"Jacobi symbol" meaning in All languages combined

{
  "etymology_text": "Named after German mathematician Carl Gustav Jakob Jacobi, who introduced the notation in 1837.",
  "forms": [
    {
      "form": "Jacobi symbols",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Jacobi symbol (plural Jacobi symbols)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "2000, Song Y. Yan, Number Theory for Computing, Springer, 2000, Softcover reprint, page 114,\nAlthough the Jacobi symbol ((1009)/(2307))=1, we still cannot determine whether or not the quadratic congruence 1009=x²(mod 2307) is soluble.\nRemark 1.6.10. Jacobi symbols can be used to facilitate the calculation of Legendre symbols."
        },
        {
          "ref": "2009, Antoine Joux, “Chapter 1: Introduction to Identity-Based Cryptography”, in Marc Joye, Gregory Neven, editors, Identity-based Cryptography, IOS Press, page 8:",
          "text": "With more than two factors, having a Jacobi symbol of 1 only means that x may be a quadratic non-residue modulo an even number of factors only. Thus in the general case, the Jacobi symbol is not enough to test for the existence of a discrete logarithm. Thanks to this efficient test, given any public process, for example based on a hash function, that transforms the identity of a user into a number x modulo N, this number can directly be used as the user's public key if its Jacobi symbol is 1.",
          "type": "quote"
        },
        {
          "text": "2014, Ibrahim Elashry, Yi Mu, Willy Susilo, Jhanwar-Barua's Identity-Based Encryption Revisited, Man Ho Au, Barbara Carminati, C.-C. Jay Kuo (editors), Network and System Security: 8th International Conference, Springer, LNCS 8792, page 279,\nFrom the above equations, guessing the Jacobi symbol ((2y_is_j_1s_j_2+2)/N) from ((2y_j_1s_j_1+2)/N) and ((2y_j_2s_j_2+2)/N) is as hard as guessing them from independent Jacobi symbols."
        }
      ],
      "glosses": [
        "A mathematical function of integer a and odd positive integer b, generally written (a/b), based on, for each of the prime factors pᵢ of b, whether a is a quadratic residue or nonresidue modulo pᵢ."
      ],
      "id": "en-Jacobi_symbol-en-noun-xwuYJJLt",
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "integer",
          "integer"
        ],
        [
          "prime factor",
          "prime factor"
        ],
        [
          "quadratic residue",
          "quadratic residue"
        ],
        [
          "nonresidue",
          "nonresidue"
        ],
        [
          "modulo",
          "modulo"
        ]
      ],
      "raw_glosses": [
        "(number theory) A mathematical function of integer a and odd positive integer b, generally written (a/b), based on, for each of the prime factors pᵢ of b, whether a is a quadratic residue or nonresidue modulo pᵢ."
      ],
      "related": [
        {
          "word": "Legendre symbol"
        },
        {
          "word": "quadratic reciprocity"
        }
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Carl Gustav Jakob Jacobi"
      ]
    }
  ],
  "word": "Jacobi symbol"
}

{
  "etymology_text": "Named after German mathematician Carl Gustav Jakob Jacobi, who introduced the notation in 1837.",
  "forms": [
    {
      "form": "Jacobi symbols",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Jacobi symbol (plural Jacobi symbols)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Legendre symbol"
    },
    {
      "word": "quadratic reciprocity"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Pages with 1 entry",
        "Pages with entries",
        "en:Number theory"
      ],
      "examples": [
        {
          "text": "2000, Song Y. Yan, Number Theory for Computing, Springer, 2000, Softcover reprint, page 114,\nAlthough the Jacobi symbol ((1009)/(2307))=1, we still cannot determine whether or not the quadratic congruence 1009=x²(mod 2307) is soluble.\nRemark 1.6.10. Jacobi symbols can be used to facilitate the calculation of Legendre symbols."
        },
        {
          "ref": "2009, Antoine Joux, “Chapter 1: Introduction to Identity-Based Cryptography”, in Marc Joye, Gregory Neven, editors, Identity-based Cryptography, IOS Press, page 8:",
          "text": "With more than two factors, having a Jacobi symbol of 1 only means that x may be a quadratic non-residue modulo an even number of factors only. Thus in the general case, the Jacobi symbol is not enough to test for the existence of a discrete logarithm. Thanks to this efficient test, given any public process, for example based on a hash function, that transforms the identity of a user into a number x modulo N, this number can directly be used as the user's public key if its Jacobi symbol is 1.",
          "type": "quote"
        },
        {
          "text": "2014, Ibrahim Elashry, Yi Mu, Willy Susilo, Jhanwar-Barua's Identity-Based Encryption Revisited, Man Ho Au, Barbara Carminati, C.-C. Jay Kuo (editors), Network and System Security: 8th International Conference, Springer, LNCS 8792, page 279,\nFrom the above equations, guessing the Jacobi symbol ((2y_is_j_1s_j_2+2)/N) from ((2y_j_1s_j_1+2)/N) and ((2y_j_2s_j_2+2)/N) is as hard as guessing them from independent Jacobi symbols."
        }
      ],
      "glosses": [
        "A mathematical function of integer a and odd positive integer b, generally written (a/b), based on, for each of the prime factors pᵢ of b, whether a is a quadratic residue or nonresidue modulo pᵢ."
      ],
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "integer",
          "integer"
        ],
        [
          "prime factor",
          "prime factor"
        ],
        [
          "quadratic residue",
          "quadratic residue"
        ],
        [
          "nonresidue",
          "nonresidue"
        ],
        [
          "modulo",
          "modulo"
        ]
      ],
      "raw_glosses": [
        "(number theory) A mathematical function of integer a and odd positive integer b, generally written (a/b), based on, for each of the prime factors pᵢ of b, whether a is a quadratic residue or nonresidue modulo pᵢ."
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Carl Gustav Jakob Jacobi"
      ]
    }
  ],
  "word": "Jacobi symbol"
}