COM5336 Cryptography Lecture 11 Euclidean Domains & Division Algorithm

1. COM5336 CryptographyLecture 11Euclidean Domains & Division Algorithm Scott CH Huang COM 5336 Cryptography Lecture 10

2. Groups • Binary operations on a set is a mapping • A set w/ an operation satisfying • Closure • Associativity • Identity • Inverse • The most fundamental algebraic structure • Semi-groups: 1 & 2 only. • Abelian groups: commutative groups. COM 5336

3. Rings • A set R with two operations: + and *. • +: commutative. • *: not necessarily commutative. • (R,+) forms an abelian group. • (R,*) forms a semi-group (i.e. no identity and inverse) • Distributivity • Ring v.s. Ring with 1 (mult. identity). COM 5336

4. Integral Domains • Domain = Ring w/o zero-divisors • ab=0 implies a=0 or b=0 • One-sided cancellation law • Integral Domain = Commutative domain w/ 1. • Two-sided cancellation law COM 5336

5. Euclidean Domains • A Euclidean Domain is an integral domain with the notion of size. • The notion of size enables us to apply the Division Algorithmand therefore Euclid’s Algorithm. • Size of a≠0, denoted by g(a) is a nonnegative integer s.t. • g(a)≤g(ab), for all b≠0. • For all a,b≠0, there exists q,r s.t. a=qb+r, w/ r=0 or g(r)<g(b) COM 5336

6. Division Algorithm • A theorem in mathematics which precisely expresses the outcome of the usual process of division of integers. • Its name is a misnomer. • It is not a true algorithm. • A well-defined procedure for achieving a specific task COM 5336

7. Division Rings • A ring with unit in which division is possible. • i.e. every nonzero element has a multiplicative inverse. • A division ring is NOT necessarily commutative. • But finite division rings must be commutative (Wedderburn's little theorem). • A field is a commutative division ring. • Therefore all finite division rings are finite fields. COM 5336

8. Relationship of Algebraic Structures Ring Ring w/ unit Division Ring Commutative ring w/ unit Integral Domain Euclidean Domain Field COM 5336

9. Division in a Euclidean Domain • a|b: ‘a’ divides ‘b’ iff there exists c s.t. b=ac • a,b,c  D, a Euclidean domain. • If a|b1, a|b2,…, then a is a common divisor of b1,b2,… • If d is a common divisor of b1,b2,…, and every common divisor divides d, then d is a greatest common divisor (GCD) of b1,b2,… • In fact, the concept of GCD can be extended to certain integral domains called Principal Ideal Domains. COM 5336

10. GCD in Algebraic Structures COM 5336

11. GCD may not exists in an Integral Domain Note that Both d1, d2 are common divisors of b1, b2 , so b1, b2 has no greatest common divisors. COM 5336

12. GCD exists in a Euclidean Domain • If , then d can be expressed as a linear combination of a,b. • If D is a Euclidean domain and , then d can be expressed as a linear combination of a,b • How to calculate the GCD? COM 5336

13. Euclid’s Inspiring Lemma • gcd(s,t)=gcd(s,t-rs) for all s,t,r in a Euclidean domain D. • This lemma directly results in Euclid’s algorithm. COM 5336

14. Euclid’s Algorithm int gcd(s,t){ while (s!=0){ u=s; s= t mod s; t=u; } return t; } COM 5336

15. Theorem #1 Let t be an element in a Euclidean domain D and m,n be two positive integers. Then *Hint: (tn-1)-tn-m (tm-1)= tn-m -1 COM 5336

16. Corollary #1 Let x be an element in a Euclidean domain D and q,n,d be positive integers. Then COM 5336

17. Conceptually COM 5336

18. Some Examples Euclidean domain ring w/ 1 finite field commutative ring w/ 1 Euclidean domain COM 5336

19. More Examples (cont’d) • The set of polynomials over an arbitrary field with polynomial addition & multiplication. • The set of polynomials with two variables x,y over an arbitrary field with polynomial addition & multiplication. COM 5336

20. Factorization in Euclidean Domains • We wish to establish a “Fundamental Theorem of Arithmetic” in Euclidean domains. • Fundamental Theorem of Arithmetic (aka Unique-Prime-Factorization Theorem) • Any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers. • In order to do that, it’s vital to introduce the idea of a “prime number” in Euclidean domains. COM 5336

21. Preliminaries • Let D be an integral domain. A unit u  D is any divisor of 1. • In the integer ring, the units are ±1. In the Gaussian integer ring, ±1, ±i are units. • a, b  D are associates if a=ub for some unit u. • In the integer ring, +3, -3 are associates. In the Gaussian integer ring, 1+ i, 1- i are associates. • A factorization of b is an expression of the form b=a1a2···ar. If each of the ai’s are either a unit or an associate of b, this is a trivial factorization. COM 5336

22. Irreducible Elements in Integral Domains • A element p  D, an integral domain, is called irreducible iff every factorization of p is trivial. • We do not consider units to be irreducible. • b  D. d|b. If d is not an associate of b, then it is called a proper divisor. • Irreducible elements have no proper divisors other than units. COM 5336

23. Primes in Integral Domains • A nonzero, non-unit element p  D, an integral domain, is called prime iff the following property holds. • If p|ab, then either p|a or p|b for a,b  D. COM 5336

24. Primes vs Irreducible Elements • In an integral domain, every prime is irreducible. • In a Principal Ideal Domain (PID), every irreducible element is prime. • In our textbook, only Euclidean domains are discussed. The author did not distinguish between primes and irreducible elements and regarded them as synonyms. COM 5336

25. Relative Primality • In a PID, two elements a,b are relatively prime iff gcd(a,b)=1. (remember that GCD must exists in a PID) • In a Euclidean domain, if p does not divide a and p is prime, then p and a are relatively prime. COM 5336

26. Some Properties • In a Euclidean domain, if p does not divide a, then there exist s,t such that ps+at=1. • In a Euclidean domain, if a is a proper divisor of b, then g(a)<g(b). COM 5336

27. Unique Factorization • Theorem 3.6: In a Euclidean domain, if b is not a unit, then b can be factorized as a product of primes: • b=p1p2···pn • If b can be factorized in another way as b=q1q2···qn , then after appropriate renumbering, pi qi are associates for all i. • In short, Euclidean domains are Unique Factorization Domains (UFD). COM 5336

28. Euclidean Domains, PIDs, UFDs Integral Domain Unique Factorization Domain *Principal Ideal Domain* Euclidean Domain Field COM 5336

29. Example of a non-UFD • Consider the integral domain • are irreducible. COM 5336