{
  "name": "Miyaji-Nakabayashi-Takano",
  "desc": "MNT (Miyaji, Nakabayashi, and Takano curves) example curves from: New explicit conditions of elliptic curve traces for FR-reduction - https://dspace.jaist.ac.jp/dspace/bitstream/10119/4432/1/73-48.pdf.",
  "curves": [
    {
      "name": "mnt1",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0x26dccacc5041939206cf2b7dec50950e3c9fa4827af",
        "bits": 170
      },
      "form": "Weierstrass",
      "params": {
        "a": {
          "raw": "0x22ffbb20cc052993fa27dc507800b624c650e4ff3d2"
        },
        "b": {
          "raw": "0x1c7be6fa8da953b5624efc72406af7fa77499803d08"
        }
      },
      "generator": {
        "x": {
          "raw": "0x25a3ae778f7ef6586abae5acde21e54b6c64edf33d0"
        },
        "y": {
          "raw": "0x05b4ace33aa53c670ce35535d6c273698a182da557d"
        }
      },
      "order": "0x0000a60fd646ad409b3312c3b23ba64e082ad7b354d",
      "cofactor": "0x3be9",
      "characteristics": {
        "discriminant": "519081282363855507291976255149499325173130091108552",
        "j_invariant": "908761003790427908077548955758380356675829025646511",
        "trace_of_frobenius": "-52213820118540299389901413",
        "embedding_degree": "3",
        "anomalous": false,
        "supersingular": false,
        "cm_disc": "3635044015161711632310195875247341545243615496026401",
        "conductor": "1",
        "torsion_degrees": [
          {
            "r": 2,
            "least": 3,
            "full": 3
          },
          {
            "r": 3,
            "least": 8,
            "full": 8
          },
          {
            "r": 5,
            "least": 2,
            "full": 4
          },
          {
            "r": 7,
            "least": 1,
            "full": 3
          },
          {
            "r": 11,
            "least": 5,
            "full": 10
          },
          {
            "r": 13,
            "least": 168,
            "full": 168
          },
          {
            "r": 17,
            "least": 4,
            "full": 8
          },
          {
            "r": 19,
            "least": 18,
            "full": 342
          }
        ]
      }
    },
    {
      "name": "mnt2/1",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0x5affffffffffff4b46081000000059bb1bf600b7",
        "bits": 159
      },
      "form": "Weierstrass",
      "params": {
        "a": {
          "raw": "0x3dd24a7e5c0bdfaccc215e22760469c73ee9d879"
        },
        "b": {
          "raw": "0x478c31a992b294e19f6e4416f958646dddede5e3"
        }
      },
      "generator": {
        "x": {
          "raw": "0x2725af3d7dea98cb9242ac6ddb9bd89bdcf38898"
        },
        "y": {
          "raw": "0x480b4184ed2c50c0230b4c73ca939c1b6b7f1103"
        }
      },
      "order": "0x5affffffffffff4b46081000000059bb1bf600b5",
      "cofactor": "0x01",
      "characteristics": {
        "discriminant": "502795212314952730723914620087437088076350616414",
        "j_invariant": "191845423032390825713658829282439417965614485182",
        "trace_of_frobenius": "3",
        "embedding_degree": "519518160144969382386592375449686021630483366068",
        "anomalous": false,
        "supersingular": false,
        "cm_disc": "2078072640579877529546369501798744086521933464281",
        "conductor": "1",
        "torsion_degrees": [
          {
            "r": 2,
            "least": 3,
            "full": 3
          },
          {
            "r": 3,
            "least": 4,
            "full": 4
          },
          {
            "r": 5,
            "least": 2,
            "full": 2
          },
          {
            "r": 7,
            "least": 6,
            "full": 42
          },
          {
            "r": 11,
            "least": 8,
            "full": 8
          },
          {
            "r": 13,
            "least": 4,
            "full": 52
          },
          {
            "r": 17,
            "least": 288,
            "full": 288
          },
          {
            "r": 19,
            "least": 2,
            "full": 18
          }
        ]
      }
    },
    {
      "name": "mnt2/2",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0x5affffffffffff4b46081000000059bb1bf600b7",
        "bits": 159
      },
      "form": "Weierstrass",
      "params": {
        "a": {
          "raw": "0x07b29491c1a02cd87844f5098d0381f6c45d6523"
        },
        "b": {
          "raw": "0x41cc630bd66ac817d43358b108ad3d214037993c"
        }
      },
      "generator": {
        "x": {
          "raw": "0x0d76b3e1f1ed76a282fa99575d29ff2e587049e9"
        },
        "y": {
          "raw": "0x36e1557ed145ad409f924420e12f74a900fab054"
        }
      },
      "order": "0x5affffffffffff4b46081000000059bb1bf600b5",
      "cofactor": "0x01",
      "characteristics": {
        "discriminant": "416330146701509871347454797378722894990374675389",
        "j_invariant": "327672737112578556672933546167246593305795867129",
        "trace_of_frobenius": "3",
        "embedding_degree": "519518160144969382386592375449686021630483366068",
        "anomalous": false,
        "supersingular": false,
        "cm_disc": "2078072640579877529546369501798744086521933464281",
        "conductor": "1",
        "torsion_degrees": [
          {
            "r": 2,
            "least": 3,
            "full": 3
          },
          {
            "r": 3,
            "least": 4,
            "full": 4
          },
          {
            "r": 5,
            "least": 2,
            "full": 2
          },
          {
            "r": 7,
            "least": 6,
            "full": 42
          },
