A binary-field curve in optimal normal basis

Parameters

Characteristics

• OID:
  1.2.840.10045.3.0.15
• Seed:
  2C04F44D696E676875615175C586B41F6CA150C9

SAGE

F.<x> = GF(2)[]
K.<z> = GF(2^239, name="z", modulus= x^239 +  x^238 +  x^236 +  x^232 +  x^224 +  x^208 +  x^207 +  x^206 +  x^204 +  x^200 +  x^192 +  x^144 +  x^143 +  x^142 +  x^140 +  x^136 +  x^128 +  x^16 +  x^15 +  x^14 +  x^12 +  x^8 + 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, 0x1ECF1B9D28D8017505E17475D3DF2982E243CA5CB5E9F94A3F36124A486E), 0, 0, fetch_int(K, 0x3EE257250D1A2E66CEF23AA0F25B12388DE8A10FF9554F90AFBAA9A08B6D)))
E.set_order(0x1555555555555555555555555555558CF77A5D0589D2A9340D963B7AD703 * 0x06)
G = E(fetch_int(K, 0x02193279FC543E9F5F7119189785B9C60B249BE4820BAF6C24BDFA2813F8B8), fetch_int(K, ))

JSON

{
  "name": "c2onb239v5",
  "desc": "A binary-field curve in optimal normal basis",
  "oid": "1.2.840.10045.3.0.15",
  "form": "Weierstrass",
  "field": {
    "type": "Binary",
    "bits": 239,
    "degree": 239,
    "poly": [
      {
        "coeff": "0x01",
        "power": 239
      },
      {
        "coeff": "0x01",
        "power": 238
      },
      {
        "coeff": "0x01",
        "power": 236
      },
      {
        "coeff": "0x01",
        "power": 232
      },
      {
        "coeff": "0x01",
        "power": 224
      },
      {
        "coeff": "0x01",
        "power": 208
      },
      {
        "coeff": "0x01",
        "power": 207
      },
      {
        "coeff": "0x01",
        "power": 206
      },
      {
        "coeff": "0x01",
        "power": 204
      },
      {
        "coeff": "0x01",
        "power": 200
      },
      {
        "coeff": "0x01",
        "power": 192
      },
      {
        "coeff": "0x01",
        "power": 144
      },
      {
        "coeff": "0x01",
        "power": 143
      },
      {
        "coeff": "0x01",
        "power": 142
      },
      {
        "coeff": "0x01",
        "power": 140
      },
      {
        "coeff": "0x01",
        "power": 136
      },
      {
        "coeff": "0x01",
        "power": 128
      },
      {
        "coeff": "0x01",
        "power": 16
      },
      {
        "coeff": "0x01",
        "power": 15
      },
      {
        "coeff": "0x01",
        "power": 14
      },
      {
        "coeff": "0x01",
        "power": 12
      },
      {
        "coeff": "0x01",
        "power": 8
      },
      {
        "coeff": "0x01",
        "power": 0
      }
    ],
    "basis": "normal"
  },
  "params": {
    "a": {
      "raw": "0x1ECF1B9D28D8017505E17475D3DF2982E243CA5CB5E9F94A3F36124A486E"
    },
    "b": {
      "raw": "0x3EE257250D1A2E66CEF23AA0F25B12388DE8A10FF9554F90AFBAA9A08B6D"
    }
  },
  "generator": {
    "x": {
      "raw": "0x02193279FC543E9F5F7119189785B9C60B249BE4820BAF6C24BDFA2813F8B8"
    },
    "y": {
      "raw": ""
    }
  },
  "order": "0x1555555555555555555555555555558CF77A5D0589D2A9340D963B7AD703",
  "cofactor": "0x06",
  "characteristics": {
    "seed": "2C04F44D696E676875615175C586B41F6CA150C9"
  }
}