Download e-book for kindle: A Computational Introduction to Number Theory and Algebra by Victor Shoup

By Victor Shoup

ISBN-10: 0511113633

ISBN-13: 9780511113635

Quantity conception and algebra play an more and more major position in computing and communications, as evidenced by means of the awesome purposes of those topics to such fields as cryptography and coding concept.

This introductory booklet emphasises algorithms and functions, akin to cryptography and mistake correcting codes, and is out there to a extensive viewers. The mathematical must haves are minimum: not anything past fabric in a regular undergraduate path in calculus is presumed, except a few adventure in doing proofs - every thing else is constructed from scratch.

Thus the publication can serve numerous reasons. it may be used as a reference and for self-study by means of readers who are looking to study the mathematical foundations of recent cryptography. it's also perfect as a textbook for introductory classes in quantity idea and algebra, specially these geared in the direction of laptop technology scholars.

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Example text

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 [1]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 .

For (ii), observe that if n divides a − b, then it also divides −(a − b) = b − a. For (iii), observe that if n divides a − b and b − c, then it also divides (a − b) + (b − c) = a − c. 3. For all positive integers n, and all a, a , b, b ∈ Z, if a ≡ a (mod n) and b ≡ b (mod n), then a + b ≡ a + b (mod n) and a · b ≡ a · b (mod n). Proof. Suppose that a ≡ a (mod n) and b ≡ b (mod n). This means that there exist integers c and d such that a = a + cn and b = b + dn. Therefore, a + b = a + b + (c + d)n, which proves the first congruence of the theorem, and a b = (a + cn)(b + dn) = ab + (ad + bc + cdn)n, which proves the second congruence.

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. 6. Let a, b, n ∈ Z with n > 0. If a is relatively prime to n, then the congruence az ≡ b (mod n) has a solution z; moreover, any integer z is a solution if and only if z ≡ z (mod n).

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A Computational Introduction to Number Theory and Algebra by Victor Shoup

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