Jacobi symbol

English

Etymology

Named after German mathematician Carl Gustav Jakob Jacobi, who introduced the notation in 1837.

Noun

Jacobi symbol (plural Jacobi symbols)

1. (number theory) A mathematical function of integer a and odd positive integer b, generally written $\left({a \over b}\right)$, based on, for each of the prime factors p_i of b, whether a is a quadratic residue or nonresidue modulo p_i.
   • 2000, Song Y. Yan, Number Theory for Computing, Springer, 2000, Softcover reprint, page 114,

     Although the Jacobi symbol $\left({\frac {1009}{2307}}\right)=1$, we still cannot determine whether or not the quadratic congruence $1009=x^{2}{\pmod {2307}}$ is soluble.

     Remark 1.6.10. Jacobi symbols can be used to facilitate the calculation of Legendre symbols.

   • 2009, Antoine Joux, “Chapter 1: Introduction to Identity-Based Cryptography”, in Marc Joye, Gregory Neven, editors, Identity-based Cryptography, IOS Press, page 8:

     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.

   • 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 $\textstyle \left({\frac {2y_{i}s_{j_{1}}s_{j_{2}}+2}{N}}\right)$ from $\textstyle \left({\frac {2y_{j_{1}}s_{j_{1}}+2}{N}}\right)$ and $\textstyle \left({\frac {2y_{j_{2}}s_{j_{2}}+2}{N}}\right)$ is as hard as guessing them from independent Jacobi symbols.

Usage notes

The value is defined as the product of Legendre symbols: if $b=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}$ is the prime factorisation of b, then

$\left({\frac {a}{b}}\right)=\left({\frac {a}{p_{1}}}\right)^{\alpha _{1}}\left({\frac {a}{p_{2}}}\right)^{\alpha _{2}}\cdots \left({\frac {a}{p_{k}}}\right)^{\alpha _{k}}$.

See also

• Legendre symbol
• quadratic reciprocity

Further reading

• Jacobi symbol on Wikipedia.Wikipedia
• Kronecker symbol on Wikipedia.Wikipedia