PPT-Cryptography Lecture 22 Fermat’s little theorem

Let G be a finite group of order m Then for any g G it holds that g m 1 Corollary Let G be a finite group of order m Then for g G and integer x it holds that g x. This theorem gave us a method to prove that a given 64258ow is optimal simply exhibit a cut with the same value This theorem for 64258ows and cuts in a graph is a speci64257c instance of the LP Duality Theorem which relates the optimal values of LP For example the graph of a di64256erentiable function has a horizontal tangent at a maximum or minimum point This is not quite accurate as we will see De64257nition Let an interval A point is a local maximum of if there is 948 0 such that wheneve 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 Ghorai Lecture V Picards existence and uniquness theorem Picards iteration 1 Existence and uniqueness theorem Here we concentrate on the solution of the 64257rst order IVP xy y 1 We are interested in the 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 Raymond Flood. Gresham Professor of Geometry. Newton’s . Laws. Tuesday . 21 October 2014 . Euler’s Exponentials. Tuesday . 18 November 2014 . Fourier’s Series. Tuesday 20 January 2015 . Möbius. Dszquphsbqiz. . Day . 9. Announcements:. Homework 2 due now. Computer quiz Thursday on chapter 2. Questions?. Today: . Finish . congruences. Fermat’s little theorem. Euler’s theorem. Important . Presenter: . Hanh. Than. FLT video. http://www.youtube.com/watch?v=SVXB5zuZRcM. Pierre de Fermat. Pierre de Fermat. . (17 August 1601– 12 January 1665): . . a French lawyer and an amateur mathematician.. Randomized Primality Testing. Carmichael Numbers. Miller-Rabin test. MA/CSSE 473 Day 08. Student questions. Fermat's Little Theorem. Implications of Fermat’s Little Theorem. What we can show and what we can’t. Chapter 4. With Question/Answer Animations. Chapter Motivation. Number theory . is the part of mathematics devoted to the study of the integers and their properties. . Key ideas in number theory include divisibility and the . Gresham Professor of Geometry. Newton’s . Laws. Tuesday . 21 October 2014 . Euler’s Exponentials. Tuesday . 18 November 2014 . Fourier’s Series. Tuesday 20 January 2015 . Möbius. . and his band . Gresham Professor of Geometry. Newton’s . Laws. Tuesday . 21 October 2014 . Euler’s Exponentials. Tuesday . 18 November 2014 . Fourier’s Series. Tuesday 20 January 2015 . Möbius. . and his band . B. 50. 4. /. I. 538. :. . Introduction to. Cryptography. (2017—03—02). Tuesday’s lecture:. One-way permutations (OWPs). PRGs from OWPs. Today’s lecture:. Basic number theory. So far:. “secret key”. 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.