Modular arithmetic: Difference between revisions

Revision as of 00:34, 27 March 2025

Revision as of 03:38, 27 March 2025

Line 157:

In pure mathematics, modular arithmetic is one of the foundations of [[number theory]], touching on almost every aspect of its study, and it is also used extensively in [[group theory]], [[ring theory]], [[knot theory]], and [[abstract algebra]]. In applied mathematics, it is used in [[computer algebra]], [[cryptography]], [[computer science]], [[chemistry]] and the [[visual arts|visual]] and [[music]]al arts.

A very practical application is to calculate checksums within serial number identifiers. For example, [[International Standard Book Number]] (ISBN) uses modulo 11 (for 10-digit ISBN) or modulo 10 (for 13-digit ISBN) arithmetic for error detection. Likewise, [[International Bank Account Number]]s (IBANs), for example, make use of modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the [[CAS registry number]] (a unique identifying number for each chemical compound) is a [[check digit]], which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the previous digit times 2, the previous digit times 3 etc., adding all these up and computing the sum modulo 10.

In cryptography, modular arithmetic directly underpins [[Public-key cryptography|public key]] systems such as [[RSA (algorithm)|RSA]] and [[Diffie–Hellman key exchange|Diffie–Hellman]], and provides [[finite field]]s which underlie [[elliptic curve]]s, and is used in a variety of [[symmetric key algorithm]]s including [[Advanced Encryption Standard]] (AES), [[International Data Encryption Algorithm]] (IDEA), and [[RC4]]. RSA and Diffie–Hellman use [[modular exponentiation]].