PPT-Cryptography Lecture 23 Cyclic groups

Let G be a finite group of order q written multiplicatively Let g be some element of G Consider the set ltggt g 0 g 1 We know g q 1 g 0 so the set has . 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 Introduction In a group we denote the cyclic group of powers of some by If then itself is cyclic with as a generator Examples of in64257nite cyclic groups include with additive generator 1 and the group of integral powers of the real numbe conjugate momentum. cyclic coordinates. Informal derivation. Applications/examples. 1. 2. Define . the . conjugate momentum. Start with the Euler-Lagrange equations. The Euler-Lagrange equations can be rewritten as. 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. Introduction I. Electrochemical methods are used . To investigate . electron transfer processes and . kinetics. To . study redox processes in organic and organometallic . chemistry. To . investigate multi-electron transfer processes in . 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. Definition. If . G. 1. , . G. 2, ..., . G. n. are groups, then their . external direct product G. 1 .  . G. 2 . ... .  . G. n. . is simply the set of all ordered . n. -tuples of elements of the groups under component-wise operations.. 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. Let G be a finite group of order . m. For any positive integer e, define . f. e. (g)=. g. e. If . gcd. (. e,m. )=1, then . f. e. is a permutation of G. Moreover, if d = e. -1. mod m then . f. d. is the inverse of . . 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.