arXiv subjects

Virendra Sule

Publications and source records attributed to Virendra Sule.

At least 19 recordsLinked to original sources

arXiv2026

Black Box Cryptanalysis of AES128

This paper presents computational results of cryptanalysis of AES using the Local Inversion by Black Box computations of the forward encryption and utilizes these results to develop a practically feasible approach for the key recovery of the full scale AES128 under Known Plaintext Attack (KPA). It is shown that complete recovery of unknown key bits is possible upto $80$ bits in a practically feasible time and memory in random KPA situation by sequential computation when remaining $48$ bits are known. The results of key recovery in $64$, $72$ and $80$ bit unknown cases are extrapolated to predict the period of the iterative sequence generated in the local inversion approach for the full $128$ bit unknown key case and a strategy is proposed to search the actual period by brute force parallel search of the sequence period with $10$ free bits defining the search space. Then it is shown that the actual key can be verified in polynomial time by fast powering of the forward encryption map. Hence this strategy shows that the key recovery problem for AES128 under KPA has a chance of success in practically feasible time for a majority of the plaintexts. Local inversion approach to cryptanalysis using black box computations is a universal method applicable to a vast variety of key recovery and map inversion problems. Hence the results presented in this paper are representative of estimates of cryptanalysis of other ciphers which can be considered almost as strong as AES128 as encryption functions.

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arXiv2025

Solving Modular Linear Systems with a Constraint by parallel decomposition of the Smith form and extended Euclidean division modulo powers of primes divisors

Integral linear systems $Ax=b$ with matrices $A$, $b$ and solutions $x$ are also required to be in integers, can be solved using invariant factors of $A$ (by computing the Smith Canonical Form of $A$). This paper explores a new problem which arises in applications, that of obtaining conditions for solving the Modular Linear System $Ax=b\rem n$ given $A,b$ in $\zz_n$ for $x$ in $\zz_n$ along with the constraint that the value of the linear function $\phi(x)=\la w,x\ra$ is coprime to $n$ for some solution $x$. In this paper we develop decomposition of the system to coprime moduli $p^{r(p)}$ which are divisors of $n$ and show how such a decomposition simplifies the computation of Smith form. This extends the well known index calculus method of computing the discrete logarithm where the moduli over which the linear system is reduced were assumed to be prime (to solve the reduced systems over prime fields) to the case when the factors of the modulus are prime powers $p^{r(p)}$. It is shown how this problem can be addressed effciently using the invariant factors and Smith form of the augmented matrix $[A,-p^{r(p)}I]$ and conditions modulo $p$ satisfied by $w$, where $p^{r(p)}$ vary over all divisors of $n$ with $p$ prime.

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arXiv2024

Polynomial Complexity of Inversion of sequences and Local Inversion of Maps

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.

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arXiv2024

Observability of Nonlinear Dynamical Systems over Finite Fields

This paper discusses the observability of nonlinear Dynamical Systems over Finite Fields (DSFF) through the Koopman operator framework. In this work, given a nonlinear DSFF, we construct a linear system of the smallest dimension, called the Linear Output Realization (LOR), which can generate all the output sequences of the original nonlinear system through proper choices of initial conditions (of the associated LOR). We provide necessary and sufficient conditions for the observability of a nonlinear system and establish that the maximum number of outputs sufficient for computing the initial condition is precisely equal to the dimension of the LOR. Further, when the system is not known to be observable, we provide necessary and sufficient conditions for the unique reconstruction of initial conditions for specific output sequences.

