Cover and Decomposition Index Calculus on Elliptic Curves made practical. Application to a seemingly secure curve over Fp6

We present a new “cover and decomposition” attack on the elliptic curve discrete logarithm problem, that combines Weil descent and decomposition-based index calculus into a single discrete logarithm algorithm. This attack applies, at least theoretically, to all composite degree extension fields, and is particularly well-suited for curves defined over Fp6 . We give a real-size example of discrete logarithm computations on a curve over a 151-bit degree 6 extension field, which would not have been practically attackable using previously known algorithms.