          {
            "r": 11,
            "least": 8,
            "full": 8
          },
          {
            "r": 13,
            "least": 4,
            "full": 52
          },
          {
            "r": 17,
            "least": 288,
            "full": 288
          },
          {
            "r": 19,
            "least": 2,
            "full": 18
          }
        ]
      }
    },
    {
      "name": "mnt3/1",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0x8afffffffffffeeb0fa77000000089f0dd49fac7",
        "bits": 160
      },
      "form": "Weierstrass",
      "params": {
        "a": {
          "raw": "0x6d01fd0a017c62075ae999977379867e07f2a6d4"
        },
        "b": {
          "raw": "0x7701535c00fd965341d38bba4cfbdcf9a4651825"
        }
      },
      "generator": {
        "x": {
          "raw": "0x1781998103c3ca14ea76b9d3a700a53e1c784789"
        },
        "y": {
          "raw": "0x53352dde04447c25c9bb332a3c7634d3b8801f34"
        }
      },
      "order": "0x8afffffffffffeeb0fa77000000089f0dd49fac5",
      "cofactor": "0x01",
      "characteristics": {
        "discriminant": "210597358905446859227534646357086406264047168945",
        "j_invariant": "15215793501150681155687505726634711188936782904",
        "trace_of_frobenius": "3",
        "embedding_degree": "793549717144513671927050677226939834588042230468",
        "anomalous": false,
        "supersingular": false,
        "cm_disc": "3174198868578054687708202708907759338352168921881",
        "conductor": "1",
        "torsion_degrees": [
          {
            "r": 2,
            "least": 3,
            "full": 3
          },
          {
            "r": 3,
            "least": 4,
            "full": 4
          },
          {
            "r": 5,
            "least": 2,
            "full": 2
          },
          {
            "r": 7,
            "least": 6,
            "full": 6
          },
          {
            "r": 11,
            "least": 5,
            "full": 5
          },
          {
            "r": 13,
            "least": 6,
            "full": 12
          },
          {
            "r": 17,
            "least": 18,
            "full": 18
          }
        ]
      }
    },
    {
      "name": "mnt3/2",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0x8afffffffffffeeb0fa77000000089f0dd49fac7",
        "bits": 160
      },
      "form": "Weierstrass",
      "params": {
        "a": {
          "raw": "0x5fbe0085bd2b23afcd5b9c7704aeed2bfdbe89e4"
        },
        "b": {
          "raw": "0x3fd4005928c76d1fde3d12fa031f48c7fe7f0698"
        }
      },
      "generator": {
        "x": {
          "raw": "0x494e297179d42c761701ab03b2e5bca98a24dfe7"
        },
        "y": {
          "raw": "0x3274201d6596252a780390a222e3763bbecfe5f1"
        }
      },
      "order": "0x8afffffffffffeeb0fa77000000089f0dd49fac5",
      "cofactor": "0x01",
      "characteristics": {
        "discriminant": "240734167439230866925769845396196631223063598668",
        "j_invariant": "246784014257171667049093683200999942046531205936",
        "trace_of_frobenius": "3",
        "embedding_degree": "793549717144513671927050677226939834588042230468",
        "anomalous": false,
        "supersingular": false,
        "cm_disc": "3174198868578054687708202708907759338352168921881",
        "conductor": "1",
        "torsion_degrees": [
          {
            "r": 2,
            "least": 3,
            "full": 3
          },
          {
            "r": 3,
            "least": 4,
            "full": 4
          },
          {
            "r": 5,
            "least": 2,
            "full": 2
          },
          {
            "r": 7,
            "least": 6,
            "full": 6
          },
          {
            "r": 11,
            "least": 5,
            "full": 5
          },
          {
            "r": 13,
            "least": 6,
            "full": 12
          },
          {
            "r": 17,
            "least": 18,
            "full": 18
          }
        ]
      }
    },
    {
      "name": "mnt3/3",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0x8afffffffffffeeb0fa77000000089f0dd49fac7",
        "bits": 160
      },
      "form": "Weierstrass",
      "params": {
        "a": {
          "raw": "0x2ddf23acb05a91bda6ba9c20d7a584aa25075ce0"
        },
        "b": {
          "raw": "0x1f8125c46a31e79fd6cc25298b23ab130cd22b5a"
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      },
      "generator": {
        "x": {
          "raw": "0x3f710d05b65b5e16ae1b946d3fc582b16a927432"
        },
        "y": {
          "raw": "0x4a30945c64fd7f85e148ba816005468447616b1f"
        }
      },
      "order": "0x8afffffffffffeeb0fa77000000089f0dd49fac5",
      "cofactor": "0x01",
      "characteristics": {
        "discriminant": "211342184030958589211568251854333361127055241928",
        "j_invariant": "531549909386191323722269488299292998191740210015",
        "trace_of_frobenius": "3",
        "embedding_degree": "793549717144513671927050677226939834588042230468",
        "anomalous": false,
        "supersingular": false,
        "cm_disc": "3174198868578054687708202708907759338352168921881",
        "conductor": "1",
        "torsion_degrees": [
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            "r": 2,
            "least": 3,
            "full": 3
          },
          {
            "r": 3,
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            "full": 4
          },
          {
            "r": 5,
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          },
          {
            "r": 7,
