;;
;; Calculates the GCD of a and b based on the Extended Euclidean Algorithm. The function also returns
;; the Bézout coefficients s and t, such that gcd(a, b) = as + bt.
;;
;; The algorithm is described on page http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Iterative_method_2
;;
(defun egcd (a b)
  (do ((r (cons b a) (cons (- (cdr r) (* (car r) q)) (car r))) ; (r+1 r) i.e. the latest is first.
       (s (cons 0 1) (cons (- (cdr s) (* (car s) q)) (car s))) ; (s+1 s)
       (u (cons 1 0) (cons (- (cdr u) (* (car u) q)) (car u))) ; (t+1 t)
       (q nil))
      ((zerop (car r)) (values (cdr r) (cdr s) (cdr u)))       ; exit when r+1 = 0 and return r s t
    (setq q (floor (/ (cdr r) (car r))))))                     ; inside loop; calculate the q

;;
;; Calculates the inverse module for a = 1 (mod m).
;;
;; Note: The inverse is only defined when a and m are coprimes, i.e. gcd(a, m) = 1.”
;;
(defun invmod (a m)
  (multiple-value-bind (r s k) (egcd a m)
    (unless (= 1 r) (error "invmod: Values ~a and ~a are not coprimes." a m))
     s))