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(defun randn (s n) | |
"Regresa un numero aleatorio en [s, n)" | |
(+ s (random (- n s)))) | |
(defun mod-exp (base exponent modulus) | |
"Calcula el exponente modular" | |
(cond ((zerop exponent) 1) | |
((evenp exponent) (mod-exp (mod (* base base) modulus) | |
(truncate exponent 2) modulus)) | |
((oddp exponent) (mod (* base (mod-exp base | |
(- exponent 1) | |
modulus)) modulus)))) | |
(defun run-witness (i n r d) | |
"Revisa si n es un numero primo con i testigos diferentes" | |
(cond ((<= i 0) t) | |
((witness (randn 2 (- n 1)) n r d) nil) | |
(t (run-witness (- i 1) n r d)))) | |
(defun inverse-modular-product (a n) | |
"Calcula el inverso de a bajo el producto modulo n" | |
(labels ((loop-fn (cur-t cur-r next-t next-r) | |
(if (= next-r 0) | |
(cond ((> cur-r 1) nil) | |
((< cur-t 0) (+ cur-t n)) | |
(t cur-t)) | |
(let ((quotient (floor cur-r next-r))) | |
(loop-fn next-t next-r | |
(- cur-t (* quotient next-t)) | |
(- cur-r (* quotient next-r))))))) | |
(loop-fn 0 n 1 a))) | |
(defun generate-keys (key-len) | |
"Genera un par de llaves RSA con longitud key-len" | |
(let* ((p (generate-prime (floor key-len 2))) | |
(q (generate-prime (floor key-len 2))) | |
(n (* p q)) | |
(cm (carmichael-totient p q)) | |
(e (coprime-in-range cm)) | |
(d (inverse-modular-product e cm))) | |
(list e d n))) | |
(defun encrypt-num (m e n) | |
"Encripta el numero m con la llave publica e n" | |
(mod-exp m e n)) | |
(defun decrypt-num (c d n) | |
"Decripta un mensaje c con la llave privada d n" | |
(mod-exp c d n)) | |
(defun coprime-in-range (n) | |
(do ((i (randn 2 n) (randn 2 n))) | |
((= (gcd i n) 1) i))) | |
(defun carmichael-totient (p q) | |
"Calcula la función de carmichael para un numero n = pq" | |
(lcm (- p 1) (- q 1))) | |
(defun generate-prime (upper-bits) | |
(let* ((lower (expt 2 (- upper-bits 1))) | |
(upper (* lower 2))) | |
(do ((n (randn lower upper) (randn lower upper))) | |
((is-prime n 20) n)))) | |
(defun split (n) | |
"Factoriza n en dos numeros tales que 2^s*d = n" | |
(do ((s 0 (+ s 1)) | |
(d n (floor d 2))) | |
((oddp d) (values s d)))) | |
(defun is-prime (n k) | |
"Indica si n es primo para n >=2" | |
(cond ((<= n 1) nil) | |
((<= n 3) t) | |
((evenp n) nil) | |
(t (multiple-value-bind (r d) (split (- n 1)) | |
(run-witness k n r d))))) | |
(defun witness-loop (i xi n) | |
(let ((x (mod-exp xi 2 n))) | |
(cond ((<= i 0) t) | |
((= x 1) t) | |
((= x (- n 1)) nil) | |
(t (witness-loop (- i 1) x n))))) | |
(defun witness (a n r d) | |
"Revisa si el testigo a demuestra que n no es primo." | |
(let ((x (mod-exp a d n))) | |
(if (or (= x 1) (= x (- n 1))) | |
nil | |
(witness-loop (- r 1) x n)))) |
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