L06_RSAAlgorithm_print

L06_RSAAlgorithm_print - COMP170 Discrete Mathematical...

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1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Sections 2.3, 2.4, pp. 72-86 The RSA Algorithm Version 2.1 Last updated, September 5, 2008 Slides c 2005 by M. J. Golin and G. Trippen
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2 2.3 The RSA Cryptosystem Exponentiation mod n The Rules of Exponents Fermat’s Little Theorem The RSA Cryptosystem The Chinese Remainder Theorem Practical Aspects of Exponentiation mod n Assorted Tools and Definitions
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3 · 7 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 1 3 5 3 3 6 2 5 1 4 4 4 1 5 2 6 3 5 5 3 1 6 4 2 6 6 5 4 3 2 1 Consider multiplication in Z 7 For every nonzero a Z 7 , the function f a ( x ) = x · 7 a is one-to-one and therefore a permutation of Z 7 - { 0 } , i.e., every row is a permutation . Lemma 2.20 : Let p be a prime number. For any nonzero number a Z p , the function f a ( x ) = x · p a is 1-to-1. In particular, the numbers, 1 · p a, 2 · p a, . . . , ( p - 1) · p a , are a permutation of the set { 1 , 2 , . . . , p - 1 } .
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4 Lemma 2.20 : Let p be a prime number. For any nonzero number a Z p , the function f a ( x ) = x · p a is 1-to-1. In particular, the numbers, 1 · p a, 2 · p a, . . . , ( p - 1) · p a , are a permutation of the set { 1 , 2 , . . . , p - 1 } . Proof: Suppose f a ( x ) is not 1-to-1. Then there are x = y with f a ( x ) = f a ( y ) . Since p is prime, Corollary 2.17 tells us that there is a - 1 Z p s.t. a · p a - 1 = 1 . x = ( x · p a ) · p a - 1 = f a ( x ) · p a - 1 Contradiction! Multiplying the two sides by a - 1 gives = f a ( y ) · p a - 1 = ( y · p a ) · p a - 1 = y Then f a ( x ) is 1-to-1
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5 A one-to-one function f : X Y is a one-way function if knowing f ( x ) does not provide you with enough information to efficiently recover x . Note that the definition of one-way function has been intentionally left quite imprecise. If f is one-to-one, then the inverse g of f with g ( f ( x )) = x always exists . For public-key cryptography, the public encoding function , P B , needs to be one-way . The secret decoding function , S B , is actually an efficient way of calculating the inverse of P B . This efficient way is only available to the “owner” who constructed P B . Knowing that g exists , though, does not always help in calcu- lating g ( u ) . For a given u , g ( u ) might be hard to calculate .
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6 Alice Bob M P B ( M ) S B ( P B ( M )) = M i) Alice wants to send M to Bob ii) In public directory, Alice looks up Bob’s Public Key, P B iii) Alice sends P B ( M ) to Bob iv) Bob uses his Secret Key, S B to decrypt M = S B ( P B ( M )) Alice P A Bob P B Candice P C Dick P D . . . . . . The Black Pages Public Key Directory Recall the Public-Key Setup
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7 2.3 The RSA Cryptosystem Exponentiation mod n The Rules of Exponents Fermat’s Little Theorem The RSA Cryptosystem The Chinese Remainder Theorem Practical Aspects of Exponentiation mod n Assorted Tools and Definitions
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8 Exponentiation mod n Last time, we considered encryption using modular addition and multiplication, and have seen weaknesses of both.
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