rename script/research/elliptic_curves/ to script/research/ec/

script/research/ec/curve.py

@@ -0,0 +1,40 @@
+from finite_fields import finitefield
+
+def add(x_1, y_1, x_2, y_2):
+    if (x_1, y_1) == (x_2, y_2):
+        if y_1 == 0:
+            return None
+
+        # slope of the tangent line
+        m = (3 * x_1 * x_1 + a) / (2 * y_1)
+        return None
+    else:
+        if x_1 == x_2:
+            return None
+
+        # slope of the secant line
+        m = (y_2 - y_1) / (x_2 - x_1)
+
+    x_3 = m*m - x_1 - x_2
+    y_3 = m*(x_1 - x_3) - y_1
+
+    return (x_3, y_3)
+
+if __name__ == "__main__":
+    # Vesta
+    q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001
+    fq = finitefield.IntegersModP(q)
+
+    a, b = fq(0x00), fq(0x05)
+
+    p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001
+
+    C = (fq(0x1ca18c7c3fcb110f9e92c694ce552238f95e9f9b911599cedaff6018cfc5ed52), fq(0x3ad6133a791e41f3e062d370b40e97e77d20effc00b7ee88c4bb097d245cb438))
+    D = (fq(0x3e544e611bb895166afe1a46c6e551c47968daf962d824f79f795cb53585b098), fq(0x2fd03c4da47baf2dfd251e85d18864d4885ddd0e8df648550565b850b79349e3))
+    C_plus_D = (fq(0x06f822cbde350215558c46aac9e60eee31afd942ca6da568845ca4f8fe911e17), fq(0x3e294e73970abc197dfff1a14e74cb20c11b81422d9f920c7b0b0c63affdf67b))
+
+    result = add(C[0], C[1], D[0], D[1])
+    print(result)
+    print(list("%x" % x.n for x in result))
+    assert result[0] == C_plus_D[0]
+    assert result[1] == C_plus_D[1]

script/research/ec/finite_fields

@@ -0,0 +1 @@
+../finite_fields/

script/research/ec/pairing/2.2.2-ecdlp.sage

@@ -0,0 +1,26 @@
+q = 1021
+K = GF(q)
+E = EllipticCurve(K, [905, 100])
+print(E)
+print(f"Group order is: {E.cardinality()}")
+P = E(1006, 416)
+assert P.additive_order() == E.cardinality()
+
+Q = E(612, 827)
+
+matches = {}
+
+for j, m in factor(E.cardinality()):
+    assert m == 1
+
+    P_j = int(E.cardinality() / j) * P
+    Q_j = int(E.cardinality() / j) * Q
+
+    for k in range(j):
+        if k * P_j == Q_j:
+            #print(f"Match found for j = {j}!")
+            matches[j] = k
+            break
+
+k = crt(list(matches.values()), list(matches.keys()))
+print(f"k = {k} mod {E.cardinality()}")

script/research/ec/pairing/2.2.4-deuring.sage

@@ -0,0 +1,22 @@
+import math
+def hasse_interval(q):
+    interval = (q + 1 - 2 * sqrt(q)).n(), (q + 1 + 2 * sqrt(q)).n()
+    return math.ceil(interval[0]), math.floor(interval[1])
+
+q = 23
+K = GF(q)
+
+low, high = hasse_interval(23)
+
+for i in range(100):
+    a = K.random_element()
+    b = K.random_element()
+
+    try:
+        E = EllipticCurve(K, [a, b])
+    except:
+        continue
+
+    assert E.cardinality() >= low
+    assert E.cardinality() <= high
+