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arXiv2023

Word Linear Complexity of sequences and Local Inversion of maps over finite fields

This paper develops the notion of \emph{Word Linear Complexity} ($WLC$) of vector valued sequences over finite fields $\ff$ as an extension of Linear Complexity ($LC$) of sequences and their ensembles. This notion of complexity extends the concept of the minimal polynomial of an ensemble (vector valued) sequence to that of a matrix minimal polynomial and shows that the matrix minimal polynomial can be used with iteratively generated vector valued sequences by maps $F:\ff^n\rightarrow\ff^n$ at a given $y$ in $\ff^n$ for solving the unique local inverse $x$ of the equation $y=F(x)$ when the sequence is periodic. The idea of solving a local inverse of a map in finite fields when the iterative sequence is periodic and its application to various problems of Cryptanalysis is developed in previous papers \cite{sule322, sule521, sule722,suleCAM22} using the well known notion of $LC$ of sequences. $LC$ is the degree of the associated minimal polynomial of the sequence. The generalization of $LC$ to $WLC$ considers vector valued (or word oriented) sequences such that the word oriented recurrence relation is obtained by matrix vector multiplication instead of scalar multiplication as considered in the definition of $LC$. Hence the associated minimal polynomial is matrix valued whose degree is called $WLC$. A condition is derived when a nontrivial matrix polynomial associated with the word oriented recurrence relation exists when the sequence is periodic. It is shown that when the matrix minimal polynomial exists $n(WLC)=LC$. Finally it is shown that the local inversion problem is solved using the matrix minimal polynomial when such a polynomail exists hence leads to a word oriented approach to local inversion.

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arXiv2023

Deciding One to One property of Boolean maps: Condition and algorithm in terms of implicants

This paper addresses the computational problem of deciding invertibility (or one to one-ness) of a Boolean map $F$ in $n$-Boolean variables. This problem is a special case of deciding invertibilty of a map $F:\mathbb{F}_{q}^n\rightarrow\mathbb{F}_{q}^n$ over the finite field $\mathbb{F}_q$ for $q=2$. Algebraic condition for invertibility of $F$ is well known to be equivalent to invertibility of the Koopman operator of $F$ as shown in \cite{RamSule}. In this paper a condition for invertibility is derived in the special case of Boolean maps $F:B_0^n\rightarrow B_0^n$ where $B_0$ is the two element Boolean algebra in terms of \emph{implicants} of Boolean equations defined by the map. This condition is then extended to the case of general maps in $n$ variables and $m\geq n$ equations. Hence this condition answers the special case of invertibility of maps $F$ defined over the binary field $\mathbb{F}_2$ alternatively, in terms of implicants instead of the Koopman operator. The problem of deciding invertibility of a map $F$ (or that of finding its Garden of Eden (GOE)) over finite fields is distinct from the satisfiability problem (SAT) or the problem of deciding consistency of polynomial equations over finite fields. Hence the well known algorithms for deciding SAT or of solvability using Grobner basis for checking membership in an ideal generated by polynomials is not known to answer the question of invertibility of a map. Similarly it appears that algorithms for satisfiability or polynomial solvability are not useful for computation of GOE of $F$ even for maps over the binary field $\mathbb{F}_2$.

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arXiv2022

Local Inversion of maps: Black box Cryptanalysis

This paper is a short summery of results announced in a previous paper on a new universal method for Cryptanalysis which uses a Black Box linear algebra approach to computation of local inversion of nonlinear maps in finite fields. It is shown that one local inverse $x$ of the map equation $y=F(x)$ can be computed by using the minimal polynomial of the sequence $y(k)$ defined by iterates (or recursion) $y(k+1)=F(y(k))$ with $y(0)=y$ when the sequence is periodic. This is the only solution in the periodic orbit of the map $F$. Further, when the degree of the minimal polynomial is of polynomial order in number of bits of the input of $F$ (called low complexity case), the solution can be computed in polynomial time. The method of computation only uses the forward computations $F(y)$ for given $y$ which is why this is called a Black Box approach. Application of this approach is then shown for cryptanalysis of several maps arising in cryptographic primitives. It is shown how in the low complexity cases maps defined by block and stream ciphers can be inverted to find the symmetric key under known plaintext attack. Then it is shown how RSA map can be inverted to find the plaintext as well as an equivalent private key to break the RSA algorithm without factoring the modulus. Finally it is shown that the discrete log computation in finite field and elliptic curves can be formulated as a local inversion problem and the low complexity cases can be solved in polynomial time.

