Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields

Since 2013 there have been several developments in algorithms for computing discrete logarithms in small-characteristic finite fields, culminating in a quasipolynomial algorithm. In this paper, we report on our successful computation of discrete logarithms in the cryptographically-interesting characteristic-three finite field F36·509 using these new algorithms; prior to 2013, it was believed that this field enjoyed a security level of 128 bits. We also show that a recent idea of Guillevic can be used to compute discrete logarithms in the cryptographically-interesting finite field F36·709 using essentially the same resources as we expended on the F36·509 computation. Finally, we argue that discrete logarithms in the finite field F36·1429 can feasibly be computed today; this is significant because this cryptographically-interesting field was previously believed to enjoy a security level of 192 bits.