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          },
          {
            "r": 11,
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            "r": 13,
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          },
          {
            "r": 17,
            "least": 18,
            "full": 18
          }
        ]
      }
    },
    {
      "name": "mnt4",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0xa2ffffffffffffffffffffffffc298b00000000000000000000005c866cf",
        "bits": 240
      },
      "form": "Weierstrass",
      "params": {
        "a": {
          "raw": "0x4be28760aa064734852cb4ff51ef2928a7a3cd75087c35cb1433714f7407"
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        "b": {
          "raw": "0x329704eb1c042f7858c878aa369f70c5c517de4e05a823dcb8224b8a4d5a"
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      },
      "generator": {
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        },
        "y": {
          "raw": "0x73afacea28dc870baa1d5b0bd4300ddd975e2eefc7c2db508fc2e92a8345"
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      },
      "order": "0xa2ffffffffffffffffffffffffc298b00000000000000000000005c866cd",
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        "j_invariant": "1124984654526861897351865205551134254199281270688380623003333782861539023",
        "trace_of_frobenius": "3",
        "embedding_degree": "1124984654526861897351865205551134254199281270688380623265871195502307020",
        "anomalous": false,
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        "cm_disc": "4499938618107447589407460822204537016797125082753522493063484782009228089",
        "conductor": "1",
        "torsion_degrees": [
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            "r": 2,
            "least": 3,
            "full": 3
          },
          {
            "r": 3,
            "least": 4,
            "full": 4
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          {
            "r": 5,
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          },
          {
            "r": 7,
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            "full": 16
          },
          {
            "r": 11,
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            "full": 120
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        ]
      }
    },
    {
      "name": "mnt5/1",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0xd2fffffffffffffffffffffffe9058d000000000000000000000a0271007",
        "bits": 240
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      "form": "Weierstrass",
      "params": {
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        "b": {
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      "generator": {
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      },
      "order": "0xd2fffffffffffffffffffffffe9058d000000000000000000000a0271005",
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        "trace_of_frobenius": "3",
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        "conductor": "1",
        "torsion_degrees": [
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            "r": 2,
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            "r": 3,
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          {
            "r": 5,
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            "r": 7,
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            "r": 13,
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    },
    {
      "name": "mnt5/2",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0xd2fffffffffffffffffffffffe9058d000000000000000000000a0271007",
        "bits": 240
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            "r": 5,
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    },
    {
      "name": "mnt5/3",
      "category": "mnt",
      "desc": "",
      "field": {
        "type": "Prime",
        "p": "0xd2fffffffffffffffffffffffe9058d000000000000000000000a0271007",
        "bits": 240
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      "form": "Weierstrass",
      "params": {
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        "b": {
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      "generator": {
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      },
      "order": "0xd2fffffffffffffffffffffffe9058d000000000000000000000a0271005",
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      "characteristics": {
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        "j_invariant": "1255033703828987787697616537237266250094419551325336715616153938849047572",
        "trace_of_frobenius": "3",
        "embedding_degree": "1456268479172808959148733486940327264608218463421238003553122264354852868",
        "anomalous": false,
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        "cm_disc": "5825073916691235836594933947761309058432873853684952014212489057419411481",
        "conductor": "1",
        "torsion_degrees": [
          {
            "r": 2,
            "least": 3,
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          {
            "r": 3,
            "least": 4,
            "full": 4
          },
          {
            "r": 5,
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          {
            "r": 7,
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          {
            "r": 11,
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          {
            "r": 13,
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        ]
      }
    }
  ]
}