Supersingular curves and explicit enumeration of irreducible polynomials over binary fields with prescribed coefficients

Title:  Supersingular curves and explicit enumeration of irreducible polynomials over binary fields with prescribed coefficients

Speaker:  Sercan Yilmaz (UCD)

Date:  Monday, 22nd February 2016

Time:  4pm

Location:  Agriculture. 1.01 (Seminar Room).


Abstract:

Using supersingular curves, we economically rederive known formulae for the number of elements of F_{2^n} of trace zero and for which the next two coefficients of the characteristic polynomial with respect to F_2 are specified. This approach makes explicit why the set of formulae for each set of traces has period 24. We extend this approach to base fields F_{2^r} with r > 1 and count the number of elements of F_{2^{rn}} for which the first three traces with respect to F_{2^r} are zero, which by a standard Mobius inversion-type argument leads to the number of irreducible polynomials of degree n over F_{2^r} for which the coefficients of x^{n-1}, x^{n-2} and x^{n-3} are zero.

 

Series: Algebra