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Internet Engineering Czesław Smutnicki Discrete Mathematics – Numbers Theory CONTENTS • • • • • • • • • • Basic notions Greatest common divisor Modular arithmetics Euclidean algorithm Modular equations Chinese theorem Modular powers Prime numbers RSA algorithm Decomposition into factors BASIC NOTIONS • • • • • • • • • • • • Natural/integer numbers Divisor d|a, a = kd for some integer k d|a if and only if -d|a Divisor: 24: 1,2,3,4,6,8,12,24 Trivial divisors 1 and a Prime number 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59 Composite number, 27 (3|27) For any integer a and any positive integer n there exist unique integers q and r, 0<=r<n so that a = qn + r Residue r = a mod n Division q = [a/n] Congruence: a b (mod n) if (a mod n) = (b mod n) Equivalence class (mod n): [a]n = {a + kn : k Z} GREATEST COMMON DIVISOR • Common divisor: if d|a and d|b • d|(ax + by) • Relatively prime numbers a and b : gcd(a,b)=1 gcd(a, b) gcd(b, a ) gcd(a, b) gcd( a, b) gcd(a, b) gcd( a , b ) gcd(a,0) a gcd(a, ka) a gcd(a, b) gcd(b, a modb) MODULAR ARITHMETIC [a]n n [b]n [a b]n [a]n n [b]n [a b]n [a]n n [b]n [a b]n [a]n n [a 1 ]n [1]n 1 [a]n / n [b]n [a]n n [b ]n + 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4 EUCLIDEAN ALGORITHM mx ny k , k gcd(m, n) a0 q1a1 a2 a1 q2 a2 a3 .......... .......... ...... as 1 qs as as 1 a0 n, a1 m, as 1 0, as gcd(m, n) x0 , x1, x2 ,..., xs , y0 , y1, y2 ,..., ys ai mxi nyi , i 0,1,2,..., s x0 0, y0 1, x1 1, y1 0, xi 1 qi xi xi 1 , yi 1 qi yi yi 1 , EUCLIDEAN ALGORITHM. INSTANCE 333 x 1234 y 1 1234 3*333 235 1234 3*333 235 333 1*235 98 333 1*235 98 235 2*98 39 235 2*98 39 98 2*39 20 98 2*39 20 39 1*20 19 39 1*20 19 20 1*19 1 20 1*19 1 19 19*1 0 19 19*1 0 EUCLIDEAN ALGORITHM. INSTANCE cont. 20 1* 39 1* 20 1 1* 39 2* 20 1 1* 39 2* 98 2* 39 1 5* 39 2*98 1 5* 235 2*98 2*98 1 5* 235 12*98 1 5* 235 12* 333 1* 235 1 17* 235 12* 333 1 17* 1234 3* 333 12* 333 1 63* 333 17*1234 1 333*63 1234*17 1 x 63 y 17 MODULAR EQUATIONS ax b (modn) d gcd(a, n) EQUATION EITHER HAS d VARIOUS SOLUTIONS mod n OR DOES NOT HAVE ANY SOLUTION CASE b = 1: MULTIPLICATIVE INVERSE (IF gcd(a,n)=1 THEN IT EXISTS AND IS UNIQUE) LEAST COMMON MULTIPLIER lcm(a, b) ab gcd(a, b) CHINESE THEOREM n n1n2 ...nk , gcd(ni , n j ) 1, i j a (a1 , a2 ,..., ak ) ai a mod ni b (b1 , b2 ,..., bk ) (a b) mod n (( a1 b1 ) mod n1 ,..., (ak bk ) mod nk ) (a b) mod n (( a1 b1 ) mod n1 ,..., (ak bk ) mod nk ) (a b) mod n (( a1 b1 ) mod n1 ,..., (ak bk ) mod nk ) MODULAR POWERS a 2c mod n (a c ) 2 mod n a 2c1 mod n a (a c ) 2 mod n RSA ALGORITHM • • • • Find two big prime numbers p and q Calculate n=p*q and z=(p-1)*(q-1) Find any number d relatively prime with z Find number e so that (e*d) mod z=1 Public key (e,n) Private key (d,n) Encryption message P Decryption C C P e mod n P C d mod n Thank you for your attention DISCRETE MATHEMATICS Czesław Smutnicki