Complexity of finite field arithmetic

For integers, the parameter n is the bit-length, and for the finite field Fq we let n = log q. In the case of polynomial root-finding, d is the degree of the polynomial and we list bounds on the expected running time since these operations are most efficiently implemented using probabilistic algorithms. In Lecture 3 we addressed the cost of addition and subtraction in both Z and Fq, and the cost of multiplication in Z. In this lecture we will fill in the rest of the table. We saw in Lecture 3 how to use Kronecker substitution to reduce polynomial multiplication to integer multiplications. Thus we can multiply elements of an arbitrary finite field Fq using integer multiplication, provided that we have a way to reduce products back to our standard representations of Fp ' Z/pZ and Fq ' Fp[x]/(f), using integers in [0, p− 1] and polynomials of degree less than deg f . For this we use Euclidean division.