By Victor Shoup

Quantity conception and algebra play an more and more major position in computing and communications, as evidenced via the awesome functions of those topics to such fields as cryptography and coding idea.

This introductory booklet emphasises algorithms and purposes, corresponding to cryptography and mistake correcting codes, and is available to a large viewers. The mathematical necessities are minimum: not anything past fabric in a regular undergraduate direction in calculus is presumed, except a few event in doing proofs - every 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 by way of readers who are looking to research the mathematical foundations of recent cryptography. it's also perfect as a textbook for introductory classes in quantity conception and algebra, in particular these geared in the direction of laptop technology scholars.

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For a positive integer n, we say that a ∈ Z with gcd(a, n) = 1 is a primitive root modulo n if the multiplicative order of a modulo n is equal to φ(n). If this is the case, then for α := [a]n , the powers αi range over all elements of Z∗n as i ranges over the interval 0, . . , φ(n) − 1. 2 that the only positive integers n for which there exists a primitive root modulo n are n = 1, 2, 4, pe , 2pe , where p is an odd prime and e is a positive integer. 15. Find an integer whose multiplicative order modulo 101 is 100.

As an example, one could write ni=1 i = n2 /2 + O(n). Analogously, Ω(g), Θ(g), and o(g) may denote anonymous functions. The expression O(1) denotes a function bounded in absolute value by a constant, while the expression o(1) denotes a function that tends to zero in the limit. ) of the notation, one may use the big-O, -Omega, and -Theta notation for functions on an arbitrary domain, in which case the relevant bound should hold throughout the entire domain. 1. Show that (a) f = o(g) implies f = O(g) and g = O(f ); (b) f = O(g) and g = O(h) implies f = O(h); (c) f = O(g) and g = o(h) implies f = o(h); (d) f = o(g) and g = O(h) implies f = o(h).

And that g is eventually positive. For n ≥ k, define F (n) := n n i=k f (i) and G(n) := i=k g(i). Show that if f = O(g) and G is eventually positive, then F = O(G). 11. Suppose that f and g are functions defined on the integers k, k + 1, . , both of which are eventually positive. For n ≥ k, define F (n) := n n i=k f (i) and G(n) := i=k g(i). Show that if f ∼ g and G(n) → ∞ as n → ∞, then F ∼ G. The following two exercises are continuous variants of the previous two exercises. ” In particular, we restrict ourselves to piecewise continuous functions (see §A3).

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