L06_RSAAlgorithm

L06_RSAAlgorithm - 1-1COMP170Discrete Mathematical Toolsfor...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSections 2.3, 2.4, pp. 72-86The RSA AlgorithmVersion 2.1 Last updated, September 5, 2008Slidesc2005 by M. J. Golin and G. Trippen2-12.3 The RSA Cryptosystem2-22.3 The RSA Cryptosystem•Assorted Tools and Definitions2-32.3 The RSA Cryptosystem•Exponentiation modn•Assorted Tools and Definitions2-42.3 The RSA Cryptosystem•Exponentiation modn•The Rules of Exponents•Assorted Tools and Definitions2-52.3 The RSA Cryptosystem•Exponentiation modn•The Rules of Exponents•Fermat’s Little Theorem•Assorted Tools and Definitions2-62.3 The RSA Cryptosystem•Exponentiation modn•The Rules of Exponents•Fermat’s Little Theorem•The RSA Cryptosystem•Assorted Tools and Definitions2-72.3 The RSA Cryptosystem•Exponentiation modn•The Rules of Exponents•Fermat’s Little Theorem•The RSA Cryptosystem•Practical Aspects of Exponentiation modn•Assorted Tools and Definitions2-82.3 The RSA Cryptosystem•Exponentiation modn•The Rules of Exponents•Fermat’s Little Theorem•The RSA Cryptosystem•The Chinese Remainder Theorem•Practical Aspects of Exponentiation modn•Assorted Tools and Definitions3-1·7123456112345622461353362514441526355316426654321Consider multiplication inZ73-2·7123456112345622461353362514441526355316426654321Consider multiplication inZ7For every nonzeroa∈Z7, thefunctionfa(x) =x·7ais one-to-one and thereforea permutation ofZ7- {}, i.e.,every row is a permutation.3-3·7123456112345622461353362514441526355316426654321Consider multiplication inZ7For every nonzeroa∈Z7, thefunctionfa(x) =x·7ais one-to-one and thereforea permutation ofZ7- {}, i.e.,every row is a permutation.Lemma 2.20: Letpbe a prime number. For any nonzeronumbera∈Zp, the functionfa(x) =x·pais 1-to-1. Inparticular, the numbers,1·pa,2·pa, . . . ,(p-1)·pa,are apermutationof the set{1,2, . . . , p-1}.4-1Lemma 2.20: Letpbe a prime number. For any nonzeronumbera∈Zp, the functionfa(x) =x·pais 1-to-1. Inparticular, the numbers,1·pa,2·pa, . . . ,(p-1)·pa,are apermutationof the set{1,2, . . . , p-1}.Proof:Supposefa(x)is not 1-to-1. Then there arex6=ywithfa(x) =fa(y). Sincepis prime, Corollary 2.17 tells usthat there isa-1∈Zps.t.a·pa-1= 1.x= (x·pa)·pa-1=fa(x)·pa-1Contradiction!Multiplying the two sides bya-1gives=fa(y)·pa-1= (y·pa)·pa-1=y4-2Lemma 2.20: Letpbe a prime number. For any nonzeronumbera∈Zp, the functionfa(x) =x·pais 1-to-1. Inparticular, the numbers,1·pa,2·pa, . . . ,(p-1)·pa,are apermutationof the set{1,2, . . . , p-1}.Proof:Supposefa(x)is not 1-to-1. Then there arex6=ywithfa(x) =fa(y). Sincepis prime, Corollary 2.17 tells usthat there isa-1∈Zps.t.a·pa-1= 1....
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L06_RSAAlgorithm - 1-1COMP170Discrete Mathematical Toolsfor...

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