A Computational Introduction to Number Theory and Algebra by Victor Shoup

By Victor Shoup

Quantity idea and algebra play an more and more major position in computing and communications, as evidenced through the remarkable functions of those matters to such fields as cryptography and coding concept.

This introductory ebook emphasises algorithms and purposes, corresponding to cryptography and mistake correcting codes, and is available to a extensive viewers. The mathematical necessities are minimum: not anything past fabric in a standard undergraduate path in calculus is presumed, except a few adventure in doing proofs - every little thing else is built from scratch.

Thus the e-book can serve a number of reasons. it may be used as a reference and for self-study through readers who are looking to study the mathematical foundations of contemporary cryptography. it's also perfect as a textbook for introductory classes in quantity idea and algebra, specifically these geared in the direction of machine technological know-how scholars.

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Let us call a function in x eventually positive if it takes positive values for all sufficiently large x. Note that by definition, if we write f = Ω(g), f = Θ(g), or f ∼ g, it must be the case that f (in addition to g) is eventually positive; however, if we write f = O(g) or f = o(g), then f need not be eventually positive. When one writes “f = O(g),” one should interpret “· = O(·)” as a binary relation between f with g. ” One may also write “O(g)” in an expression to denote an anonymous function f such that f = O(g).

Then we obtain: pe11 µ(d)/d = (1 − 1/p1 ) · · · (1 − 1/pr ). 8) 30 Congruences As another example, suppose f = J. Then we obtain r (1 − 1), µ(d) = (µ J)(n) = i=1 d|n which is 1 if n = 1, and is zero if n > 1. Thus, we have µ J = I. 18 (M¨ obius inversion formula). Let f and F be arithmetic functions. Then we have F = J f if and only if f = µ F . Proof. If F = J f , then µ F = µ (J f ) = (µ J) f = I f = f, and conversely, if f = µ F , then J f =J (µ F ) = (J µ) F = I F = F. ✷ The M¨ obius inversion formula says this: F (n) = f (d) for all positive integers n d|n if and only if µ(d)F (n/d) for all positive integers n.

10, −4, 2, 8, 14, . } [3] = {. . , −9, −3, 3, 9, 15, . } [4] = {. . , −8, −2, 4, 10, 16, . } [5] = {. . , −7, −1, 5, 11, 17, . } 22 Congruences Let us write down the addition and multiplication tables for Z6 . 10. Let n be a positive integer, and consider the set Zn of residue classes modulo n with addition and multiplication of residue classes as defined above. For all α, β, γ ∈ Zn , we have (i) α + β = β + α (addition is commutative), (ii) (α + β) + γ = α + (β + γ) (addition is associative), (iii) α + [0]n = α (existence of additive identity), (iv) α − α = [0]n (existence of additive inverses), (v) α · β = β · α (multiplication is commutative), (vi) (α · β) · γ = α · (β · γ) (multiplication is associative), (vii) α · (β + γ) = α · β + α · γ (multiplication distributes over addition) (viii) α · [1]n = α (existence of multiplicative identity).

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