A new method is proposed for the construction of pairing-friendly elliptic curves. For any fixed embedding degree, it can transform the problem to solving equation systems instead of exhaustive searching, thus it's more targeted and efficient. Via this method, we obtain various families including complete families, complete families with variable discriminant and sparse families. Specifically, we generate a complete family with important application prospects which has never been given before as far as we know.

Braids have been studied by mathematicians for more than one century. Because they are so practical as to be used for cryptography, many cryptographers have been interested in them. For the last five years, there have been proposed some cryptographic applications and cryptanalyses in the area of braids. We survey the main examples of these results.