# modular arithmetic

## English

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### Noun

modular arithmetic (countable and uncountable, plural modular arithmetics)

1. Any system of arithmetic for integers which, for some given positive integer n, is equivalent to the set of integers being mapped onto the finite set {0, ... n} according to congruence modulo n, and in which addition and multiplication are defined consistently with the results of ordinary arithmetic being so mapped.
• 1969, Joseph Landin, An Introduction to Algebraic Structures, Dover, page 153:
The reader now has examined, in some detail, several specific modular arithmetics, namely, ${\displaystyle \mathbb {Z} _{2},\mathbb {Z} _{5},\mathbb {Z} _{6},\mathbb {Z} _{7},\mathbb {Z} _{8}}$ and ${\displaystyle \mathbb {Z} _{9}}$.
• 2010, Christof Paar, Jan Pelzl, Understanding Cryptography: A Textbook for Students and Practitioners, Springer, page 13:
In this section we use two historical ciphers to introduce modular arithmetic with integers. Even though the historical ciphers are no longer relevant, modular arithmetic is extremely important in modern cryptography, especially for asymmetric algorithms.
• 1997, Robert E. Jamison, “Rhythm and Pattern: Discrete Mathematics with an Artistic Connection for Elementary School Teachers”, in Joseph G. Rosenstein, Deborah S. Franzblau, Fred S. Roberts, editors, Discrete Mathematics in the Schools, American Mathematical Society, page 215:
Hence for prime moduli, modular arithmetic is very similar to regular rational arithmetic with all four operations defined.