From 2b5f3beb316a6924dbaba211d53d5f03287df3e6 Mon Sep 17 00:00:00 2001
From: Michael Hamburg <mike@shiftleft.org>
Date: Thu, 18 Jun 2015 18:36:32 -0700
Subject: [PATCH] sagelike and clike routines for decaffeinating curve25519

---
 aux/decaffeinate_curve25519.sage | 61 +++++++++++++++++++++++++++++++-
 1 file changed, 60 insertions(+), 1 deletion(-)

diff --git a/aux/decaffeinate_curve25519.sage b/aux/decaffeinate_curve25519.sage
index 05594d1..57cc9bd 100644
--- a/aux/decaffeinate_curve25519.sage
+++ b/aux/decaffeinate_curve25519.sage
@@ -23,7 +23,6 @@ def M_to_E(P):
     if maybe(): X,Y = Y,-X
     # OK, have point in ed
     return X,Y
-    
 
 def decaf_encode_from_E(X,Y):
     assert X^2 + Y^2 == 1 + d*X^2*Y^2
@@ -38,6 +37,43 @@ def decaf_encode_from_E(X,Y):
     if not qpositive(s): s = -s
     
     return s
+
+def isqrt(x):
+    assert(x.is_square())
+    if x == 0: return 0
+    else: return 1/sqrt(x)
+
+def decaf_encode_from_E_c(X,Y):
+    Z = F.random_element()
+    T = X*Y*Z
+    X = X*Z
+    Y = Y*Z
+    assert X^2 + Y^2 == Z^2 + d*T^2
+    
+    # Precompute
+    sd = sqrt(F(1-d))
+    
+    zx = Z^2-X^2
+    TZ = T*Z
+    assert zx.is_square
+    ooAll = isqrt(zx*TZ^2)
+    osx = ooAll * TZ
+    ooTZ = ooAll * zx * osx
+    
+    floop = qpositive(T^2 * ooTZ)
+    if floop:
+        frob = zx * ooTZ
+    else:
+        frob = sd
+        Y = -X
+        
+    osx *= frob
+    
+    if qpositive(-2*osx*Z) != floop: osx = -osx
+    s = Y*(ooTZ*Z + osx)
+    if not qpositive(s): s = -s
+    
+    return s
     
 def is_rotation((X,Y),(x,y)):
     return x*Y == X*y or x*X == -y*Y
@@ -49,13 +85,36 @@ def decaf_decode_to_E(s):
     X,Y = 2*s / (1+s^2), (1-s^2) / t
     assert qpositive(X*Y)
     return X,Y
+    
+def decaf_decode_to_E_c(s):
+    assert qpositive(s)
+    
+    s2 = s^2
+    s21 = 1+s2
+    t2 = s21^2 - 4*d*s2
+    
+    alt  = s21*s
+    the  = isqrt(t2*alt^2)
+    oot  = the * alt
+    the *= t2
+    tos  = the * s21
+    X = 2 * (tos-the) * oot
+    Y = (1-s2) * oot
+    
+    if not qpositive(tos): Y = -Y
+    assert qpositive(X*Y)
+    
+    return X,Y
 
 def test():
     P = 2*M.random_point()
     X,Y = M_to_E(P)
     s = decaf_encode_from_E(X,Y)
+    assert s == decaf_encode_from_E_c(X,Y)
     XX,YY = decaf_decode_to_E(s)
+    XX2,YY2 = decaf_decode_to_E_c(s)
     assert is_rotation((X,Y),(XX,YY))
+    assert is_rotation((X,Y),(XX2,YY2))
 
 
     
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