Exclusive-or Sum Research Articles

Genetic algorithm for minimization of ESOP representations for multiple-output logic functions

This article concerns ESOP representations for multiple-output logic functions. Exclusive-or sum-of-products expression (ESOP) is a formula representing a logic function as an exclusive-or sum of products of variables and their negations. EXOR-based realization leads to reduction in size of logic circuits for many practical logic functions and improves their testability. Minimization of a logic function is a problem of constructing ESOP representation for the given function having the minimal number of summands. We introduce a correspondence between ESOPs for multiple-output logic functions and single-output logic functions and propose genetic algorithm obtaining close to minimal ESOP representations for multiple-output logic functions.

A Nonlinear Integer Programming Approach for the Minimization of Boolean Expressions

A novel approach is suggested in this paper for the minimization of Boolean expressions. This is particularly useful in logic synthesis, since it leads to simpler logic circuit implementations. Although the proposed method is general, emphasis is given on Exclusive-or Sum Of Products (ESOPs) functions. A transformation is derived to convert the problem from the Boolean algebra area to the classical algebraic area. The resulting problem becomes a nonlinear, integer program and an original branch-and-bound procedure with several relaxations is developed for its solution. The suggested methodology is especially suitable for the minimization of incompletely specified functions, which is a difficult problem in the Boolean area. Numerical examples are provided to demonstrate the applicability and performance of the approach and possible future directions are described. The resulting nonlinear problems can sometimes be very demanding but their challenging solutions could solve open ESOP problems.

Product Transformation and Heuristic EXOR-AND-OR Logic Synthesis of Incompletely Specified Functions

Previously, the synthesis of reversible and quantum permutative circuits based on products of Exclusive-OR (EXOR) sums of inputs (POE) was explored as an approach to reduce quantum cost. Herein, the mathematical representation used in POE expressions is formally detailed and contrasted with the pseudoproduct-based theory of Luccio and Pagli. The theory and practice of product transformation, or computation of equivalent POE expressions, is presented. To demonstrate the generality of the mathematical representations outside of the reversible domain, a new method to synthesize incompletely specified functions as EXOR-AND-OR circuits is introduced. The method is heuristic. It first uses spectral information from the Walsh–Hadamard transform to compute a set of POE implicants for a function, and then uses product transformation to minimize both the total number of literals and unique EXOR sums (ignoring polarity) used in the POE implicants. The product transformation minimization extends to multioutput functions. The new method has no performance guarantees, i.e., it does not have a fallback mode of minimum or near-minimum sum of products; therefore, it is suited to a tournament synthesis approach. When synthesizing benchmark functions of up to 26 inputs, the new method is shown to typically produce lower cost circuits than heuristic pseudoproduct-based methods.

Upper bound for the length of functions over a finite field in the class of pseudopolynomials

An exclusive-OR sum of pseudoproducts (ESPP), or a pseudopolynomial over a finite field is a sum of products of linear functions. The length of an ESPP is defined as the number of its pairwise distinct summands. The length of a function f over this field in the class of ESPPs is the minimum length of an ESPP representing this function. The Shannon length function L k ESPP (n) on the set of functions over a finite field of k elements in the class of ESPPs is considered; it is defined as the maximum length of a function of n variables over this field in the class of ESPPs. It is proved that L k ESPP (n) = O(k n /n 2).

Order of the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

Order of the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

Synthesis of reversible PLA using products sharing

Reversible logic is a computing design, where the ideal implementation would produce zero entropy gain. This unique feature causes prominent use of reversible computing. At the same time, more integration capability and regular structure for synthesizing large number of logic functions made programmable devices enthusiastic to use. In this paper, we propose design algorithm of one of the programmable logic devices, Programmable Logic Array (PLA) with a newly designed low cost 3 $$\times $$× 3 reversible Tara Babu (TB) gate, which can realize multi-output Exclusive-OR Sum of Product (ESOP) functions. In addition, we present a heuristic algorithm to sort and realize the product terms of ESOP functions in order to share the internal sub-products to reduce the number of gates in the proposed circuit. Proposed algorithms make the design more efficient with improvement 9.83 % in number of gates, 21.3 % in garbage outputs count and 14.75 % quantum cost parameters than the existing techniques averagely. Moreover, the area and power consumption of the proposed PLA are shown. Performance is also analyzed by using MCNC benchmark functions.

