Is DL difficult under the group of Unimodular matrices? Is discrete logarithm assumed to be computationally hard in a non-abelian group as the  subgroup of the general linear group under matrix multiplication formed by the unimodular matrices? The two assumed unimodular matrices have the form $$A=\begin{bmatrix}
    X       &amp; 1  \\
    1       &amp; 0 
\end{bmatrix},B=\begin{bmatrix}
    X       &amp; X+1  \\
    1       &amp; 1  
\end{bmatrix}$$, for a irreducible polynomial P_n of degree $n$. 
Elements belong to a quotient field $\mathbb{F}_{2^n}=\mathbb{F}_2[X]/P_n$
You will need to solve discrete log(s) over the field itself at some point to solve the matrix equation. Now, discrete logs over $F_{2^n}$, as opposed to $F_p$ ($p$ prime) are much easier as shown in the work by Joux, but no full proof yet. 

However see, e.g.,here where the abstract states 
"A new index calculus algorithm with complexity L(1/4+o(1)) in very small characteristic" is presented.

The L-notation is clarified in wikipedia, but the notation above essentially means that the complexity is essentially $\exp{[ c_n (\log n)^{1/4} (\log\log n)^{3/4}}]$ so closer to polynomial than exponential and the coefficient $c_n$ goes to zero as $n$ goes to infinity.