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arXiv2022

Local inversion of maps: A new attack on Symmetric encryption, RSA and ECDLP

This paper presents algorithms for local inversion of maps and shows how several important computational problems such as cryptanalysis of symmetric encryption algorithms, RSA algorithm and solving the elliptic curve discrete log problem (ECDLP) can be addressed as local inversion problems. The methodology is termed as the \emph{Local Inversion Attack}. It utilizes the concept of \emph{Linear Complexity} (LC) of a recurrence sequence generated by the map defined by the cryptanalysis problem and the given data. It is shown that when the LC of the recurrence is bounded by a bound of polynomial order in the bit length of the input to the map, the local inversion can be accomplished in polynomial time. Hence an incomplete local inversion algorithm which searches a solution within a specified bound on computation can estimate the density of weak cases of cryptanalysis defined by such data causing low LC. Such cases can happen accidentally but cannot be avoided in practice and are fatal insecurity flaws of cryptographic primitives which are wrongly assumed to be secure on the basis of exponential average case complexity. An incomplete algorithm is proposed for solving problems such as key recovery of symmetric encryption algorithms, decryption of RSA ciphertext without factoring the modulus, decrypting any ciphertext of RSA given one plaintext ciphertext pair created with same public key in chosen ciphertext attack and solving the discrete logarithm on elliptic curves over finite fields (ECDLP) as local inversion problems. It is shown that when the LCs of the respective recurrences for given data are small, solutions of these problems are possible in practically feasible time and memory resources.

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arXiv2022

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

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 $\ff_{2^n}$.

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arXiv2020

Robust stabilization of multiport networks

This paper formulates and solves the problem of robust compensation of multiport active network. This is an important engineering problem as networks designed differ in parameter values due to tolerance during manufacture from their actual realizations in chips and hardware. Parameters also undergo changes due to environmental factors. Hence, practical use of networks requires compensation which is only possible by connecting compensating network at the ports. The resulting interconnection is then required to be stable over a range of parameter values. This is called robust compensation. This paper formulates such a problem using an extension of the coprime factorization theory well known in feedback control theory to the situation of multiport network interconnection developed in \cite{msm1} and formulates the robust stabilization problem as an $H_{\infty}$ optimization problem. The port interconnection of networks does not confirm with computation of the function of the interconnected network analogous to that of the feedback interconnection using signal flow graph. Hence the well known stabilization and stability theory of feedback systems cannot be utilized for such a problem. A new formulation of stabilization theory of network interconnection was formulated and developed by the authors in \cite{msm1}. The variations of parameters of the network are used to define a worst case neighborhood of the network in terms of its coprime fractions at the nominal values of parameters. The solution of the optimization problem is then carried out by the standard procedure of converting such a problem to the Nehari optimization problem \cite{fran}. This methodology of solving the robust compensation of multiport networks using feedback control theory is believed to be novel.

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arXiv2020

On Linear Representation, Complexity and Inversion of maps over finite fields

This paper defines a linear representation for nonlinear maps $F:\mathbb{F}^n\rightarrow\mathbb{F}^n$ where $\mathbb{F}$ is a finite field, in terms of matrices over $\mathbb{F}$. This linear representation of the map $F$ associates a unique number $N$ and a unique matrix $M$ in $\mathbb{F}^{N\times N}$, called the Linear Complexity and the Linear Representation of $F$ respectively, and shows that the compositional powers $F^{(k)}$ are represented by matrix powers $M^k$. It is shown that for a permutation map $F$ with representation $M$, the inverse map has the linear representation $M^{-1}$. This framework of representation is extended to a parameterized family of maps $F_{\lambda}(x): \mathbb{F} \to \mathbb{F}$, defined in terms of a parameter $\lambda \in \mathbb{F}$, leading to the definition of an analogous linear complexity of the map $F_{\lambda}(x)$, and a parameter-dependent matrix representation $M_\lambda$ defined over the univariate polynomial ring $\mathbb{F}[\lambda]$. Such a representation leads to the construction of a parametric inverse of such maps where the condition for invertibility is expressed through the unimodularity of this matrix representation $M_\lambda$. Apart from computing the compositional inverses of permutation polynomials, this linear representation is also used to compute the cycle structures of the permutation map. Lastly, this representation is extended to a representation of the cyclic group generated by a permutation map $F$, and to the group generated by a finite number of permutation maps over $\mathbb{F}$.

