PPT-Cryptography Lecture 21 Groups

Introduce the notion of a group Provides a way of reasoning about objects that share the same mathematical structure Not absolutely needed to understand crypto applications but does make it conceptually easier. However computational aspects of lattices were not investigated much until the early 1980s when they were successfully employed for breaking several proposed cryptosystems among many other applications It was not until the late 1990s that lattices w 897 Special Topics in Cryptography Instructors Ran Canetti and Ron Rivest Lecture 25 PairingBased Cryptography May 5 2004 Scribe Ben Adida 1 Introduction The 64257eld of PairingBased Cryptography has exploded Denition 001 The smallest class of groups that contain nite and abelian groups and is closed under taking subgroups quotients extensions and di rected unions is called the class of elementary amenable groups We denote this class by EG As we proved i Intro to IT. . COSC1078 Introduction to Information Technology. . Lecture 22. Internet Security. James Harland. james.harland@rmit.edu.au. Lecture 20: Internet. Intro to IT. . Introduction to IT. Week two!. The Game. 8 groups of 2. 5 rounds. Math 1. Modern history. Math 2. Computer Programming. Analyzing and comparing Cryptosystems. 10 questions per round. Each question is worth 1 point. Math Round 1. 1. Part I: Crypto. Chapter 2: Crypto Basics. MXDXBVTZWVMXNSPBQXLIMSCCSGXSCJXBOVQXCJZMOJZCVC. TVWJCZAAXZBCSSCJXBQCJZCOJZCNSPOXBXSBTVWJC. JZDXGXXMOZQMSCSCJXBOVQXCJZMOJZCNSPJZHGXXMOSPLH. JZDXZAAXZBXHCSCJXTCSGXSCJXBOVQX. Symmetric Encryption. Key exchange . Public-Key Cryptography. Key exchange. Certification . Why Cryptography. General Security Goal. - . Confidentiality . (. fortrolig. ). - . End-point Authentication . 1. Administrative Note. Professor Blocki is traveling and will be back on Wednesday. . E-mail: . jblocki@purdue.edu. . Thanks to Professor Spafford for covering the first lecture!. 2. https://www.cs.purdue.edu/homes/jblocki/courses/555_Spring17/index.html. We have discussed two classes of cryptographic assumptions. Factoring-based (factoring, RSA assumptions). Dlog. -based (. dlog. , CDH, and DDH assumptions). In two classes of groups. A. ll these problems are believed to be “hard,” i.e., to have no polynomial-time algorithms. Cyclic group G of order q with generator g.  G.  . G = {g. 0. , g. 1. , …, g. q-1. }. For any h .  G, define . log. g. h .  {0, …, q-1} as. . log. g. h = x  . The . art and science of concealing the messages to introduce secrecy in . information security . is recognized as cryptography. .. The word ‘cryptography’ was coined by combining two Greek words, ‘Krypto’ . Crypto is amazing. Can do things that initially seem impossible. Crypto is important. It impacts each of us every day. Crypto is fun!. Deep theory. Attackers’ mindset. Necessary administrative stuff. . Patra. Quick Recall and Today’s Roadmap. >> Hash . Functions- stands in between public and private key world. >. > . Key Agreement. >> Assumptions in Finite Cyclic groups - DL, CDH, DDH. Let G be a finite group of order . q . (written multiplicatively). Let g be some element of G. Consider the set <g> = {g. 0. , g. 1. , …}. We know . g. q. . = 1 = g. 0. , so the set has ≤ .