A Novel Method for Increasing the Entropy of a Sequence of Independent, Discrete Random Variables

In this paper, we propose a novel method for increasing the entropy of a sequence of independent, discrete random variables with arbitrary distributions. The method uses an auxiliary table and a novel theorem that concerns the entropy of a sequence in which the elements are a bitwise exclusive-or sum of independent discrete random variables.

Novel Robust Single Layer Wire Crossing Approach for Exclusive OR Sum of Products Logic Design with Quantum-Dot Cellular Automata

Novel Robust Single Layer Wire Crossing Approach for Exclusive OR Sum of Products Logic Design with Quantum-Dot Cellular Automata

On the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

Representations of Boolean functions by exclusive-OR sums (modulo 2) of pseudoproducts is studied. An ExOR-sum of pseudoproducts (ESPP) is the sum modulo 2 of products of affine (linear) Boolean functions. The length of an ESPP is defined as the number of summands in this form, and the length of a Boolean function in the class of ESPPs is defined as the minimum length of an ESPP representing this function. The Shannon function LESPP(n) of the length of Boolean functions in the class of ESPPs is considered, which equals the maximum length of a Boolean function of n variables in this class. Lower and upper bounds for the Shannon function LESPP(n) are found. The upper bound is proved by using an algorithm which can be applied to construct representations by ExOR-sums of pseudoproducts for particular Boolean functions.

Minimizing ESCT forms for two-variable multiple-valued input binary output functions

As EXOR (Exclusive-OR) expansions of binary output functions, the ESOP (EXOR Sum of Products) form and its extension, the ESCT (EXOR Sum of Complex Terms) form, have been studied extensively in the literature. An efficient algorithm for minimizing ESOP forms is known for two-variable multiple-valued input functions. On the other hand, no ESCT minimization algorithm is known for such functions. In this paper, we give an efficient algorithm for minimizing ESCT forms of two-variable multiple-valued input functions, showing that the number of terms can be reduced by at most one.

Modified Alternating Step Generators with Non-Linear Scrambler

Pseudorandom generators, which produce keystreams for stream ciphers by the exclusive-or sum of output bits from alternately clocked linear feedback shift registers, are vulnerable to cryptanalysis. In order to increase their resistance to attacks, we introduce a nonlinear scrambler at the output of these generators. The role of the scrambler plays the nonlinear feedback shift register. In addition, we propose the Modified Alternating Step Generator (MASG1S) built with the nonlinear scrambler and regularly or irregularly clocked linear feedback shift registers with nonlinear filtering functions.

A PARALLEL ALGORITHM FOR MINIMIZING ESOP EXPRESSIONS

Two parallel algorithms are proposed in this paper for solving the problem of finding exact exclusive-or sum of products (ESOP) expressions for an arbitrary Boolean function. This minimization problem is a very difficult one and solutions have been proposed only for up to seven variables. The processing time for some symmetric functions of seven variables is of the order of weeks. The proposed algorithm is a hybrid one (OpenMP, MPI) and a speed-up of more than nine could be achieved, for a cluster of three nodes with four cores each.

OR-Bridging Fault Identification and Diagnosis for Exclusive-OR Sum of Products Reed-Muller Canonical Circuits

Problem statement: The faults in digital circuit can be classified broadly as single stuck-atfaults, multiple stuck-ay-faults, stuck-open faults, stuck-on faults, path delay faults, transient faults. Extensive research had been carried out in the field of testing of digital circuits to limit the number of input vectors. The cardinality of the test vectors proposed by many authors was quite high for large number of input variables. In this study a testable circuit with a small test set for detection and diagnosis of OR-bridging type fault in Reed-Muller canonical Exclusive-OR Sum of Products logic circuits, independent of the function for a given number of inputs had been proposed. Approach: A network structure comprising a set of Exclusive-OR gates and gates and a couple of auxiliary outputs were considered. The circuit as well as the test vectors were simulated by MATLAB coding. The faultfree and OR-bridging faults involving any two lines of control and data lines were then simulated. The outputs were represented in a compact decimal form for ease of tabulation. Two quantitative indices for comparison of results had also been discussed. Simulation and analysis for various random functions had been presented. Results: From the test results it was found that the identifiability for the set of random functions tested was more than 90% with just n + 5 test vectors compared to 2n test vectors required for conventional testing. It was also observed that even though the overall distinguishabililty factor was in the range of 45-80%, the individual set distinguishability was more than 90%. Conclusion: The proposed scheme had reduced the possibility of unidentifiable faults for the specified type of function. The location was also diagnosed through the output set. The analysis and diagnosis had been done through compact tabulation and two quantification indices.

