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Modular arithmetic - Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus. Modular arithmetic was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae published in 1801.
Quadratic reciprocity - In mathematics, in number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows us to determine the solvability of any quadratic equation in modular arithmetic, even though it does not provide an efficient method for actually finding a solution.
Residue number system - A residue number system (RNS) represents a large integer using a set of smaller integers, so that computation may be performed more efficiently. It relies on the Chinese Remainder Theorem of modular arithmetic for its operation, a mathematical idea from Sun Tsu Suan-Ching (Master Sun’s Arithmetic Manual) in the 4th century AD.
Shanks-Tonelli algorithm - The Shanks-Tonelli algorithm is used within modular arithmetic to solve a congruence of the form x^2 \equiv n \mod p, where n is a quadratic residue (mod p), and typically p \equiv 1 \mod 4.
Long, Ling - Iowa State University. Modular forms for non-congruence subgroups; Modularity of elliptic and K3 surfaces; Arithmetic of Calabi-Yau varieties and applications.
SAGE - System for Arithmetic Geometry Experimentation: a framework for number theory, algebra, and geometry, initially designed for computing with elliptic curves and modular forms, by William Stein. Open source under GPL.
Chang, Sungkong - Armstrong Atlantic State University. Arithmetic of elliptic curves; arithmetic of Jacobian varieties/abelian varieties; modular curves; theory of complex multiplication; algebraic number theory. Thesis, publications.
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