# A Complete algorithm for local inversion of maps: Application to Cryptanalysis

For a map (function) F(x):^n→^n and a given y in the image of F the problem of local inversion of F is to find all inverse images x in ^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(√(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 _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),…} to get one possible solution when this sequence is periodic. All other solutions are obtained in chains {e,F(e),…} 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 _2^n.

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