Simple modern algebra

Simple modern algebra Groups, rings, and fields Modular arithmetic Euclid’s algorithm Polynomials and Galois multiplication Conventional crypto - Noack

Elementary terms and notation • Set – a collection of objects – not otherwise defined in naïve set theory • Correspondence – can be one-to-one or many-to-one or one-to-many • Common symbols Conventional crypto - Noack

Common relationships and definitions • Equality – relationship is an equality relationship if: • Reflexive a = a • Transitive a = b and b = c imply a = c • Symmetric a = b implies b = a • Objects do not need to be equal numerically to satisfy an equivalence relationship – example, similar triangles • Closure a,b  S implies a  b  S • Associativity a  (b  c) = (a  b)  c – can be written a  b  c • Identity e  S such that a  S e a = a, a e = a • Inverse a  S a’  S such that a’ a = e, a a’ = e • Commutativity a,b  S a b = b a • Distributivity a(b + c) = ab + ac • This is notational, the two operations are + and implied * even though they are not necessarily numerical addition or multiplication – examples are Boolean Conventional crypto - Noack

The hierarchy from group to field • Group • Set (S) and operation () over S • Satisfies closure, associativity, identity (e) and inverse (a’) • Also cyclic group if every element is a power of some possibly unique element • Abelian group • Group with commutativity • Ring • Set with two operations called addition (+) and multiplication () or (*) • Identity is 0, inverse is -a • Abelian group under addition • Satisfies closure, associativity, distributivity (* over +) for multiplication • Integral domain • Ring with identity (1) and no zero divisors • Field • Integral domain with defined inverse (a-1) Conventional crypto - Noack

Some notation and examples • Common numeric sets are called • Z (integers), Q (rationals), R (reals), C (complex) • Common subsets • Z + (positive), Z* (nonzero),Zp`{0, 1, … p-1} • Examples • Z is a group under +, Z + is not (why) • Book says Z + is an infinite cyclic group generated by 1 and + (why isn’t this true) • Definitions for division and divisibility • b|a means a = mb for some c  Z and b  Z*, meaning b divides a • Also for any a  Z and n  Z + , a = cn + r, with r  Zn and c  Z • r is called the residue or remainder Conventional crypto - Noack

Modulo definition and operations • Definition of a mod n • The remainder in a = cn + r • Properties • a = b mod n means n|(a-b) the equal sign followed by mod means modulo equality. • Modulo equality is an equality relationship • a mod n mod n = a mod n • Addition, subtraction, and multiplication, but not division mod n carry over into modular arithmetic • Division-like issues depend on whether n is prime • Test yourself • What algebraic structure does Zn under under addition and multiplication modulo n form? – ring, integral domain, field? • What is –a in modulo arithmetic • Under what conditions does ab=ac mod n imply b=c mod n? Conventional crypto - Noack

Euclid’s algorithm • This ancient algorithm; • Finds the gcd of two integer-like quantities • Euclid (365BC?-275BC?) worked in Alexandria and wrote the Elements at about age 40 • The algorithm itself • gcd (a,b) = max(k such that k|a and k|b), k  Z + and a,b  Z * • based on repeated application of gcd (a,b) = gcd (b,a mod b) • It is easy to prove it terminates in 2 log2 steps. • Proof is slightly indirect – • Can be used with polynomials and also to find multiplicative inverses in finite fields Conventional crypto - Noack

Polynomials • Polynomial in X with coefficients in some field • anXn + an-1Xn-1 + an-2Xn-2 + … a0X0 • Defined operations • Addition – coefficient by coefficient addition – the coefficients remain in the same field • Multiplication by a scalar – multiply the coefficients by the scalar • Multiplication of two polynomials – the high-school method • Division – the high-school method – note that A(X)/B(Z) is really A(X) mod B(X) and is “smaller” than B(X) • gcd exists and is found by Euclid’s algorithm • Some interesting equivalences • Polynomial – array • Polynomial in Z2 – binary register contents – bit sequence • Polynomial in Zn – positional representation of number in base n • But note that the numeric addition and multiplication algorithms are not the standard polynomial operations Conventional crypto - Noack

Galois field multiplication • Motivation • We need another invertible operation over Zp where p = 2n • Ordinary multiplication in a non-prime sized field doesn’t result in a unique inverse • Galois fields with size 256 are easily constructed and are used in a number of block encryption algorithms • Motivation for putting the rest of this on the board • Try doing equations in PowerPoint Conventional crypto - Noack

Simple modern algebra