lecture5 - Topics in Cryptography Lecture 5: Basic Number...

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1 page 1 March 11, 2008 Introduction to Cryptography, Benny Pinkas Topics in Cryptography Lecture 5: Basic Number Theory Benny Pinkas
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2 page 2 March 11, 2008 Introduction to Cryptography, Benny Pinkas Classical symmetric ciphers Alice and Bob share a private key k. System is secure as long as k is secret. Major problem: generating and distributing k. Alice Bob k k
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3 page 3 March 11, 2008 Introduction to Cryptography, Benny Pinkas Diffie and Hellman: “New Directions in Cryptography”, 1976. “We stand today on the brink of a revolution in cryptography. The development of cheap digital hardware has freed it from the design limitations of mechanical computing… …such applications create a need for new types of cryptographic systems which minimize the necessity of secure key distribution… …theoretical developments in information theory and computer science show promise of providing provably secure cryptosystems, changing this ancient art into a science.”
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4 page 4 March 11, 2008 Introduction to Cryptography, Benny Pinkas Diffie-Hellman Came up with the idea of public key cryptography Alice Bob public key Bob secret key Bob Everyone can learn Bob’s public key and encrypt messages to Bob. Only Bob knows the decryption key and can decrypt. Key distribution is greatly simplified.
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5 page 5 March 11, 2008 Introduction to Cryptography, Benny Pinkas But before we get to public key cryptography… Basic number theory Divisors, modular arithmetic The GCD algorithm Groups References: Many books on number theory Almost all books on cryptography Cormen, Leiserson, Rivest, (Stein), “Introduction to Algorithms”, chapter on Number-Theoretic Algorithms.
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6 page 6 March 11, 2008 Introduction to Cryptography, Benny Pinkas Divisors, prime numbers We work over the integers A non-zero integer b divides an integer a if there exists an integer c s.t. a=c·b . Denoted as b|a I.e. b divides a with no remainder Examples Trivial divisors: 1|a, a|a Each of {1,2,3,4,6,8,12,24} divides 24 5 does not divide 24 Prime numbers An integer a is prime if it is only divisible by 1 and by itself. 23 is prime, 24 is not.
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7 page 7 March 11, 2008 Introduction to Cryptography, Benny Pinkas Modular Arithmetic Modular operator: a mod b, (or a%b ) is the remainder of a when divided by b I.e., the smallest r 0 s.t. integer q for which a = qb+r. (Thm: there is a single choice for such q,r ) Examples 12 mod 5 = 2 10 mod 5 = 0 -5 mod 5 = 0 -1 mod 5 = 4
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8 page 8 March 11, 2008 Introduction to Cryptography, Benny Pinkas Modular congruency a is congruent to b modulo n (a b mod n) if (a-b) = 0 mod n Namely, n divides a-b In other words, (a mod n) = (b mod n) E.g., 23 12 mod 11 4 -1 mod 5
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9 page 9 March 11, 2008 Introduction to Cryptography, Benny Pinkas Modular congruency Modular congruency is an equivalence relation:
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lecture5 - Topics in Cryptography Lecture 5: Basic Number...

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