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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-toone or one-to-many Common symbols Belongs to – is a member of For all There exists (at least one) Not equal 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 Inversea 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 442 286 286 156 156 130 130 26 26 0 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