By Ian F. Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone, Tomik Yaghoobian (auth.), Alfred J. Menezes (eds.)
The conception of finite fields, whose origins might be traced again to the works of Gauss and Galois, has performed an element in quite a few branches in arithmetic. Inrecent years we now have witnessed a resurgence of curiosity in finite fields, and this can be partially because of vital functions in coding concept and cryptography. the aim of this ebook is to introduce the reader to a couple of those contemporary advancements. it may be of curiosity to quite a lot of scholars, researchers and practitioners within the disciplines of computing device technology, engineering and arithmetic. we will concentration our cognizance on a few particular contemporary advancements within the concept and purposes of finite fields. whereas the subjects chosen are taken care of in a few intensity, we've not tried to be encyclopedic. one of the subject matters studied are diversified tools of representing the weather of a finite box (including general bases and optimum common bases), algorithms for factoring polynomials over finite fields, equipment for developing irreducible polynomials, the discrete logarithm challenge and its implications to cryptography, using elliptic curves in developing public key cryptosystems, and the makes use of of algebraic geometry in developing reliable error-correcting codes. to restrict the scale of the quantity we've got been pressured to fail to remember a few very important functions of finite fields. a few of these lacking functions are in short pointed out within the Appendix besides a few key references.
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Extra resources for Applications of Finite Fields
FACTORING POLYNOMIALS The running time of the Berlekamp Q-matrix method is O( n 3 q) Fq operations. The method is thus only efficient if q is small. There are several methods to get around this problem. The first method we will describe is due to P. , they are non-zero idempotents which cannot be decomposed into a sum of two non-zero orthogonal idempotents) . Observe that ei(:V) has vector representation with a 1 in position i and O's elsewhere. It is easy to see that the only primitive idempotents in n are ei(:V)' 1 ~ i ~ t.
Given a finite field F q and a polynomial f( x) = N L AiXi , Ai E Fq i=O determine the roots of f( x) in Fq • In light of the distinct degree factorization described in the preceding section we can assume that f( x) has N distinct roots in F q • If q is small then the problem is easily solved by doing an exhaustive search for roots. For large values of q more sophisticated methods are required. R. Berlekamp  and is a powerful randomized algorithm which can be used when q = pm for any odd prime and any integer m ~ 1.
36 (1981), 587-592.  B. CHOR AND R . RIVEST, "A knapsack-type public key cryptosystem based on arithmetic in finite fields", IEEE Trans. Info . , 34 (1988), 901-909.  J. VON ZUR GATHEN, "Irreducibility of multivariate polynomials", J. Comput . , 31 (1985), 225-264. 38 REFERENCES  J. VON ZUR GATHEN, "Factoring polynomials and primitive elements for special primes", Theoretical Computer Science , 52 (1987), 77-89.  J. VON ZUR GATHEN AND E. KALTOFEN, "Factoring sparse multivariate polynomials", J.
Applications of Finite Fields by Ian F. Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone, Tomik Yaghoobian (auth.), Alfred J. Menezes (eds.)