By Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof

This booklet collects the result of the workshops on functions of Algebraic Curves and functions of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main fashionable researchers within the region of finite fields and their purposes around the globe, addressing outdated and new difficulties on curves and different features of finite fields, with emphasis on their different purposes to many parts of natural and utilized arithmetic.

**Read or Download Algebraic Curves and Finite Fields: Cryptography and Other Applications PDF**

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**Extra info for Algebraic Curves and Finite Fields: Cryptography and Other Applications**

**Example text**

5 ] generated by the above two sets of polynomials, but we can simplify the system of polynomial equations first. Taking for example the last of each set of polynomials, 256 64 ????1 := ℎ64 + ????116 + ????1 and ????2 := ℎ16 1 + ℎ1 + ????1 1 + ℎ1 + ????1 + ????1 , we find that ????3 := ????1 − ????24 = ℎ41 + ℎ1 + ????116 + ????14 + ????1 is an element of the ideal ????. Moreover, since ????2 = ????3 + ????34 and ????1 = ????3 + ????34 + ????316 , we can replace ????1 and ????2 by ????3 when generating the ideal ????. Also we can eliminate the variables ℎ???? altogether, since they can be expressed in terms of ????1 , ????2 , ????3 using the first five generators of ????.

Satisfying: ???????? (????0 , ????1 ) = 0 , with ???????? (????, ????) the Drinfeld modular polynomial corresponding to ???? and ???????? (????????−1 , ???????? , ????????+1 ) = 0, for 1 ≤ ???? < ???? , Good towers of function fields | 27 with ???????? (????, ????, ????) a suitable trivariate polynomial of ????-degree ????deg ???? . Consequently, the tower of function fields G := (???????? )????≥0 can be recursively defined by a recursion of depth 2 in the following way: ????0 := ????(????0 ) , ????1 := ????(????0 , ????1 ) , where ???????? (????0 , ????1 ) = 0 and for ???? ≥ 1 ????????+1 := ???????? (????????+1 ) , where ???????? (????????−1 , ???????? , ????????+1 ) = 0 .

235–265, 1997. V. G. Drinfeld and S. G. Vladut, The number of points of an algebraic curve (translated from the Russian paper in Funktsional. Anal. i Prilozhen), Functional Anal. Appl. 17, pp.