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Task/Modular-inverse/Common-Lisp/modular-inverse.lisp

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+;;
+;; 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))