A computational introduction to number theory and algebra by Victor Shoup PDF

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 purposes of those matters to such fields as cryptography and coding thought. This introductory booklet emphasises algorithms and functions, resembling cryptography and mistake correcting codes, and is offered to a wide viewers. The mathematical must haves are minimum: not anything past fabric in a standard undergraduate path in calculus is presumed, except a few adventure in doing proofs - every thing else is constructed from scratch. therefore the e-book can serve numerous 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 contemporary cryptography. it's also perfect as a textbook for introductory classes in quantity concept and algebra, specially these geared in the direction of machine technology scholars.

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Thus, if n = pe11 · · · perr is the prime factorization of n, then µ(n) = 0 if ei > 1 for some i, and otherwise, µ(n) = (−1)r . Here are some examples: µ(1) = 1, µ(2) = −1, µ(3) = −1, µ(4) = 0, µ(5) = −1, µ(6) = 1. It is easy to see (verify) that for any function f , f I = f , and that (f J)(n) = d|n f (d). Also, the functions I, J, and µ are multiplicative (verify). 28 For any multiplicative function f , if n = pe11 · · · perr is the prime factorization of n, we have d|n µ(d)f (d) = (1 − f (p1 )) · · · (1 − f (pr )).

If a ∈ Z, we can convert a to an object α ∈ Zn by writing α ← [a mod n], and if a ∈ {0, . . , n − 1}, this type conversion is purely conceptual, involving no actual computation. Conversely, if α ∈ Zn , we can convert α to an object a ∈ {0, . . , n − 1}, by writing a ← rep(α); again, this type conversion is purely conceptual, and involves no actual computation. Another interesting problem is exponentiation in Zn : given α ∈ Zn and a non-negative integer e, compute αe ∈ Zn . Perhaps the most obvious way to do this is to iteratively multiply by α a total of e times, requiring time O(e len(n)2 ).

In lines 6–10, we essentially compute (ri+ · · · ri )B ← (ri+ · · · ri )B − qi b. In each loop iteration, the value of tmp lies between −(B 2 − B) and B − 1, and the value carry lies between −(B − 1) and 0. 4. If the estimate qi is too large, this is manifested by a negative value of ri+ at line 11. 1, except that we ignore the carry out of that algorithm. 3. 13 Work out the details of an algorithm that computes the quotient and remainder for signed integers, using an algorithm for unsigned integers as a subroutine.

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