Add support for extension field curves.

9 files changed, 569 insertions, 392 deletions

diff --git a/.gitignore b/.gitignore
index 85a3099..c6682af 100644
--- a/.gitignore
+++ b/.gitignore
@@ -2,4 +2,6 @@
 node_modules/
 package-lock.json
 public/
-src/
\ No newline at end of file
+src/
+.idea/
+*.sage.py
\ No newline at end of file
diff --git a/analyze.sage b/analyze.sage
index c3f45a0..5f82677 100755
--- a/analyze.sage
+++ b/analyze.sage
@@ -110,12 +110,25 @@ def get_curve_group(data):
 		return E, G
 	elif data["field"]["type"] == "Binary":
 		F.<x> = GF(2)[]
-		poly = sum(x^i for i in [data["field"]["poly"]["m"], data["field"]["poly"].get("e1"), data["field"]["poly"].get("e2"), data["field"]["poly"].get("e3"), 0] if i is not None)
-		K = GF(2^data["field"]["poly"]["m"], name="x", modulus=poly)
+		poly = sum(F(int(elem["coeff"], 16)) * x^elem["power"] for elem in data["field"]["poly"])
+		K = GF(2^data["field"]["degree"], name="x", modulus=poly)
 		E = EllipticCurve(K, (1, K.fetch_int(int(data["params"]["a"], 16)), 0, 0, K.fetch_int(int(data["params"]["b"], 16))))
 		E.set_order(int(data["order"], 16) * int(data["cofactor"], 16))
 		G = E(K.fetch_int(int(data["generator"]["x"], 16)), K.fetch_int(int(data["generator"]["y"], 16)))
 		return E, G
+	elif data["field"]["type"] == "Extension":
+		base = int(data["field"]["base"], 16)
+		F.<x> = GF(base)[]
+		poly = sum(F(int(elem["coeff"], 16)) * x^elem["power"] for elem in data["field"]["poly"])
+		K = GF(base^data["field"]["degree"], name="x", modulus=poly)
+		a = K(sum(F(int(elem["coeff"], 16)) * x^elem["power"] for elem in data["params"]["a"]))
+		b = K(sum(F(int(elem["coeff"], 16)) * x^elem["power"] for elem in data["params"]["b"]))
+		E = EllipticCurve(K, (a, b))
+		E.set_order(int(data["order"], 16) * int(data["cofactor"], 16))
+		gx = K(sum(F(int(elem["coeff"], 16)) * x^elem["power"] for elem in data["generator"]["x"]))
+		gy = K(sum(F(int(elem["coeff"], 16)) * x^elem["power"] for elem in data["generator"]["y"]))
+		G = E(gx, gy)
+		return E, G
 	else:
 		raise ValueError()
 
diff --git a/nist/curves.json b/nist/curves.json
index e50efcd..67d9957 100644
--- a/nist/curves.json
+++ b/nist/curves.json
@@ -242,16 +242,18 @@
     {
       "name": "K-163",
       "category": "nist",
-      "desc": "",
+      "desc": "Koblitz curve.",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 163,
-          "e1": 7,
-          "e2": 6,
-          "e3": 3
-        },
-        "bits": 163
+        "poly": [
+          {"power": 163, "coeff": "0x01"},
+          {"power": 7, "coeff": "0x01"},
+          {"power": 6, "coeff": "0x01"},
+          {"power": 3, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 163,
+        "degree": 163
       },
       "form": "Weierstrass",
       "params": {
@@ -283,13 +285,15 @@
       "desc": "",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 163,
-          "e1": 7,
-          "e2": 6,
-          "e3": 3
-        },
-        "bits": 163
+        "poly": [
+          {"power": 163, "coeff": "0x01"},
+          {"power": 7, "coeff": "0x01"},
+          {"power": 6, "coeff": "0x01"},
+          {"power": 3, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 163,
+        "degree": 163
       },
       "form": "Weierstrass",
       "params": {
@@ -319,14 +323,16 @@
     {
       "name": "K-233",
       "category": "nist",
-      "desc": "",
+      "desc": "Koblitz curve.",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 233,
-          "e1": 74
-        },
-        "bits": 233
+        "poly": [
+          {"power": 233, "coeff": "0x01"},
+          {"power": 74, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 233,
+        "degree": 233
       },
       "form": "Weierstrass",
       "params": {
@@ -359,11 +365,13 @@
       "desc": "",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 233,
-          "e1": 74
-        },
-        "bits": 233
+        "poly": [
+          {"power": 233, "coeff": "0x01"},
+          {"power": 74, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 233,
+        "degree": 233
       },
       "form": "Weierstrass",
       "params": {
@@ -394,16 +402,18 @@
     {
       "name": "K-283",
       "category": "nist",
-      "desc": "",
+      "desc": "Koblitz curve.",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 283,
-          "e1": 12,
-          "e2": 7,
-          "e3": 5
-        },
-        "bits": 283
+        "poly": [
+          {"power": 283, "coeff": "0x01"},
+          {"power": 12, "coeff": "0x01"},
+          {"power": 7, "coeff": "0x01"},
