The Adapted Modular Number System (AMNS) is an integer number system which aims to speed up arithmetic operations modulo a prime p. Such a system is defined by a tuple (p, n, γ, ρ, E), where p, n, γ and ρ are integers and E ∈ Z[X]. In [12] conditions required to build AMNS with E(X) = X n + 1 are provided. In this paper, we generalise their approach and provide a method to generate multiple AMNS for a given prime p with E(X) = X n − λ and λ ∈ Z \ {0}. Moreover, we propose a complete set of algorithms without conditional branching to perform arithmetic and conversion operations in the AMNS, using a Mont-gomery-like method described in [26]. We show that our implementation outperforms GNU MP and OpenSSL libraries. Finally, we highlight some properties of the AMNS which state that it could lead to a helpful countermeasure against some side channel attacks.