A computational introduction to number theory and algebra by Victor Shoup

By Victor Shoup

Quantity idea and algebra play an more and more major position in computing and communications, as evidenced through the remarkable functions of those topics to such fields as cryptography and coding idea. This introductory ebook emphasises algorithms and purposes, corresponding to cryptography and blunder correcting codes, and is available to a extensive viewers. The mathematical necessities are minimum: not anything past fabric in a customary undergraduate direction in calculus is presumed, except a few adventure in doing proofs - every thing else is constructed from scratch. hence the ebook can serve numerous reasons. it may be used as a reference and for self-study by means of readers who are looking to research the mathematical foundations of contemporary cryptography. it's also excellent as a textbook for introductory classes in quantity idea and algebra, in particular these geared in the direction of machine technological know-how scholars.

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This yields a running time of O( 3 len(M )2 + 3 len( )). , it is O( 3 len(M )2 ). Using the Chinese Remainder Theorem, we can actually do much better than this, as follows. For any integer n > 1, and for all 1 ≤ r, t ≤ , we have crt ≡ ars bst (mod n). 4) then we must have crt = crt . , that n divides (crt − crt ). 4), we obtain |crt − crt | ≤ |crt | + |crt | ≤ M + n/2 < n/2 + n/2 = n. So we see that the quantity (crt − crt ) is a multiple of n, while at the same time this quantity is strictly less than n in absolute value; hence, this quantity must be zero.

29 Suppose we have an algorithm that computes the square of an -bit integer in time S( ), where S is a well-behaved complexity function. Show how to use this algorithm to compute the product of two arbitrary integers of at most bits in time O(S( )). 6 Notes The “classical” algorithms presented here for integer multiplication and division are by no means the best possible. com/gmp). Moreover, there are algorithms whose running time is asymptotically faster. 23, which was originally invented by Karatsuba [41] (although Karatsuba is one of two authors on this paper, the paper gives exclusive credit for this particular result to Karatsuba).

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 )). 3) Proof. 3) are those corresponding to divisors d of the form pi1 · · · pi , where pi1 , . . , pi are distinct; the value contributed to the sum by such a term is (−1) f (pi1 · · · pi ) = (−1) f (pi1 ) · · · f (pi ).

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