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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.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ ...
where the modulus m is a prime number or a power of a prime number, the multiplier a is an element of high multiplicative order modulo m (e.g., a primitive root modulo n), and the seed X 0 is coprime to m. Other names are multiplicative linear congruential generator (MLCG) [2] and multiplicative congruential generator (MCG).
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1]
Primitive root modulo m: A number g is a primitive root modulo m if, for every integer a coprime to m, there is an integer k such that g k ≡ a (mod m). A primitive root modulo m exists if and only if m is equal to 2, 4, p k or 2p k, where p is an odd prime number and k is a positive integer.
The formula above is then a consequence of the fact that the th roots of unity sum to 0, since each th root of unity is a primitive th root of unity for exactly one divisor of . However it is also possible to prove this identity from first principles.
Let G p denote the group of non-zero integers modulo p under multiplication ... which is greater than the square root of P and is a primitive root ... 25 (1): 111 ...