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arXiv2020

Koopman operator approach for computing structure of solutions and Observability of non-linear finite state system

Given a discrete dynamical system defined by a map in a vector space over a finite field called Finite State Systems (FSS), a dual linear system over the space of functions on the state space is constructed using the dual map. This system constitutes the well known Koopman linear system framework of dynamical systems, hence called the Koopman linear system (KLS). It is first shown that several structural properties of solutions of the FSS can be inferred from the solutions of the KLS. The problems of computation of structural parameters of solutions of non-linear FSS are computationally hard and hence become infeasible as the number of variables increases. In contrast, it has been well known that these problems can be solved by linear algebra for linear FSS in terms of elementary divisors of matrices and their orders. In the next step, the KLS is reduced to the smallest order (called RO-KLS) while still retaining all the information of the parameters of structure of solutions of the FSS. Hence when the order of the RO-KLS is sufficiently small, the above computational problems of non-linear FSS are practically feasible. Next, it is shown that the observability of the non-linear FSS with an output function is equivalent to that of the RO-KLS with an appropriate linear output map. Hence, the problem of non-linear observability is solved by an observer design for the equivalent RO-KLS. Such a construction should have striking applications to realistic FSS arising in Cryptology and Biological networks.

eess.SY↗

arXiv2019

Analysis of Periodic Feedback Shift Registers

This paper develops methods for analyzing periodic orbits of states of linear feedback shift registers with periodic coefficients and estimating their lengths. These shift registers are among the simplest nonlinear feedback shift registers (FSRs) whose orbit lengths can be determined by feasible computation. In general such a problem for nonlinear FSRs involves infeasible computation. The dynamical systems whose model includes such FSRs are termed as Periodic Finite State systems (PFSS). This paper advances theory of such dynamical systems. Due to the finite field valued coefficients, the theory of such systems turns out to be radically different from that of linear continuous or discrete time periodic systems with real coefficients well known in literature. A special finite field version of the Floquet theory of such periodic systems is developed and the structure of trajectories of the PFSS is analyzed through that of a shift invariant linear system after Floquet transformation. The concept of extension of a dynamical system is proposed for such systems whenever the equivalent shift invariant system can be obtained over an extension field.

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arXiv2017

Projective cofactor decompositions of Boolean functions and the satisfiability problem

Given a CNF formula $F$, we present a new algorithm for deciding the satisfiability (SAT) of $F$ and computing all solutions of assignments. The algorithm is based on the concept of \emph{cofactors} known in the literature. This paper is a fallout of the previous work by authors on Boolean satisfiability \cite{sul1, sul2,sude}, however the algorithm is essentially independent of the orthogonal expansion concept over which previous papers were based. The algorithm selects a single concrete cofactor recursively by projecting the search space to the set which satisfies a CNF in the formula. This cofactor is called \emph{projective cofactor}. The advantage of such a computation is that it recursively decomposes the satisfiability problem into independent sub-problems at every selection of a projective cofactor. This leads to a parallel algorithm for deciding satisfiability and computing all solutions of a satisfiable formula.

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arXiv2017

Implicant based parallel all solution solver for Boolean satisfiability

This paper develops a parallel computational solver for computing all satifying assignments of a Boolean system of equations defined by Boolean functions of several variables. While there are we known solvers for satisfiability of Boolean formulas in CNF form, these are designed primarily for deciding satisfiability of the formula and do not address the problem of finding all satisfying solutions. Moreover development of parallel solvers for satisfiability problems is still an unfinished problem of Computer Science. The solver proposed in this paper is aimed at representing all solutions of Boolean formulas even without the CNF form with a parallel algorithm. Algorithm proposed is applied to Boolean functions in algebraic normal form (ANF). The algorithm is based on the idea to represent the satisfying assignments in terms of a complete set of implicants of the Boolean functions appearing as factors of a Boolean formula. The algorithm is effective mainly in the case when the factors of the formula are sparse (i.e. have a small fraction of the total number of variables). This allows small computation of a complete set of implicants of individual factors one at a time and reduce the formula at each step. An algorithm is also proposed for finding a complete set of orthogonal implicants of functions in ANF. An advantages of this algorithm is that all solutions can be represented compactly in terms of implicants. Finally due to small and distributed computation at every step as well as computation in terms of independent threads, the solver proposed in this paper is expected to be useful for developing heuristics for a well scalable parallel solver for large size problems of Boolean satisfiability over large number of processors.

