New PDF release: A Computational Introduction to Number Theory and Algebra

By Victor Shoup

ISBN-10: 0511113633

ISBN-13: 9780511113635

Quantity idea and algebra play an more and more major function in computing and communications, as evidenced by means of the outstanding functions of those topics to such fields as cryptography and coding idea.

This introductory ebook emphasises algorithms and functions, reminiscent of cryptography and blunder correcting codes, and is obtainable to a large viewers. The mathematical necessities are minimum: not anything past fabric in a customary undergraduate path in calculus is presumed, except a few adventure in doing proofs - every little thing else is built from scratch.

Thus the booklet 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 recent cryptography. it's also excellent as a textbook for introductory classes in quantity thought and algebra, specially these geared in the direction of computing device technology scholars.

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Extra info for A Computational Introduction to Number Theory and Algebra

Example text

3. Let a, b, n, n ∈ Z with n > 0, n > 0, and gcd(n, n ) = 1. Show that if a ≡ b (mod n) and a ≡ b (mod n ), then a ≡ b (mod nn ). 4. Let a, b, n ∈ Z such that n > 0 and a ≡ b (mod n). Show that gcd(a, n) = gcd(b, n). 5. Prove that for any prime p and integer x, if x2 ≡ 1 (mod p) then x ≡ 1 (mod p) or x ≡ −1 (mod p). 6. Let a be a positive integer whose base-10 representation is a = (ak−1 · · · a1 a0 )10 . Let b be the sum of the decimal digits of a; that is, let b := a0 + a1 + · · · + ak−1 . Show that a ≡ b (mod 9).

12. Let p be a prime and k an integer 0 < k < p. Show that the binomial coefficient p p! (p − k)! which is an integer, of course, is divisible by p. 13. An integer a ∈ Z is called square-free if it is not divisible by the square of any integer greater than 1. Show that any integer n ∈ Z can be expressed as n = ab2 , where a, b ∈ Z and a is square-free. 14. Show that any non-zero x ∈ Q can be expressed as x = ±pe11 · · · perr , where the pi are distinct primes and the ei are non-zero integers, and that this expression in unique up to a reordering of the primes.

6. Let a be a positive integer whose base-10 representation is a = (ak−1 · · · a1 a0 )10 . Let b be the sum of the decimal digits of a; that is, let b := a0 + a1 + · · · + ak−1 . Show that a ≡ b (mod 9). From this, justify the usual “rules of thumb” for determining divisibility by 9 and 3: a is divisible by 9 (respectively, 3) if and only if the sum of the decimal digits of a is divisible by 9 (respectively, 3). 7. Show that there are 14 distinct, possible, yearly (Gregorian) calendars, and show that all 14 calendars actually occur.

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A Computational Introduction to Number Theory and Algebra by Victor Shoup


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