Virendra Sule

Mayuresh Bakshi, Virendra Sule, Maryam Shoejai Baghini

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.

Virendra Sule

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.

Virendra Sule

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 $\ff_{2^n}$.

Ramachandran Anantharaman, Virendra Sule

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.