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Lecture Slides

# Lecture Slides - Number Theory Reminder Introduction...

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Introduction Number Theory Reminder Cryptography and Protocols Andrei Bulatov

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Cryptography and Protocols – Number Theory Reminder 13-2 Divisibility, Primes, etc. Divisibility, residues Prime numbers Primality tests Prime decomposition Greatest common divisor Relatively prime numbers Euler totient function
Cryptography and Protocols – Number Theory Reminder 13-3 Residues For a positive integer n , we denote - the set {0,1,2,…,n –1} - the set {1,2,…,n –1} - addition, multiplication and exponentiation modulo n with these operations is called the set of residues modulo n Every integer m , positive or negative, has a corresponding residue — m mod n For example, 17 mod 5 = 2, 20 mod 5 = 0, -1 mod 5 = 4 n Z + n Z y x , , × + n Z -3

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Cryptography and Protocols – Number Theory Reminder 13-4 Modular Arithmetic We define addition, subtraction, and multiplication of residues: Let a,b . Then a + b (mod n) is the element c such that c a + b (mod m) a – b (mod n) is the element c such that c a – b (mod m) a b (mod n) is the element c such that c a b (mod m) Example.
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