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arXiv2016

Stabilization Theory for Active Multi Port Networks

This paper proposes a theory for designing stable interconnection of linear active multi-port networks at the ports. Such interconnections can lead to unstable networks even if the original networks are stable with respect to bounded port excitations. Hence such a theory is necessary for realising interconnections of active multiport networks. Stabilization theory of linear feedback systems using stable coprime factorizations of transfer functions has been well known. This theory witnessed glorious developments in recent past culminating into the $H_{\infty}$ approach to design of feedback systems. However these important developments have seldom been utilized for network interconnections due to the difficulty of realizing feedback signal flow graph for multi-port networks with inputs and outputs as port sources and responses. This paper resolves this problem by developing the stabilization theory directly in terms of port connection description without formulation in terms of signal flow graph of the implicit feedback connection. The stable port interconnection results into an affine parametrized network function in which the free parameter is itself a stable network function and describes all stabilizing port compensations of a given network.

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arXiv2014

Generalized cofactors and decomposition of Boolean satisfiability problems

We propose an approach for decomposing Boolean satisfiability problems while extending recent results of \cite{sul2} on solving Boolean systems of equations. Developments in \cite{sul2} were aimed at the expansion of functions $f$ in orthonormal (ON) sets of base functions as a generalization of the Boole-Shannon expansion and the derivation of the consistency condition for the equation $f=0$ in terms of the expansion co-efficients. In this paper, we further extend the Boole-Shannon expansion over an arbitrary set of base functions and derive the consistency condition for $f=1$. The generalization of the Boole-Shannon formula presented in this paper is in terms of \emph{cofactors} as co-efficients with respect to a set of CNFs called a \emph{base} which appear in a given Boolean CNF formula itself. This approach results in a novel parallel algorithm for decomposition of a CNF formula and computation of all satisfying assignments when they exist by using the given data set of CNFs itself as the base.

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arXiv2014

An algorithm for Boolean satisfiability based on generalized orthonormal expansion

This paper proposes an algorithm for deciding consistency of systems of Boolean equations in several variables with co-efficients in the two element Boolean algebra $B_{0}=\{0,1\}$ and find all satisfying assignments. The algorithm is based on the application of a well known generalized Boole-Shannon orthonormal (ON) expansion of Boolean functions. A necessary and sufficient consistency condition for a special class of functions was developed in \cite{sule} using such an expansion. Paper \cite{sule} develops a condition for consistency of the equation $f(X)=0$ for the special classes of Boolean functions 1) $f$ in $B(Φ(X))$ for an ON set $Φ$ of Boolean functions in $X$ over a general Boolean algebra $B$ and 2) $f$ in $B(X_{2})(Φ(X_{1}))$. The present paper addresses the problem of obtaining the consistency conditions for arbitrary Boolean functions in $B_{0}(X)$. Next, the consistency for a single equation is shown equivalent to another system of Boolean equations which involves the ON functions and characterizes all solutions. This result is then extended for Boolean systems in several variables over the algebra $B_{0}=\{0,1\}$ which does not convert the system into a single equation. This condition leads to the algorithm for computing all solutions of the Boolean system without using analogous resolution and determine satisfiability. For special systems defined by CNF formulas this algorithm results into an extension of the DPLL algorithm in which the \emph{splitting rule} is generalized to several variables in terms of ON terms in the sense that splitting of CNF set in a single variable $x$ is equivalent to ON terms $x,x'$.

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