Generating Hard Problems of Cellular Automata

We propose two hard problems in cellular automata. In particular the problems are: [DDP$^M_{n,p}$] Given two \emph{randomly} chosen configurations $t$ and $s$ of a cellular automata of length $n$, find the number of transitions $\tau$ between $s$ and $t$. [SDDP$^\delta_{k,n}$] Given two \emph{randomly} chosen configurations $s$ of a cellular automata of length $n$ and $x$ of length $k<n$, find the configuration $t$ such that $k$ number of cells of $t$ is fixed to $x$ and $t$ is reachable from $s$ within $\delta$ transitions. We show that the discrete logarithm problem over the finite field reduces to DDP$^M_{n,p}$ and the short integer solution problem over lattices reduces to SDDP$^\delta_{k,n}$. The advantage of using such problems as the hardness assumptions in cryptographic protocols is that proving the security of the protocols requires only the reduction from these problems to the designed protocols. We design one such protocol namely a proof-of-work out of SDDP$^\delta_{k,n}$.