By Victor Shoup
Quantity concept and algebra play an more and more major function in computing and communications, as evidenced by means of the awesome functions of those topics to such fields as cryptography and coding concept.
This introductory e-book emphasises algorithms and functions, comparable to cryptography and mistake correcting codes, and is out there to a large viewers. The mathematical must haves are minimum: not anything past fabric in a customary undergraduate path in calculus is presumed, except a few adventure in doing proofs - every 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 through readers who are looking to research the mathematical foundations of contemporary cryptography. it's also perfect as a textbook for introductory classes in quantity thought and algebra, particularly these geared in the direction of laptop technology scholars.
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Additional info for A Computational Introduction to Number Theory and Algebra
5 in the language of residue classes. For α ∈ Zn and positive integer k, the expression αk denotes the product α · α · · · · · α, where there are k terms in the product. One may extend this deﬁnition to k = 0, deﬁning α0 to be the multiplicative identity n . If α has a multiplicative inverse, then it is easy to see that for any integer k ≥ 0, αk has a multiplicative inverse as well, namely, (α−1 )k , which we may naturally write as α−k . In general, one has a choice between working with congruences modulo n, or with the algebraic structure Zn ; ultimately, the choice is one of taste and convenience, and it depends on what one prefers to treat as “ﬁrst class objects”: integers and congruence relations, or elements of Zn .
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). 12. Suppose that f and g are piece-wise continuous on [a, ∞), x and that g is eventually positive. For x ≥ a, deﬁne F (x) := a f (t)dt and x G(x) := a g(t)dt. Show that if f = O(g) and G is eventually positive, then F = O(G). 13. Suppose that f and g are piece-wise continuous [a, ∞), both x of which are eventually positive.
Ak exists and is unique, and moreover, we have νp (gcd(a1 , . . , ak )) = min(νp (a1 ), . . , νp (ak )) for all p. 9) Analogously, for a1 , . . , ak ∈ Z, with k ≥ 1, we call m ∈ Z a common multiple of a1 , . . , ak if ai | m for i = 1, . . , k; moreover, such an m is called the least common multiple of a1 , . . , ak if m divides all common multiples of a1 , . . , ak . It is clear that the least common multiple of a1 , . . , ak exists and is unique, and moreover, we have νp (lcm(a1 , .
A Computational Introduction to Number Theory and Algebra by Victor Shoup