39 Slides--RSA

39 Slides--RSA - CS103A HO #39 RSA 2/20/08 RSA...

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CS103A HO #39 RSA 2/20/08 1 CS103A 2/20/08 Midterm Exam, Question 1, Part 3: We will accept answer (d) or (f). If you answered (d) and did not get credit, please bring in your paper! Alice Bob RSA Cryptography: Motivation Alice Bob RSA Cryptography: Motivation Alice Bob RSA Cryptography: Motivation Alice Bob We need a design such that Eve also gets a supply of Bob’s locks, but cannot deduce the key. RSA Cryptography: Motivation plaintext Compute ciphertext Compute plaintext Public Key K e Private Key K d

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CS103A HO #39 RSA 2/20/08 2 We want a function E(m, K e ) which converts a message m into some ciphertext c which appears meaningless to any observer—E(m, K e ) must be a one-way function. Alice (or anybody else) can use this function to encrypt her message before sending it to Bob. Of course, Bob must be able to apply some secret function D(c, K d ) to recover m—For any m, it must hold that m = D(E(m, K e ), K d ). What we need is called a trapdoor function. RSA Cryptography: Motivation Recall that f( x ) g x (mod p) is an effective one-way function, with a fixed g and prime modulus p. Unfortunately, this is believed to be very difficult to invert for anybody. (There are no known trapdoors) RSA Cryptography: Motivation What about f( x ) = x e (mod N) with a fixed e and composite modulus N? RSA Cryptography: Motivation What about f( x ) = x e (mod N) with a fixed e and composite modulus N? This function is one-way, for carefully chosen e, N, and yet it has a trapdoor! RSA Cryptography: Motivation To understand why, we will need the Euler totient (or ϕ ) function. For any n, ϕ (n) is defined as the number of positive integers x in the range 1 x n such that x n. (Note: x n is shorthand for “x and n are relatively prime”) ϕ (15) = 8 since {1, 2, 4, 7, 8, 11, 13, 14} are 15 Leonhard Euler 1707-1783 RSA Cryptography: Mathematical Principles Theorem #1: If p is prime, then ϕ (p) = p – 1. Theorem #2: If x = p n for some prime p, then ϕ (x) = p n –p n-1 . Theorem #3: If x = m n, with m n, then ϕ (x) = ϕ (m) ⋅ϕ (n). RSA Cryptography: Mathematical Principles
CS103A HO #39 RSA 2/20/08 3 Given the factorization of a number N: N = p 1 e1 p 2 e2 p n en ϕ (N) = ϕ (p 1 e1 ) ⋅ϕ (p 2 e2 ) ⋅ϕ (p n en ) ϕ (N) = (p 1 e1 -p 1 e1-1 ) (p 2 e2 –p 2 e2-1 ) (p n en -p n en-1 ) So, computing ϕ (N) is easy if the factorization of N is known. Euler proved this in the 1730’s.

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This note was uploaded on 10/01/2011 for the course CS 103A taught by Professor Plummer,r during the Winter '07 term at Stanford.

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39 Slides--RSA - CS103A HO #39 RSA 2/20/08 RSA...

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