Jacobi symbol

Etymology
Named after German mathematician, who introduced the notation in 1837.

Noun

 * 1)  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 pi of b, whether a is a quadratic residue or nonresidue modulo pi.
 * 2) * 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.
 * 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.
 * 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


 * $$\left(\frac{a}{b}\right) = \left(\frac{a}{p_1}\right)^{\alpha_1}\left(\frac{a}{p_2}\right)^{\alpha_2}\cdots \left(\frac{a}{p_k}\right)^{\alpha_k}$$.