A binary-field curve in optimal normal basis

Parameters

Characteristics

• OID:
  1.2.840.10045.3.0.8
• Seed:
  A399387EAE54D696E6768756151750E58B416D57

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, 0x65903E04E1E4924253E26A3C9AC28C758BD8184A3FB680E8), 0, 0, fetch_int(K, 0x54678621B190CFCE282ADE219D5B3A065E3F4B3FFDEBB29B)))
E.set_order(0x4000000000000000000000009CF2D6E3901DAC4C32EEC65D * 0x2)
G = E(fetch_int(K, 0x025A2C69A32E8638E51CCEFAAD05350A978457CB5FB6DF994A), fetch_int(K, ))

JSON

{
  "name": "c2onb191v4",
  "desc": "A binary-field curve in optimal normal basis",
  "oid": "1.2.840.10045.3.0.8",
  "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": "0x65903E04E1E4924253E26A3C9AC28C758BD8184A3FB680E8"
    },
    "b": {
      "raw": "0x54678621B190CFCE282ADE219D5B3A065E3F4B3FFDEBB29B"
    }
  },
  "generator": {
    "x": {
      "raw": "0x025A2C69A32E8638E51CCEFAAD05350A978457CB5FB6DF994A"
    },
    "y": {
      "raw": ""
    }
  },
  "order": "0x4000000000000000000000009CF2D6E3901DAC4C32EEC65D",
  "cofactor": "0x2",
  "characteristics": {
    "seed": "A399387EAE54D696E6768756151750E58B416D57"
  }
}