High Quality Content by WIKIPEDIA articles! The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used within modular arithmetic to solve a congruence of the form x^2 equiv n pmod p where n is a quadratic residue (mod p), and p is an odd prime.Tonelli–Shanks cannot be used for composite moduli; finding square roots modulo composite numbers is a computational problem equivalent to integer factorization.An equivalent, but slightly more redundant version of this algorithm was developed by Alberto Tonelli in 1891. The version discussed here was developed independently by Daniel Shanks in 1973, who explained: "My tardiness in learning of these historical references was due to the fact that I had lent Volume 1 of Dickson's History to a friend and it was never returned."

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