A Computational Introduction to Number Theory and Algebra - download pdf or read online

By Victor Shoup

ISBN-10: 0511113633

ISBN-13: 9780511113635

Quantity idea and algebra play an more and more major position in computing and communications, as evidenced by way of the amazing purposes of those matters to such fields as cryptography and coding thought.

This introductory ebook emphasises algorithms and functions, resembling cryptography and mistake correcting codes, and is out there to a wide viewers. The mathematical must haves are minimum: not anything past fabric in a customary undergraduate direction in calculus is presumed, except a few event in doing proofs - every little thing else is built from scratch.

Thus the publication can serve a number of reasons. it may be used as a reference and for self-study by way of readers who are looking to study the mathematical foundations of contemporary cryptography. it's also perfect as a textbook for introductory classes in quantity conception and algebra, particularly these geared in the direction of machine technology scholars.

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Extra info for A Computational Introduction to Number Theory and Algebra

Example text

Let a, b, n, n ∈ Z with n > 0 and n | n. Show that if a ≡ b (mod n), then a ≡ b (mod n ). 3. Let a, b, n, n ∈ Z with n > 0, n > 0, and gcd(n, n ) = 1. Show that if a ≡ b (mod n) and a ≡ b (mod n ), then a ≡ b (mod nn ). 4. Let a, b, n ∈ Z such that n > 0 and a ≡ b (mod n). Show that gcd(a, n) = gcd(b, n). 5. Prove that for any prime p and integer x, if x2 ≡ 1 (mod p) then x ≡ 1 (mod p) or x ≡ −1 (mod p). 6. Let a be a positive integer whose base-10 representation is a = (ak−1 · · · a1 a0 )10 . Let b be the sum of the decimal digits of a; that is, let b := a0 + a1 + · · · + ak−1 .

5 guarantees, we do indeed have 5 ≡ 3 (mod 2). ✷ Next, we consider the problem of determining the solutions z to congruences of the form az + c ≡ b (mod n), for given integers a, b, c, n. Since we may both add and subtract c from both sides of a congruence modulo n, it is clear that z is a solution to the above congruence if and only if az ≡ b − c (mod n). Therefore, it suffices to consider the problem of determining the solutions z to congruences of the form az ≡ b (mod n), for given integers a, b, n.

Since n1 , . . , nk are pairwise relatively prime, this implies that n | (z − z), or equivalently, z ≡ z (mod n). 4. The following table illustrates what the above theorem says for n1 = 3 and n2 = 5. 20 Congruences z 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 z mod 3 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 z mod 5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 We see that as z ranges from 0 to 14, the pairs (z mod 3, z mod 5) range over all pairs (a1 , a2 ) with a1 ∈ {0, 1, 2} and a2 ∈ {0, . . , 4}, with every pair being hit exactly once.

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A Computational Introduction to Number Theory and Algebra by Victor Shoup


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