Tem emaa 4: Ar Arit itm´ m´etic eticaa Mo Modu dula larr Magdalena Fern´ andez andez Lebr´ on on
[email protected] Dpto. Matem´ atica atica Aplicada I
Curso 2013-2014
Introducci´on on a la Matem´atica atica Discreta Grad Gradoo en I.I. I.I.-T´ -T´ecno ecnolo logg´ıa ıass Info Inform rm´ aticas a´ticas
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N´umeros umeros congruentes Ejemplo Si contamos 100 d´ıas ıas a partir de hoy, ¿en qu´e d´ıa ıa de la semana semana caer´ a? Como 100 = 14 × 7 + 2 , dentro dentro de 100 d´ıas ıas ser´ a el mismo mis mo d´ıa ıa de la semana sem ana que dentro dentro de dos d´ıas. ıas. Aqu´ Aqu´ı hemos hemos tomado n = 7 y hemos reemplazado 100 por el resto de su divisi´ on entre 7, es decir, por 2. Definici´on on ´ meros congruentes] [Numeros u Sea Sea n un entero positivo y sean a y b dos enteros cualesquiera. Se dice que es congruente con b m´ odulo y lo denotamos por
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Caracterizaci´on on de los n´umeros umeros congruentes
Ejemplos 1
16 ≡ 11 (mod mod 5) ya que al dividir ambos (16 y 11) por 5 obtenemos de resto 1: 16 = 3 × 5 + 1 y 11 = 2 × 5 + 1
2
8 ≡ 5 (mod mod 3) ya que 8 = 2 × 3 + 2 y 5 = 1 × 3 + 2 .
Lema
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Relaci´on on de equivalencia
Lema Para cualquier entero fijo n ≥ 1 se verifican las propiedades: 1
Reflexiva:
a ≡ a (mod mod n) para cualquier entero a
a ≡ b (mod mod n) =⇒ b ≡ a (mod mod n) a ≡ b (mod mod n) 3 Transitiva: =⇒ a ≡ c (mod mod n). b ≡ c (mod mod n) Estas tres propiedades definen una relaci´ on de equivalencia, por 2
Sim´etrica:
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Clases de equivalencia Cada elemento a ∈ Z define la clase de equivalencia [a] = {x ∈ Z : x ≡ a (mod n)} = {..,a−2n, a−n,a,a+ n,a,a+n, a+2 +2n,.. n,..} quedando Z dividido en las n clases de equivalencia correspondientes a los n posibles restos de dividir un entero entre n [0], [0], [1], [1], [2], [2], . . . , [n−1]
ya que
a ≡ b (mod n) ⇐⇒ [a] = [b]
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Enteros m´odulo odulo n
Definici´on on Para cada n ≥ 1, el conjunto de las n clases de congruencia m´ odulo n lo denotamos por Zn y se conoce como el conjunto de los enteros m´ odulo n. Zn
= {0, 1, 2, . . . , n − 1}
donde los elementos a ∈ Zn representan a sus respectivas clases
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La ar arit itm´ m´etic eticaa de
Zn
Definici´on on Si a y b son elementos de Zn (ie, representan a las clases de equivalencia [a] y [b] m´ odulo n respectivamente), definimos [a] + [b [ b] = [a + b] [ ] [b] [ b]
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La ar arit itm´ m´etic eticaa en
Zn
Lema Para cualquier entero n ≥ 1, si a ≡ a (mod mod n) y b ≡ b (mod mod n), entonces a + b ≡ a + b, a − b ≡ a − b y a b ≡ ab.
Ejemplo
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La ar arit itm´ m´etic eticaa en
Zn
Si intentamos definir la exponencial de clases en
Zn como
[a][b] = [ab ], limit´andonos andonos a los valores no negativos de b con el fin de que ab sea entero.
