curves in sage (#39)

math/curve.sage

@@ -0,0 +1,61 @@
+# our prime modulus
+F101 = IntegerModRing(101)
+
+# A number 5 in our prime modulus, should be 5
+print(IntegerMod(F101, 5))
+
+# Should be 96
+print(IntegerMod(F101, -5))
+
+# should be 81
+print(IntegerMod(F101, 1/5))
+
+# should be 20
+print(IntegerMod(F101, -1/5))
+
+# should be 100
+print(IntegerMod(F101, -1))
+
+# Lets make our elliptic curve
+E = EllipticCurve(F101, [0, 3])
+
+# lets print out the points, notice they print (x,y,z) the difference between homogenious points and affine points is that to use affine you just divide x,y by z. 
+# We can see here that for all points in the curve group z = 1 except the zero point at infinity. So for this field they are the same
+print(E.points())
+
+# Define polynomial ring
+R.<X> = PolynomialRing(F101)
+
+# Lets make an extension field
+# niavely: we could pick x^2 + 1 but
+# x^2 + 1 = x^2 + 100 = (x+10)(x-10) -> There is a root in the field
+# lets pick x^2 + 2 which is irreducible in our field
+
+# Extended polynomial ring
+K.<X> = GF(101**2, modulus = x^2 + 2)
+
+# Curve group over polynomial ring
+E2 = EllipticCurve(K, [0, 3])
+print(E2.points())
+
+# G1 is the generator for E1
+G1 = E(1,2)
+print(G1)
+
+# N is the order of the group E1    
+N = 17
+
+# G2 is the generator for E2
+G2 = E2([36, 31 *X])
+print(G2)
+
+# Now Lets generate the structured refrence string (SRS), 
+# we will use the "random" number 2 for the example but in practice it should be strong random. 
+# a circuit with n gates requires an SRS with at least
+# n + 5 elements as below
+# We will let it be of length 9, pythagorean triple uses 4 gates
+g1SRS = [(2**i)*G1 for i in range(7)]
+print(g1SRS)
+
+g2SRS = [(2**i)*G2 for i in range(2)]
+print(g2SRS)