basic_arithmetics.ml

(** Basic arithmetics with built-in integers *)

open Builtin

(* Greater common divisor and smaller common multiple
   implemetations.
*)

(** Greater common (positive) divisor of two non-zero integers.
    @param a non-zero integers
    @param b non-zero integer
*)
let rec gcd a b = if b = 0 then a else gcd b (modulo a b);;

(* Extended Euclidean algorithm. Computing Bezout Coefficients. *)

(** Extended euclidean division of two integers NOT OCAML DEFAULT.
    Given non-zero entries a b computes triple (u, v, d) such that
    a*u + b*v = d and d is gcd of a and b.
    @param a non-zero integer
    @param b non-zero integer.
*)
let rec bezout a b = if b = 0 then 1, 0, a else let (x, y, d) = bezout b (modulo a b) in (y, x - a/b*y, d);;

let modinv a m = let (u, _, d) = bezout a m in if d <> 1 then invalid_arg "a and m not coprime" else modulo u m;;