Polynomial Complexity of Inversion of sequences and Local Inversion of Maps
Virendra Sule
TL;DR
This work extends sequence inversion from linear recurrences to nonlinear polynomial recurrences over the binary field by introducing the Polynomial Complexity of Inversion $\mathrm{PCI}(d)$, the smallest recurrence order $m$ for which a degree-$d$ polynomial $f\in P(m,d)$ yields a unique inverse. Central to the framework is the Hankel-type matrix $H(m,d)(S)$ (and its vector-extended form for multi-coordinate sequences) used to impose linear RR constraints and a bilinear inversion constraint, enabling a rigorous existence and uniqueness theory (Fundamental Lemma). The authors develop a comprehensive theory: (i) conditions for the existence of associated polynomials, (ii) a decomposition of $f$ into $X_0h(\cdot)+g(\cdot)$ to separate affine and higher-degree parts, (iii) the projection and rank criteria governing the number of admissible polynomials and inverses, and (iv) extended results for restricted monomial sets $\mathcal M$, including linear and quadratic cases and Golomb-type non-singularity. They further discuss vector-sequence generalizations, computational bounds, and practical implications for local inversion of maps in cryptography, including conjectures about average-case polynomial-time inversion with higher-degree RRs. The overall contribution provides a nonlinear generalization of LC-based inversion theory, with potential disruptive implications for cryptanalysis when $\mathrm{PCI}(d)$ is small on average. The work also delineates an array of open questions and conjectures related to partial-sequence inversion probabilities and the role of monomial structure in practical inversion performance.
Abstract
This Paper defines and explores solution to the problem of \emph{Inversion of a finite Sequence} over the binary field, that of finding a prefix element of the sequence which confirms with a \emph{Recurrence Relation} (RR) rule defined by a polynomial and satisfied by the sequence. The minimum number of variables (order) in a polynomial of a fixed degree defining RRs is termed as the \emph{Polynomial Complexity} of the sequence at that degree, while the minimum number of variables of such polynomials at a fixed degree which also result in a unique prefix to the sequence and maximum rank of the matrix of evaluation of its monomials, is called \emph{Polynomial Complexity of Inversion} at the chosen degree. Solutions of this problems discovers solutions to the problem of \emph{Local Inversion} of a map $F:\ftwo^n\rightarrow\ftwo^n$ at a point $y$ in $\ftwo^n$, that of solving for $x$ in $\ftwo^n$ from the equation $y=F(x)$. Local inversion of maps has important applications which provide value to this theory. In previous work it was shown that minimal order \emph{Linear Recurrence Relations} (LRR) satisfied by the sequence known as the \emph{Linear Complexity} (LC) of the sequence, gives a unique solution to the inversion when the sequence is a part of a periodic sequence. This paper explores extension of this theory for solving the inversion problem by considering \emph{Non-linear Recurrence Relations} defined by a polynomials of a fixed degree $>1$ and satisfied by the sequence. The minimal order of polynomials satisfied by a sequence is well known as non-linear complexity (defining a Feedback Shift Register of smallest order which determines the sequences by RRs) and called as \emph{Maximal Order Complexity} (MOC) of the sequence. However unlike the LC there is no unique polynomial recurrence relation at any degree.