Download Algebraic Curves and Finite Fields: Cryptography and Other by Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne PDF

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

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the main well known researchers within the region of finite fields and their purposes around the globe, addressing outdated and new difficulties on curves and different elements of finite fields, with emphasis on their varied purposes to many components of natural and utilized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

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

Sample text

Finding a uniformizing element ????0 of ????(????0 (????2 +????+1)) is somewhat more elaborate. Since such a uniformizing element fills several pages, it is omitted. Below we will state the reduction of ????0 modulo ???? and ???? + 1, so the reader can get an impression of its form. Once ????0 is found, ????0 and ????1 can be expressed in terms of it. In this case we find (????0 + 1)3 (????20 + ????0 + ????2 + ???? + 1) ????0 = ????0 and (????0 + ????2 + ???? + 1)3 (????20 + ????0 + ????2 + ???? + 1) . ????40 To find the polynomial ????????2 +????+1 (????, ????), we need to factor the polynomial ????1 = (????5 + (????2 + ???? + 1)????3 + (????2 + ???? + 1)????2 + (????2 + ????)???? + (????2 + ???? + 1))????4 + ????(????5 + (????2 + ????)????4 + (????2 + ???? + 1)2 ????3 + (????2 + ???? + 1)3 ????2 + (????2 + ???? + 1)4 ) , whose factors are ???????? + ????2 + ???? + 1 and ????????2 +????+1 (????, ????) = ????4 ????3 + (????2 + ???? + 1)(????3 ????2 + ????2 ????3 + (????2 + ???? + 1)????2 ???? + ????????3 + (????2 + ???? + 1)????????2 + (????2 + ???? + 1)2 ????) + ????4 .

To describe how to obtain (unreduced) towers, we will use the language of Drinfeld modules. Let ???? be a Drinfeld module of rank 2 with ????-invariant ????0 and ???????? be an ????-isogenous Drinfeld module with ????-invariant ????1 . The Drinfeld modular polynomial ???????? (????, ????) relates these ????-invariants, more precisely it holds that ???????? (????0 , ????1 ) = 0. Thinking of ????0 as a transcendental element, we can use this equation to define a so-called Drinfeld modular curve ????0 (????). If we want to emphasize the role of ????, we will write ????1 = ????1 (????).

The Drinfeld modular polynomial of level ????2 + ???? + 1 is given by ????????2 +????+1 (????, ????) = ????5 + ????5 + ????4 ????4 + (????2 + ???? + 1)(????4 ????2 + ????2 ????4 ) + (????2 + ???? + 1)(????4 ???? + ????????4 ) + ????3 (???? + 1)3 (????2 + ???? + 1)(????4 + ????4 ) + ????2 (???? + 1)2 (????2 + ???? + 1)????3 ????3 + (????2 + ????)(????2 + ???? + 1)(????3 + ???? + 1)(????3 + ????2 + 1)(????3 ????2 + ????2 ????3 ) + ????3 (???? + 1)3 (????2 + ???? + 1)(????3 ???? + ????????3 ) + ????6 (???? + 1)6 (????2 + ???? + 1)2 (????3 + ????3 ) + ????5 (???? + 1)5 (????2 + ???? + 1)(????4 + ???? + 1)????2 ????2 + ????6 (???? + 1)6 (????2 + ???? + 1)(????4 + ???? + 1)(????2 ???? + ????????2 ) + ????9 (???? + 1)9 (????2 + ???? + 1)3 (????2 + ????2 ) + ????11 (???? + 1)11 ???????? .

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