A Complete algorithm for local inversion of maps: Application to Cryptanalysis
This provides a theoretical advance in cryptanalysis by offering a deterministic method for local inversion, though it is incremental as it builds on existing probabilistic approaches like TMTO attacks.
The paper tackles the problem of local inversion of maps over finite fields, which is relevant in cryptanalysis of one-way functions, by proposing a complete algorithm that finds all inverse images for a given output. The algorithm achieves polynomial-time online computation for solutions in periodic orbits or chains from Garden of Eden points, with complexity dependent on linear complexity or chain length.
For a map (function) $F(x):\ftwo^n\rightarrow\ftwo^n$ and a given $y$ in the image of $F$ the problem of \emph{local inversion} of $F$ is to find all inverse images $x$ in $\ftwo^n$ such that $y=F(x)$. In Cryptology, such a problem arises in Cryptanalysis of One way Functions (OWFs). The well known TMTO attack in Cryptanalysis is a probabilistic algorithm for computing one solution of local inversion using $O(\sqrt N)$ order computation in offline as well as online for $N=2^n$. This paper proposes a complete algorithm for solving the local inversion problem which uses linear complexity for a unique solution in a periodic orbit. The algorithm is shown to require an offline computation to solve a hard problem (possibly requiring exponential computation) and an online computation dependent on $y$ that of repeated forward evaluation $F(x)$ on points $x$ in $\ff_{2^n}$ which is polynomial time at each evaluation. However the forward evaluation is repeated at most as many number of times as the Linear Complexity of the sequence $\{y,F(y),\ldots\}$ to get one possible solution when this sequence is periodic. All other solutions are obtained in chains $\{e,F(e),\ldots\}$ for all points $e$ in the Garden of Eden (GOE) of the map $F$. Hence a solution $x$ exists iff either the former sequence is periodic or a solution occurs in a chain starting from a point in GOE. The online computation then turns out to be polynomial time $O(L^k)$ in the linear complexity $L$ of the sequence to compute one possible solution in a periodic orbit or $O(l)$ the chain length for a fixed $n$. Hence this is a complete algorithm for solving the problem of finding all rational solutions $x$ of the equation $F(x)=y$ for a given $y$ and a map $F$ in $\ff_{2^n}$.