FYI -- Almost everything you ever wanted to know about bit-hacking on Intel processors for implementing binary finite fields. Yes, squaring and square roots are easy -- even in a polynomial (rather than a normal basis) representation. Aranha talks about multiple-precision operations on 64-bit and 128-bit registers, so we're talking about big fields here with hundreds and thousands of bits. Notice that the number of views of this video today is only *41* (including mine)! Clearly, it is far easier to get around crypto than to break crypto; only 41 people cared enough to find out what's going on! https://www.youtube.com/watch?v=Zr7htuOurI4 Efficient Software Implementation of Binary Field Arithmetic Using Vector Instruction Sets Microsoft Research 41 views Published on Jul 27, 2016