Internet Engineering Czesław Smutnicki Discrete Mathematics – Residual Arithmetic

Internet Engineering Czesław Smutnicki Discrete Mathematics – Residual Arithmetic

CONTENTS • Basic notions • Modular arithmetic • Congruence • Chinese theorem • Residual coutning system RNS • Signed RNS • Conversions

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], q = a div 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 • Relatively prime numbers a and b : gcd(a,b)=1

MODULAR ARITHMETIC

CHINESE THEOREM

MODULAR POWERS

CARRY INDEPENDENT SYSTEM

DIVISIONS

OTHER PROPERTIES

RNS BASE SELECTION

CONVERSION FROM RNS

Thank you for your attention DISCRETE MATHEMATICS Czesław Smutnicki

Internet Engineering Czesław Smutnicki Discrete Mathematics – Residual Arithmetic