Indifferentiable Hashing to Barreto-Naehrig Curves

A number of recent works have considered the problem of constructing constant-time hash functions to various families of elliptic curves over finite fields. In the relevant literature, it has been occasionally asserted that constant-time hashing to certain special elliptic curves, in particular so-called BN elliptic curves, was an open problem. It turns out, however, that a suitably general encoding function was constructed by Shallue and van de Woestijne back in 2006. In this paper, we show that, by specializing the construction of Shallue and van de Woestijne to BN curves, one obtains an encoding function that can be implemented rather efficiently and securely, that reaches about 9/16ths of all points on the curve, and that is well-distributed in the sense of Farashahi et al., so that one can easily build from it a hash function that is indifferentiable from a random oracle.