+          {"power": 5, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 283,
+        "degree": 283
       },
       "form": "Weierstrass",
       "params": {
@@ -436,13 +446,15 @@
       "desc": "",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 283,
-          "e1": 12,
-          "e2": 7,
-          "e3": 5
-        },
-        "bits": 283
+        "poly": [
+          {"power": 283, "coeff": "0x01"},
+          {"power": 12, "coeff": "0x01"},
+          {"power": 7, "coeff": "0x01"},
+          {"power": 5, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 283,
+        "degree": 283
       },
       "form": "Weierstrass",
       "params": {
@@ -472,14 +484,16 @@
     {
       "name": "K-409",
       "category": "nist",
-      "desc": "",
+      "desc": "Koblitz curve.",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 409,
-          "e1": 87
-        },
-        "bits": 409
+        "poly": [
+          {"power": 409, "coeff": "0x01"},
+          {"power": 87, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 409,
+        "degree": 409
       },
       "form": "Weierstrass",
       "params": {
@@ -510,11 +524,13 @@
       "desc": "",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 409,
-          "e1": 87
-        },
-        "bits": 409
+        "poly": [
+          {"power": 409, "coeff": "0x01"},
+          {"power": 87, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 409,
+        "degree": 409
       },
       "form": "Weierstrass",
       "params": {
@@ -542,16 +558,18 @@
     {
       "name": "K-571",
       "category": "nist",
-      "desc": "",
+      "desc": "Koblitz curve.",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 571,
-          "e1": 10,
-          "e2": 5,
-          "e3": 2
-        },
-        "bits": 571
+        "poly": [
+          {"power": 571, "coeff": "0x01"},
+          {"power": 10, "coeff": "0x01"},
+          {"power": 5, "coeff": "0x01"},
+          {"power": 2, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 571,
+        "degree": 571
       },
       "form": "Weierstrass",
       "params": {
@@ -582,13 +600,15 @@
       "desc": "",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 571,
-          "e1": 10,
-          "e2": 5,
-          "e3": 2
-        },
-        "bits": 571
+        "poly": [
+          {"power": 571, "coeff": "0x01"},
+          {"power": 10, "coeff": "0x01"},
+          {"power": 5, "coeff": "0x01"},
+          {"power": 2, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 571,
+        "degree": 571
       },
       "form": "Weierstrass",
       "params": {
diff --git a/other/curves.json b/other/curves.json
index a67be8a..d886142 100644
--- a/other/curves.json
+++ b/other/curves.json
@@ -364,6 +364,38 @@
       }
     },
     {
+      "name": "Fp254n2BNa",
+      "category": "other",
+      "desc": "Curve used in: https://eprint.iacr.org/2010/354.pdf",
+      "field": {
+        "type": "Extension",
+        "base": "0x2370fb049d410fbe4e761a9886e502417d023f40180000017e80600000000001",
+        "bits": 508,
+        "degree": 2,
+        "poly": [
+          {"power": 2, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x05"}
+        ]
+      },
+      "form": "Weierstrass",
+      "params": {
+        "a": [{"power": 0, "coeff": "0x00"}],
+        "b": [{"power": 1, "coeff": "0x2370fb049d410fbe4e761a9886e502417d023f40180000017e80600000000000"}]
+      },
+      "generator": {
+        "x": [
+          {"power": 1, "coeff": "0xa1cf585585a61c6e9880b1f2a5c539f7d906fff238fa6341e1de1a2e45c3f72"},
+          {"power": 0, "coeff": "0x19b0bea4afe4c330da93cc3533da38a9f430b471c6f8a536e81962ed967909b5"}
+        ],
+        "y": [
+          {"power": 1, "coeff": "0x0ee97d6de9902a27d00e952232a78700863bc9aa9be960C32f5bf9fd0a32d345"},
+          {"power": 0, "coeff": "0x17abd366ebbd65333e49c711a80a0cf6d24adf1b9b3990eedcc91731384d2627"}
+        ]
+      },
+      "order": "0x2370fb049d410fbe4e761a9886e502411dc1af70120000017e80600000000001",
+      "cofactor": "0x2370fb049d410fbe4e761a9886e50241dc42cf101e0000017e80600000000001"
+    },
+    {
       "name": "Fp254BNb",
       "category": "other",
       "desc": "Curve used in https://www.iacr.org/archive/eurocrypt2011/66320047/66320047.pdf",
diff --git a/schema.json b/schema.json
index d1ff298..6a91758 100644
--- a/schema.json
+++ b/schema.json
@@ -5,35 +5,51 @@
 			"type": "string",
 			"pattern": "^(0x[0-9a-fA-F]+|[0-9]+)$"
 		},
+		"Polynomial": {
+			"type": "array",
+			"items": {
+				"type": "object",
+				"properties": {
+					"power": {"type":  "integer"},
