By Arkadii Slinko

This publication examines the connection among arithmetic and knowledge within the sleek global. certainly, sleek societies are awash with facts which has to be manipulated in lots of alternative ways: encrypted, compressed, shared among clients in a prescribed demeanour, protected against an unauthorised entry and transmitted over unreliable channels. All of those operations may be understood simply by way of someone with wisdom of fundamentals in algebra and quantity theory.

This ebook presents the mandatory heritage in mathematics, polynomials, teams, fields and elliptic curves that's adequate to appreciate such real-life functions as cryptography, mystery sharing, error-correcting, fingerprinting and compression of data. it's the first to hide many fresh advancements in those issues. according to a lecture direction given to third-year undergraduates, it's self-contained with a variety of labored examples and routines supplied to check figuring out. it could possibly also be used for self-study.

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Prαr 1− 1 pr 1 1 ... 1 − p2 pr 1 ... 1 − , pr 1− as required. 2 φ(264) = φ(23 · 3 · 11) = 264 φ(269) = 268 as 269 is prime. 1 2 2 3 10 11 = 80. We also have The following corollary will be important in the cryptography section. 1 If n = pq, where p and q are primes, then φ(n) = ( p−1)(q −1) = pq − p − q + 1. There are no known methods for computing φ(n) in situations where the prime factorisation of n is not known. If n is so big that modern computers cannot factorise it, you can publish n and keep φ(n) secret.

How many solutions in Z11 does the equation x 102 = 4 have? List them all. 6. Given an odd number m > 1, find the remainder when 2φ(m)−1 is divided by m. This remainder should be expressed in terms of m. 7. (Wilson’s Theorem) Let p be an integer greater than one. Prove that p is prime if and only if ( p −1)! = −1 in Z p . ) 8. Prove that any commutative finite ring R (unity is not assumed) without zero divisors is a field. 5 Representation of Numbers There is an important distinction between numbers and their representations.

2. 3. 4. 5. Compute φ(125), φ(180) and φ(1001). Factor n = 4386607, which is a product of two primes, given φ(n) = 4382136. Find m = p 2 q 2 , given that p and q are primes and φ(m) = 11424. 2013 Find the remainder of 2(2 ) on division by 5. Using Fermat’s Little Theorem find the remainder on dividing by 7 the number 333555 + 555333 . 6. Let n = 1234567890987654321 and a = 111111111. Calculate a n−1 mod n. Is the result consistent with the hypothesis that n is prime? 7. Let p > 2 be a prime. Prove that all prime divisors of 2 p − 1 have the form 2kp + 1.

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