Normalmod

Normalmod is a mathematical operation used to normalize integers with respect to a modulus into a canonical residue class. In its standard form, for any integer a and positive integer m, normalmod(a, m) denotes the unique integer r with 0 ≤ r < m such that a ≡ r (mod m). The function is widely used in modular arithmetic to produce consistent, nonnegative residues and to facilitate comparisons across different computations or representations.

Formal properties: For fixed m > 0, normalmod maps the integers to the residue set {0, ..., m−1}. It satisfies normalmod(a, m) ≡ a (mod m) and normalmod(a, m) ∈ [0, m−1]. It is compatible with addition and multiplication: normalmod(a+b, m) = normalmod(normalmod(a, m) + normalmod(b, m), m), and normalmod(ca, m) = normalmod(c · normalmod(a, m), m) for integers c. It is idempotent in the sense that applying normalmod to an already normalized value yields the same value.

Computation and examples: For example, normalmod(-1, 5) = 4, normalmod(7, 5) = 2, normalmod(12, 4) = 0. Some software libraries implement normalmod directly, while others define the operation via a floor-based formula: a − m ⌊a/m⌋. Variants exist that map residues to the interval [1, m] instead of [0, m−1] by a modification r = ((a − 1) mod m) + 1.

Applications and variants: Normalmod is used to ensure consistent residue representations in cryptographic routines, hash functions, random number generators, and modular algorithms where negative values should not occur. In multi-modulus settings, residues can be combined via the Chinese Remainder Theorem in a way that preserves the normal form. See also the modulo operation, residue class, Chinese Remainder Theorem, normal form, and normalization.