+					"coeff": {"$ref":  "#/definitions/Number"}
+				}
+			}
+		},
+		"Element": {
+			"anyOf": [
+				{"$ref":  "#/definitions/Number"},
+				{"$ref":  "#/definitions/Polynomial"}
+			]
+		},
 		"Weierstrass": {
 			"type": "object",
 			"properties": {
-				"a": {"$ref": "#/definitions/Number"},
-				"b": {"$ref": "#/definitions/Number"}
+				"a": {"$ref": "#/definitions/Element"},
+				"b": {"$ref": "#/definitions/Element"}
 			},
 			"required": ["a", "b"]
 		},
 		"Edwards": {
 			"type": "object",
 			"properties": {
-				"c": {"$ref": "#/definitions/Number"},
-				"d": {"$ref": "#/definitions/Number"}
+				"c": {"$ref": "#/definitions/Element"},
+				"d": {"$ref": "#/definitions/Element"}
 			},
 			"required": ["c", "d"]
 		},
 		"TwistedEdwards": {
 			"type": "object",
 			"properties": {
-				"a": {"$ref": "#/definitions/Number"},
-				"d": {"$ref": "#/definitions/Number"}
+				"a": {"$ref": "#/definitions/Element"},
+				"d": {"$ref": "#/definitions/Element"}
 			},
 			"required": ["a", "d"]
 		},
 		"Montgomery": {
 			"type": "object",
 			"properties": {
-				"a": {"$ref": "#/definitions/Number"},
-				"b": {"$ref": "#/definitions/Number"}
+				"a": {"$ref": "#/definitions/Element"},
+				"b": {"$ref": "#/definitions/Element"}
 			},
 			"required": ["a", "b"]
 		},
@@ -46,43 +62,33 @@
 			},
 			"required": ["p", "bits"]
 		},
-		"Trinomial": {
-			"type": "object",
-			"properties": {
-				"m": {"type": "integer"},
-				"e1": {"type": "integer"}
-			},
-			"required": ["m", "e1"]
-		},
-		"Pentanomial": {
+		"BinaryField": {
 			"type": "object",
 			"properties": {
-				"m": {"type": "integer"},
-				"e1": {"type": "integer"},
-				"e2": {"type": "integer"},
-				"e3": {"type": "integer"}
+				"type": {"const": "Binary"},
+				"base": {"const": "0x02"},
+				"degree": {"type": "integer"},
+				"poly": {"$ref": "#/definitions/Polynomial"},
+				"bits": {"type": "integer"}
 			},
-			"required": ["m", "e1", "e2", "e3"]
+			"required": ["degree", "poly", "bits"]
 		},
-		"BinaryField": {
+		"ExtenstionField": {
 			"type": "object",
 			"properties": {
-				"type": {"const": "Binary"},
-				"poly": {
-					"anyOf": [
-						{"$ref": "#/definitions/Trinomial"},
-						{"$ref": "#/definitions/Pentanomial"}
-					]
-				},
+				"type": {"const": "Extension"},
+				"base": {"$ref": "#/definitions/Number"},
+				"degree": {"type": "integer"},
+				"poly": {"$ref": "#/definitions/Polynomial"},
 				"bits": {"type": "integer"}
 			},
-			"required": ["poly", "bits"]
+			"required": ["base", "degree", "poly", "bits"]
 		},
 		"Point": {
 			"type": "object",
 			"properties": {
-				"x": {"$ref": "#/definitions/Number"},
-				"y": {"$ref": "#/definitions/Number"}
+				"x": {"$ref": "#/definitions/Element"},
+				"y": {"$ref": "#/definitions/Element"}
 			},
 			"required": ["x", "y"]
 		},
@@ -96,7 +102,8 @@
 				"field": {
 					"oneOf": [
 						{"$ref": "#/definitions/PrimeField"},
-						{"$ref": "#/definitions/BinaryField"}
+						{"$ref": "#/definitions/BinaryField"},
+						{"$ref": "#/definitions/ExtenstionField"}
 					]
 				},
 				"form": {
diff --git a/secg/curves.json b/secg/curves.json
index 982b03d..a7d2952 100644
--- a/secg/curves.json
+++ b/secg/curves.json
@@ -899,10 +899,12 @@
       "field": {
         "type": "Binary",
         "bits": 113,
-        "poly": {
-          "m": 113,
-          "e1": 9
-        }
+        "poly": [
+          {"power": 113, "coeff": "0x01"},
+          {"power": 9, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "degree": 113
       },
       "params": {
         "a": "0x003088250ca6e7c7fe649ce85820f7",
@@ -927,10 +929,12 @@
       "field": {
         "type": "Binary",
         "bits": 113,
-        "poly": {
-          "m": 113,
-          "e1": 9
-        }
+        "poly": [
+          {"power": 113, "coeff": "0x01"},
+          {"power": 9, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "degree": 113
       },
       "params": {
         "a": "0x00689918dbec7e5a0dd6dfc0aa55c7",
@@ -952,12 +956,14 @@
       "field": {
         "type": "Binary",
         "bits": 131,
-        "poly": {
-          "m": 131,
-          "e1": 8,
-          "e2": 3,
-          "e3": 2
-        }
+        "poly": [
+          {"power": 131, "coeff": "0x01"},
+          {"power": 8, "coeff": "0x01"},
+          {"power": 3, "coeff": "0x01"},
+          {"power": 2, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "degree": 131
       },
       "params": {
         "a": "0x07a11b09a76b562144418ff3ff8c2570b8",
@@ -979,12 +985,14 @@