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La ar arit itm´ m´etic eticaa en
Zn
Para cualquier entero k ≥ 2, podemos definir sumas finitas, productos y potencias de clases en Zn por [a1 ] + [a2 ] + · · · + [a [ak ] = [a1 + a2 + · · · + ak ] [a1 ][a ][a2 ] · · · [ak ] = [a1 a2 · · · ak ] [a]k = [ak ]
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Propiedades de la suma y el producto Internas:
∀x, y ∈ Zn =⇒ x + y, xy ∈ Zn . Asociativas:
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Propiedades de la suma y el producto
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Tablas de la suma y el producto en
Z4
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Tablas de la suma y el producto en
Z5
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Unidades de
Zn
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Observaciones
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Para un n´ umero umero de 100 d´ıgitos ıgit os n ≈ 10100 habr habr´´ıa que que hace hacer, r, en el peor de los casos 2 · (1 + log2 10100 ) = 666 operaciones y nuestro ordenador que realiza un bill´ on on de operaciones por segundo tardar´ tardar´ıa menos de una milmillon´ esima esima de segundo en realizarlas, frente a los 3 · 1078 siglos sigl os que tardar tard ar´´ıa multiplic multiplicand andoo a por a sucesivamente. Luego el test de pseudoprimalidad de Fermat es f´acilmente acilmente aplicable. Si para una determinada base el test no puede asegurarnos
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N´umeros umeros de Carmichael ¿Ser´ ¿Ser´ıa una buena estrategia seguir probando bases hasta encontrar alguna en la que, si el n´ umero umero es compuesto, se detecte como compuesto? La respuesta evidentemente es que no. Definici´on on Se denominan n´ umeros de Carmichael a aquellos n´ umeros que siendo compuestos superan los test de pseudoprimalidad de Fermat cualquiera que sea la base que se tome. n de Carmichael ⇐⇒
n compuesto an ≡ a (mod mod n) ∀ a ∈ Z+
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N´umeros umeros de Carmichael Teorema ´ n de los numeros ´ meros de Carmichael] [Caracterizacion o u Un n´ umero compuesto n es de Carmichael si, y s´ olo si, es libre de cuadrados (un producto de primos distintos) y p − 1 divide a n − 1 para cada primo p que divide a n. Ejemplo 561 = 3 · 11 · 17 es compuesto y libre de cuadrados y adem´ as
3 − 1 = 2 | (561 − 1) = 560
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Test de pseudoprimalidad de Fermat
Elegimos al azar una base a del rango {2, 3, . . . , n − 1}, resultando: Si mcd (a, (a, n) = d > 1, como d divide a n sabemos que n es compuesto. Si mcd (a, (a, n) = 1 y 1 (mod n) =⇒ n es compuesto. an 1 ≡ Obs´ Obs´ervese ervese que que al ser a ⊥ n podemos hacer uso del teorema de Fermat en vez de su corolario. −
Si mcd (a, (a, n) = 1 y an
1
−
≡ 1 (mod n) nos quedaremos
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1
Escog coger, ale aleator toriamen mente una una bas base a ∈ {2, . . . , n − 1}
2
Calcular el mcd (a, (a, n) Si mcd (a, (a, n) > 1 , devol devolver ver compuesto
1 ( mod n), devo si no, si an−1 ≡ devolve lver r compuesto si no, no, devo devolv lver er posible primo
Obs´ Obs´ervese ervese que al tomar tom ar a ∈ {2, . . . , n − 1} existen n − φ(n) − 1 posibilidades de que mcd (a, (a, n) > 1 > 1 en cuyo caso sabemos que el n´ umero umero n es compuesto y s´olo olo φ(n) 1 casos en los que
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Otros test de primalidad Para cualquier entero positivo elegido al azar, la funci´on on isprime de isprime de Maple determina si n es primo con una probabilidad de error inferior a 10 15 . −
Para la obtenci´on on de n´ umeros primos muy grandes se umeros utilizan los n´ umeros umeros de Mersenne (M (M p = 2 p − 1 con p
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Test de Lucas-Lehmer Este test, que resulta en apariencia demasiado largo de realizar, es ideal para ordenadores, ya que las congruencias se realizan m´odulo odulo 2 p − 1 que en binario son muy f´aciles aciles de obtener. Adem´ as se ha refinado computacionalmente con el as
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Congruencias lineales
Para dar sentido al cociente [b [b]/[a] de dos clases de congruencias [a [a], [b] ∈ Zn , tenemos que considerar la soluci´on on de la congruencia lineal ax ≡ b (mod n).
