Computing Hasse-Witt matrices of hyperelliptic curves in average   polynomial time
  We present an efficient algorithm to compute the Hasse-Witt matrix of a hyperelliptic curve C/Q modulo all primes of good reduction up to a given bound N, based on the average polynomial-time algorithm recently introduced by Harvey. An implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for N = 2^26. 
David Harvey and Andrew V. Sutherland
math.NT