USING SIMPLE DISJOINT DECOMPOSITION TO PERFORM SECURE COMPUTATIONS

This paper deals with the use of a minimal model for performing secure computations. The communication is based on a protocol which makes use of disjoint function decomposition and more precisely of minimal ESCT (Exclusive-or Sum of Complex Terms) expressions in order to perform a secure computation. The complexity of this protocol is directly proportional to the size of the ESCT expression in use, which is much smaller in comparison to other proposed minimal models (e.g., ESOP). Moreover, quantum algorithms are discussed that provide significant speedup to the process of producing the ESCT expressions, when compared to conventional ones. Hence, this paper provides a very useful application of the ESCT expressions in the field of cryptographic protocols.

Single Stuck-At Fault Diagnosing Circuit of Reed-Muller Canonical Exclusive-Or Sum of Product Boolean Expressions

A testable design with a universal test set for single stuck-at zero and stuck-at one faults of Reed-Muller canonical form of Exclusive-OR sum of product logic expressions is proposed. The test circuit detects almost all the single stuck-at faults and needs only simple modifications for variations in the circuit under test. The number of test vectors is also quite small compared with the classical method. The factor of un-identifiability is discussed and a new quantification parameter for the fault diagnosis has also been introduced. Results of Matlab simulations for a few logic functions are included.

Two-level logic circuits using EXOR sums of products

Two-level logic is most often implemented as an inclusive-OR sum of product terms, e.g. with PLAs. Using exclusive-OR (EXOR) sums may simplify the representation and manipulation of Boolean functions and result in more easily testable implementations requiring fewer product terms. However, due to the lack of relevant algorithms and efficient implementation structures, it has not been possible to translate these theoretical advantages into practical benefits. In this paper solutions for the two main problems associated with the use of EXOR sums are presented. On the one hand we describe a new method to minimise functions using two-level EXOR sums of products, on the other hand we present an implementation structure called the XPLA to map the minimisation results to efficient circuit layouts. We show, for a set of benchmark examples, that the minimisation algorithm results in representations with considerably smaller product term counts than previous EXOR minimisation algorithms or sum-of-product minimisation algorithms. We also show, although the EXOR operator is more expensive to implement in today's technologies, that XPLA implementations can be considerably more compact than PLAs in some cases, and give increased testability.

Canonical restricted mixed-polarity exclusive-OR sums of products and the efficient algorithm for their minimisation

The concept of canonical restricted mixed polarity (CRMP) exclusive-OR sum of products forms is introduced. The CRMP forms include the inconsistent canonical Reed-Muller forms and the fixed-polarity Reed-Muller (FPRM) forms as special cases. The set of CRMP forms is included in the set of exclusive-OR sum-of-product (ESOP) expressions. An attempt to characterise minimal CRMP forms for completely specified Boolean functions is presented as well as an insight into the complexity of computation needed to find such a form. Some fundamental properties unique to CRMPs are proven. It is also proven that the upper bound on the number of terms in the CRMP form is smaller than that in the conventional normal forms and equal to that of the ESOPs. A theorem providing a lower bound on the number of CRMP terms is given. Finally, based on these theoretical results, a heuristic algorithm and its implementation to obtain a quasiminimal CRMP form for a multioutput function are presented.

Low-complexity synthesis of incompletely specified multiple-output mod-2 sums

A new method for (quasi) minimisation of incompletely specified multiple-output mod-2 sum expressions is presented. It uses a new low-complexity algorithm for the Reed-Muller transform (RMT). Heuristically choosing a (quasi-) optimal polarity of the transform, its coefficients are combined so as to form a locally disjoint RM coefficient cover. Inverse RMT yields a (quasi)minimal exclusive-OR sum of products. To exploit common product terms, multiple-output functions are combined to form a single-output hyperfunction. ‘don't cares’ are (quasi-)optimally allotted during the procedure. Since throughout the algorithm only cubes are to be handled, the method is fast and requires only moderate storage. It compares favourably with other algorithms both in the average number of products required and in its computational complexity. The program HEALEX can handle functions of hundreds of variables on an AT personal computer.