Number-Theoretic Algorithms Modern cryptography is based on number theory and, in particular, the inability to factor the product of large prime numbers The major topics include modular arithmetic Euclid’s algorithm to solve a x  b (mod n) the Chinese remainder theorem efficient computing of ab mod n the RSA public key cryptosystem finding large prime numbers efficiently factoring integers

Operations on large integers Basic concepts We measure the size in the number of bits for integer a1a2…ak a polynomial time algorithm is polynomial in lg a1, lg a2, … , lg ak we count the number of bit operations, such as multiplication of two b-bit integers takes O(b2) time Set notation Z is the set of integers { …, -2, -1, 0, 1, 2, …} N is the set of natural numbers {0, 1, 2, 3, … }

Basic notation and concepts d | a means that “d divides a” or “a is a multiple of d” 1 and a are “trivial” divisors; a number is prime if its only divisors are the trivial divisors; a non-prime number is composite q = a/n is the quotient r = a mod n is the remainder

Modular Notation Notation Equivalence class modulo n Common divisors a  b (mod n) means (a mod n) = (b mod n) a  b (mod n) iff n | (b - a) Equivalence class modulo n [a]n = {a + k n : k  Z} Z n = { [a] n : 0 <= a <= n-1 } Common divisors d | a and d | b implies d | (ax + by) for integers x,y a | b and b | a implies a =  b gcd(a,b) where a and b are not both zero is the largest of the common divisors of a and b

Greatest Common Divisors

Relatively Prime & Unique Factorization Two integers are relatively prime if their only common divisor is one Unique Factorization