       "field": {
         "type": "Binary",
         "bits": 131,
-        "poly": {
-          "m": 131,
-          "e1": 8,
-          "e2": 3,
-          "e3": 2
-        }
+        "poly": [
+          {"power": 131, "coeff": "0x01"},
+          {"power": 8, "coeff": "0x01"},
+          {"power": 3, "coeff": "0x01"},
+          {"power": 2, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "degree": 131
       },
       "params": {
         "a": "0x03e5a88919d7cafcbf415f07c2176573b2",
@@ -1004,13 +1012,15 @@
       "oid": "1.3.132.0.1",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 163,
-          "e1": 7,
-          "e2": 6,
-          "e3": 3
-        },
-        "bits": 163
+        "poly": [
+          {"power": 163, "coeff": "0x01"},
+          {"power": 7, "coeff": "0x01"},
+          {"power": 6, "coeff": "0x01"},
+          {"power": 3, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 163,
+        "degree": 163
       },
       "form": "Weierstrass",
       "params": {
@@ -1045,13 +1055,15 @@
       "oid": "1.3.132.0.2",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 163,
-          "e1": 7,
-          "e2": 6,
-          "e3": 3
-        },
-        "bits": 163
+        "poly": [
+          {"power": 163, "coeff": "0x01"},
+          {"power": 7, "coeff": "0x01"},
+          {"power": 6, "coeff": "0x01"},
+          {"power": 3, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 163,
+        "degree": 163
       },
       "form": "Weierstrass",
       "params": {
@@ -1084,13 +1096,15 @@
       "oid": "1.3.132.0.15",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 163,
-          "e1": 7,
-          "e2": 6,
-          "e3": 3
-        },
-        "bits": 163
+        "poly": [
+          {"power": 163, "coeff": "0x01"},
+          {"power": 7, "coeff": "0x01"},
+          {"power": 6, "coeff": "0x01"},
+          {"power": 3, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 163,
+        "degree": 163
       },
       "form": "Weierstrass",
       "params": {
@@ -1126,10 +1140,12 @@
       "field": {
         "type": "Binary",
         "bits": 193,
-        "poly": {
-          "m": 193,
-          "e1": 15
-        }
+        "poly": [
+          {"power": 193, "coeff": "0x01"},
+          {"power": 15, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "degree": 193
       },
       "params": {
         "a": "0x0017858feb7a98975169e171f77b4087de098ac8a911df7b01",
@@ -1154,10 +1170,12 @@
       "field": {
         "type": "Binary",
         "bits": 193,
-        "poly": {
-          "m": 193,
-          "e1": 15
-        }
+        "poly": [
+          {"power": 193, "coeff": "0x01"},
+          {"power": 15, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "degree": 193
       },
       "params": {
         "a": "0x0163f35a5137c2ce3ea6ed8667190b0bc43ecd69977702709b",
@@ -1180,11 +1198,13 @@
       "oid": "1.3.132.0.26",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 233,
-          "e1": 74
-        },
-        "bits": 233
+        "poly": [
+          {"power": 233, "coeff": "0x01"},
+          {"power": 74, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 233,
+        "degree": 233
       },
       "form": "Weierstrass",
       "params": {
@@ -1217,11 +1237,13 @@
       "oid": "1.3.132.0.27",
       "field": {
         "type": "Binary",
-        "poly": {
-          "m": 233,
-          "e1": 74
-        },
-        "bits": 233
+        "poly": [
+          {"power": 233, "coeff": "0x01"},
+          {"power": 74, "coeff": "0x01"},
+          {"power": 0, "coeff": "0x01"}
+        ],
+        "bits": 233,
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-        "bits": 409
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-        "bits": 409
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diff --git a/wtls/curves.json b/wtls/curves.json
index 3b9ac5f..3b3ef17 100644
--- a/wtls/curves.json
+++ b/wtls/curves.json
@@ -6,10 +6,12 @@
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@@ -30,14 +32,16 @@
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@@ -63,10 +67,12 @@
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@@ -126,7 +134,7 @@
       },
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+        "x": "0x09487239995a5ee76b55f9c2f098",
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@@ -151,7 +159,7 @@
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+        "x": "0x4a96b5688ef573284664698968c38bb913cbfc82",
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       "name": "wap-wsg-idm-ecid-wtls7",
