1 / 20

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.

tyj
Download Presentation

Discrete Math II - Introduction -

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related