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Discrete Math II - Introduction -

Study number theory, graph theory, automata, and mathematical techniques essential for cryptography, security, and coding theory. Explore key concepts in computer science and engineering.

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Discrete Math II - Introduction -

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  1. Discrete Math II- Introduction - Howon Kim 2019.9.2

  2. About this course… • Course name : Discrete Math II (CP21697) • Study the basics on number theory, graph theory, automata, and mathematical techniques for computer science & engineering • Number theory (finite fields and ring) is the fundamental knowledge for the cryptography & security and Coding Theory etc. • Graph theory & automata is the basic mathematical techniques to understand the computer science, networks and many topics in computer engineering

  3. About this course… • About Instructor • Office : A06-503 • Office hours : 12:30 ~ 13:30 PM(Monday, Wednesday) • Email: howonkim@gmail.com, howonkim@pusan.ac.kr • Phone: 010-8540-6336 • Homepage : http://infosec.pusan.ac.kr • Major Research Interests • 지능형 사물인터넷(Intelligent IoT) 연구 • 블록체인(Blockchain) 연구 • 딥러닝, 산업용 AI 연구 • 암호(Cryptography), 정보보호(Information Security), IoT 보안 • 암호 칩, 보안 칩 설계 연구, TLS 전용 칩 개발 • 국가보안기술연구소, ETRI, KISA와공동연구 • 부산대학교 사물인터넷 연구센터, 블록체인 전문연구실 운영 중

  4. About this course… • Textbook • “Discrete and combinatorial mathematics ” (5th Ed), R.P. Grimaldi, 2004 • Selected Materials for mathematical techniques • Time & Classroom • 13:30 ~ 14:45 PM (Monday, Wednesday), A6-514호 • References • Discrete mathematics by Richard Johnsonbaugh • Introduction to Automata Theory, Languages, and Computation by John E. Hopcroft • Introduction to Graph Theory by Douglas B West • A Course in Number Theory and Cryptography by Neal Koblitz

  5. Course Materials http://infosec.pusan.ac.kr About this course…

  6. About this course

  7. About this course • Grading Policy (수업시간 참여 충실도 반영)

  8. Algebra Definition Tuple <K, op1, op2, …, opn> < R, ,, ,  > < {T,F}, ,,  > ;Boolean algebra K : a set of data |K| : order finite or infinite Operator opj Closure opj : Ki K Unary if i=1, Binary if i=2, … 8

  9. Identity and Zero  : K  K  K Identity element e for  in K(항등원) ea = a e =a for all a ∈ K Zero elementzfor  in K(영원) za = a z =z for all a ∈ K Examples < Z, + > Identity : 0, Zero : none < Z,  > Identity : 1, Zero : 0 9

  10. Inverse  : K  K  K Let e be the identity element for  in K. Left inverse a’La = e , a ∈ K Right inverse a a’R =e,a ∈ K Ifa’L=a’R=a’,a’is the inverse of a. Example < Z, + > Identity 0, (-x) is the inverse of x : x + (-x) = (-x) + x = 0 10

  11. Properties of Operator Let  : K  K  K be a binary operator. (1) Closure (2) Associative (a  b) c = a (b  c) for all a, b, c ∈ K. (3) Identity There is an identity element e ∈ K for . (4) Inverse For each a ∈ K, there is an inverse a’∈ Kfor . (5) Commutative a  b = b  a for all a,b∈K. 11

  12. Binary Algebra < K,  > for binary operator : K  K  K Semigroup (반군) : Associative < Z+, + > A semigroup is a set with an associativebinary operation which satisfies closure and associative law. Monoid (단위반군) : Associative, Identity < N, + >, < Z,  >, < {T,F},  > A monoid is a set that is closed under an associativebinary operation and has an identity element Group (군) : Associative, Identity, Inverse < Z, + > Abelian group (대수군) : Associative, Identity, Inverse, Commutative < Z, + > 12

  13. Binary Algebra Properties Closure Associative Identity Inverse Commutative (1), (2) Semigroup Abelian Semigroup (5) (3) Monoid Abelian Monoid (5) (4) Group Abelian Group (5) • < K,  > Set 13

  14. Binary Algebra Set Closure Semigroup Associative Abelian Semigroup Monoid Identity Abelian Monoid Group Inverse Abelian Group Commutative 14

  15. Ring ( Two operators ) < K, , > Two binary operators,  : K  K  K Conditions for Ring < K, > is an abelian group.  is associative  is distributive over  a  (b  c) = (a  b)  (a  c) and (a b) c = (a  c)  (b  c) for all a,b,c ∈ K. 15

  16. Definitions < K, , > < K, > : abelian group, and distribution laws hold Conditions for operator  : Ring (환) : Associative Ring with Unity : Associative, Identity Commutative Ring : Associative, Commutative Commutative Ring with Unity Associative, Identity, Commutative Field (체) Associative, Identity, Commutative, Inverse 16

  17. Ring and Field Properties for  (0) Distributive (1) Closure (2) Associative (3) Identity (4) Inverse (5) Commutative (0), (1), (2) Ring Commutative Ring (5) (3) (3) Ring with Unity Commutative Ring with Unity (5) (4) Field • < K, , > Set 17

  18. Ring and Field < K, , > Closure Ring Associative Distributive Field Ring with Unity Commutative Ring Inverse Commutative Identity Commutative Ring with Unity 18

  19. Next… • Basics on Number Theory…

  20. Q&A

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