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Congruencias lineales Teorema (a, n), entonces la congruencia lineal Si d = mcd (a, ax ≡ b (mod mod n)
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Ejemplos congruencias lineales
Ejemplo
Consideremos la congruencia
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Propiedades de las congruencias lineales
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C´alculo alculo de una soluci´on on particular de ax ≡ b (mod n)
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Algoritmo de resoluci´on on de un sistema de congruencias lineales P1
P2 P3 P4 P5
Reso Resolv lver er ca cada da ecua ecuaci ci´ on o´n ai x ≡ bi (mod ni ) del sistema para convertirlo en uno del tipo x ≡ ai (mod ni ) Si alguna carece de soluci´on on el sistema no la tiene. Comp Compro roba barr que que mc mcd d (ni , n j ) divide a ai − a j Si falla alg´ un caso, el sistema no tiene soluci´on. un on. Desc Descom ompon poner er ca cada da ecua ecuaci ci´ on o´n en un sistema. Elim Elimin inar ar las las ecua ecuaci cion ones es que que no son son nece necesa sari rias as.. Calcular n (producto de los m´odulos odulos resultantes), c n/n y d soluciones de las ecuaciones
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Criterios de divisibilidad Teorema Sea Sea P ( P (x) un polinomio con coeficientes enteros y sea n ≥ 1. a ≡ b (mod mod n) =⇒ P ( P (a) ≡ P ( P (b) (mod mod n) Aplicaci´ on: on: Obtenci´on on de criterios de divisibilidad . Sea N = a n an 1 . . . a1 a0 con 0 ≤ ai ≤ 9 la expresi´on on decimal de un entero N . N . −
N = an 10n +· · · a1 10+ 10+a a0 = P (10) P (10) con P ( P (x) = an xn +· · · a1 x+a0 Divisibilidad por 3:
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Criterios de divisibilidad Criterio de divisibilidad por 9 n
10 ≡ 1 (mod 9) ⇒ N = P (10) P (10) ≡ P (1) P (1) =
ai (mod 9)
i=0
Un n´ umero es divisible por 9 si, y s´olo umero olo si, lo es el n´ umero umero formado por la suma de sus cifras. Criterio de divisibilidad por 11 10 ≡ −1 (mod 11) ⇒ N ≡ P ( P (−1) = a0 −a1 +··+(−1)n an (mod 11)
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Aplicaci´on on congruencias
Como n|m ⇔ m ≡ 0 (mod n), se sigue que los problemas sobre divisibilidad son equivalentes a los problemas sobre congruencias y, estos ultimos u ´ ltimos son, a veces, m´as as f´ aciles aciles de resolver. EJEMPLO
Probar, mediante congruencias, que 32n+5 + 24n+1 es divisible por 7 cualquiera que sea el entero n ≥ 1. Trabajando m´ odulo odulo 7 se tiene que 32n+5 +24n+1
35 32n +2 24n = 243 9n +2 16n ≡ 5 2n +2 2n
7 2n ≡ 0
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Aplicaci´on on a Cr Crip ipto togr graf af´´ıa
Numeramos las letras del alfabeto (en la pr´actica, actica, los 256 c´ odigos odigos ASCII) = 00
F = 06 L = 12 Q = 18 W = 24
A = 01 G = 07 M = 13 R = 19 X = 25
B = 02 H = 08 N = 14 S = 20 Y = 26
C = 03 I = 09 ˜ = 15 N T = 21 Z = 27
D = 04 J = 10 O = 16 U = 22
E = 05 K = 11 P = 17 V = 23
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Criptograf´ıa asim´etrica La constituyen los c´odigos odigos criptogr´aficos aficos que utilizan claves diferentes para cifrar y descifrar. Esto nos permite que la clave que se usa para cifrar pueda ser p´ ublica siempre que la clave para descifrar sea privada , es decir, conocida unicamente ´ por el destinatario , ni siquiera por el remitente
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RSA ´ blica: Es una pareja de enteros (n, La clave publica: u (n, e) tales que n sea primo con cualquier posible palabra palabra de un texto. Esta condici´on on se garantiza si cualquier divisor primo de n es mayor que la mayor palabra posible: para r = 4,
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El cifrado y el descifrado ( n, e) es la El cifrado: cifrado: Si N es N es una palabra del texto y (n, clave p´ ublica a utilizar, la palabra cifrada C se ublica C se obtiene N → C = N e (mod n) El descifrado: Si C es C es una palabra cifrada del texto y (n, d) es la clave privada, la palabra descifrada N se N se obtiene
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La elecci´on on de una clave segura Una clave segura es aquella que garantiza la privacidad de la clave privada (n, (n, d). Como los enteros (n, ( n, e) son p´ ublicos, ublicos, la seguridad de la clave consiste en la imposibilidad de la obtenci´ on on del entero d que forma parte de la clave privada. Dado que d = e 1 (mod φ(n)) y tanto e como n son p´ ublicos, parece imposible obtener una clave segura. ublicos, −
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El proceso
Supongamos que tres personas A, B y C hacen C hacen p´ ublicas ublicas sus respectivas claves (n (nA , eA ), (nB , eB ) y (nC , eC ). Si A quiere enviar un mensaje M a B lo cifra con la clave
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La firma