A binary-field curve in optimal normal basis

Parameters

Characteristics

• OID:
  1.2.840.10045.3.0.9
• Seed:
  2D88F7BC545794D696E676875615175973391555

SAGE

F.<x> = GF(2)[]
K.<z> = GF(2^191, name="z", modulus= x^191 +  x^190 +  x^188 +  x^184 +  x^176 +  x^160 +  x^128 +  x^64 +  x^63 +  x^62 +  x^60 +  x^56 +  x^48 +  x^32 + 1)
def fetch_int(K, h):
	return sum(map(lambda ix: K(Integer(ix[1]) * z^(2^ix[0])), enumerate(h.binary())), K(0))
E = EllipticCurve(K, (1, fetch_int(K, 0x25F8D06C97C822536D469CD5170CDD7BB9F500BD6DB110FB), 0, 0, fetch_int(K, 0x75FF570E35CA94FB3780C2619D081C17AA59FBD5E591C1C4)))
E.set_order(0x0FFFFFFFFFFFFFFFFFFFFFFFEEB354B7270B2992B7818627 * 0x8)
G = E(fetch_int(K, 0x032A16910E8F6C4B199BE24213857ABC9C992EDFB2471F3C68), fetch_int(K, ))

JSON

{
  "name": "c2onb191v5",
  "desc": "A binary-field curve in optimal normal basis",
  "oid": "1.2.840.10045.3.0.9",
  "form": "Weierstrass",
  "field": {
    "type": "Binary",
    "bits": 191,
    "degree": 191,
    "poly": [
      {
        "coeff": "0x01",
        "power": 191
      },
      {
        "coeff": "0x01",
        "power": 190
      },
      {
        "coeff": "0x01",
        "power": 188
      },
      {
        "coeff": "0x01",
        "power": 184
      },
      {
        "coeff": "0x01",
        "power": 176
      },
      {
        "coeff": "0x01",
        "power": 160
      },
      {
        "coeff": "0x01",
        "power": 128
      },
      {
        "coeff": "0x01",
        "power": 64
      },
      {
        "coeff": "0x01",
        "power": 63
      },
      {
        "coeff": "0x01",
        "power": 62
      },
      {
        "coeff": "0x01",
        "power": 60
      },
      {
        "coeff": "0x01",
        "power": 56
      },
      {
        "coeff": "0x01",
        "power": 48
      },
      {
        "coeff": "0x01",
        "power": 32
      },
      {
        "coeff": "0x01",
        "power": 0
      }
    ],
    "basis": "normal"
  },
  "params": {
    "a": {
      "raw": "0x25F8D06C97C822536D469CD5170CDD7BB9F500BD6DB110FB"
    },
    "b": {
      "raw": "0x75FF570E35CA94FB3780C2619D081C17AA59FBD5E591C1C4"
    }
  },
  "generator": {
    "x": {
      "raw": "0x032A16910E8F6C4B199BE24213857ABC9C992EDFB2471F3C68"
    },
    "y": {
      "raw": ""
    }
  },
  "order": "0x0FFFFFFFFFFFFFFFFFFFFFFFEEB354B7270B2992B7818627",
  "cofactor": "0x8",
  "characteristics": {
    "seed": "2D88F7BC545794D696E676875615175973391555"
  }
}