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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801.
Modular arithmetic for a modulus defines any two elements and that differ by a multiple of to be equivalent, denoted by . Every integer is equivalent to one of the integers from 0 {\displaystyle 0} to n − 1 {\displaystyle n-1} , and the operations of modular arithmetic modify normal arithmetic by replacing the result of any ...
In modular arithmetic, the integers coprime (relatively prime) to n from the set {,, …,} of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n.
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
In mathematics, modular arithmetic is a system of arithmetic for certain equivalence classes of integers, called congruence classes. Sometimes it is suggestively called 'clock arithmetic', where numbers 'wrap around' after they reach a certain value (the modulus). For example, when the modulus is 12, then any two numbers that leave the same ...
The modular inverse of aR mod N is REDC((aR mod N) −1 (R 3 mod N)). Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, that is, to R mod N, and by replacing the multiply and square steps by Montgomery multiplies.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
Using a residue numeral system for arithmetic operations is also called multi-modular arithmetic. Multi-modular arithmetic is widely used for computation with large integers, typically in linear algebra , because it provides faster computation than with the usual numeral systems, even when the time for converting between numeral systems is ...