0 = enc_le(specials[i],classes[0].encLen)
+        else: r0 = randombytes(classes[0].encLen)
+        
         rets = [bytes((cls.elligator(r0)*i).encode()) for cls in classes]
         if len(set(rets)) != 1:
             print "Divergence in elligator at %d" % i