Read e-book online A computational introduction to number theory and algebra PDF

By Victor Shoup

Quantity conception and algebra play an more and more major function in computing and communications, as evidenced through the awesome purposes of those topics to such fields as cryptography and coding conception. This introductory booklet emphasises algorithms and purposes, corresponding to cryptography and mistake correcting codes, and is obtainable to a wide viewers. The mathematical must haves are minimum: not anything past fabric in a standard undergraduate direction in calculus is presumed, except a few adventure in doing proofs - every thing else is constructed from scratch. therefore the booklet can serve a number of 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 recent cryptography. it's also perfect as a textbook for introductory classes in quantity thought and algebra, specially these geared in the direction of laptop technological know-how scholars.

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Note that in Step 2, if our Chinese Remainder algorithm happens to be implemented to return an integer z with 0 ≤ z < n, we can easily get a balanced remainder by just subtracting n from z if z ≥ n/2. The correctness of the above algorithm has already been established. Let us now analyze its running time. 18) to be O( 2 len(M ) len(M )). Under our assumption about the cost of arithmetic modulo small primes, the cost of Step 1c is 50 Chapter 4. Euclid’s Algorithm O( 3 k), and since k = O(len(M )) = O(len(M ) + len( )), the cost of this step is O( 3 (len(M ) + len( ))).

4. Set n ← nn . 5. Output z , n . Show that the output z , n of the algorithm satisfies the conditions stated above, and estimate its running time. 11 Using the algorithm in the previous exercise as a subroutine, give a simple O(len(n)2 ) algorithm that takes as input integers n1 , . . , nk , and a1 , . . , ak , with ni > 1, gcd(ni , nj ) = 1 for i = j, and 0 ≤ ai < ni , and outputs z, n such that 0 ≤ z < n, z ≡ ai (mod ni ), and n = i ni . The algorithm should be “incremental,” in that it processes the pairs (ni , ai ) one at a time, using time O(len(n) len(ni )) to process each such pair.

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 ).

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A computational introduction to number theory and algebra by Victor Shoup

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