Jacobi symbol en · NOUN
Etymology
Named after German mathematician Carl Gustav Jakob Jacobi, who introduced the notation in 1837.
Meanings
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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.
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, From 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.
2000, Song Y. Yan, Number Theory for Computing, Springer, 2000, Softcover reprint, page 114, Although the Jacobi symbol ((1009)/(2307))=1, we still cannot determine whether or not the quadratic congruence 1009=x²(mod 2307) is soluble. Remark 1.6.10. Jacobi symbols can be used to facilitate the calculation of Legendre symbols.
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.
2009, Antoine Joux, “Chapter 1: Introduction to Identity-Based Cryptography”, in Marc Joye, Gregory Neven, editors, Identity-based Cryptography, IOS Press, page 8:
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| Jacobi symbols | Number=Plur | lexicographic |