@@ -215,16 +223,18 @@
       "cofactor": "0x4",
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+        "x": "0x017232ba853a7e731af129f22ff4149563a419c26bf50a4c9d6eefad6126",
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@@ -245,12 +255,14 @@
       "cofactor": "0x2",
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diff --git a/x962/curves.json b/x962/curves.json
index 3431df4..1058ded 100644
--- a/x962/curves.json
+++ b/x962/curves.json
@@ -352,12 +352,14 @@
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@@ -379,12 +381,14 @@
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@@ -406,12 +410,14 @@
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@@ -460,12 +468,14 @@
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@@ -487,10 +497,12 @@
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-        "poly": {
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+        "degree": 191,
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@@ -512,10 +524,12 @@
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@@ -537,10 +551,12 @@
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+        "degree": 191,
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@@ -562,10 +578,12 @@
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+        "degree": 239,
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@@ -587,10 +605,12 @@
       "field": {
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-        "poly": {
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+        "degree": 239,
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@@ -612,10 +632,12 @@
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+        "degree": 239,
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@@ -637,12 +659,14 @@
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@@ -664,12 +688,14 @@
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@@ -691,12 +717,14 @@
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-        "poly": {
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@@ -718,10 +746,12 @@
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+        "degree": 359,
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@@ -743,10 +773,12 @@
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-        "poly": {
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+        "degree": 431,
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diff --git a/x963/curves.json b/x963/curves.json
index 54293dd..8004a2b 100644
--- a/x963/curves.json
+++ b/x963/curves.json
@@ -10,13 +10,15 @@
       "oid": "",
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@@ -36,13 +38,15 @@
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@@ -67,13 +71,15 @@
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@@ -99,10 +105,11 @@
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@@ -127,10 +134,11 @@
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@@ -154,11 +162,13 @@
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@@ -184,11 +194,13 @@
       "oid": "",
       "field": {
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+          {"power": 74, "coeff": "0x01"},
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-        "poly": {
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@@ -215,10 +227,11 @@
       "field": {
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-        "poly": {
-          "m": 239,
-          "e1": 158
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+        "poly": [
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@@ -243,12 +256,13 @@
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-        "poly": {
-          "m": 283,
-          "e1": 12,
-          "e2": 7,
-          "e3": 5
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@@ -274,12 +288,13 @@
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-        "poly": {
-          "m": 283,
-          "e1": 12,
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@@ -304,11 +319,13 @@
       "oid": "",
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-        "poly": {
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@@ -333,11 +350,13 @@
       "oid": "",
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+          {"power": 87, "coeff": "0x01"},
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-        "poly": {
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@@ -362,13 +381,15 @@
       "oid": "",
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@@ -393,13 +414,15 @@
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