Probabilistic verification of Boolean functions

1999

This paper describes a probabilistic method for verifying the equivalence of two multiple-valued functions. Each function is hashed to an integer code by transforming it to a integer-valued polynomial and the equivalence of two polynomials is checked probabilistically. The hash codes for two equivalent functions are always the same. Thus, the equivalence of two functions can be verified with a known probability of error, arising from collisions between inequivalent functions. Such a probabilistic verification can be an attractive alternative for verifying functions that are too large to be handled by deterministic verification methods.

IEEE Transactions on Computers, 1997

A new Boolean function representation scheme, the Indexed Binary Decision Diagram (IBDD), is proposed to provide a compact representation for functions whose Ordered Binary Decision Diagram (OBDD) representation is intractably large. We explain properties of IBDDs and present algorithms for constructing IBDDs from a given circuit. Practical and effective algorithms for satisfiability testing and equivalence checking of IBDDs, as well as their implementation results, are also presented. The results show that many functions, such as multipliers and the hidden-weighted-bit function, whose analysis is intractable using OBDDs, can be efficiently accomplished using IBDDs. We report efficient verification of Booth multipliers, as well as a practical strategy for polynomial time verification of some classes of unsigned array multipliers.

Computer and Information Science, 2008

An efficient method of finding optimal (OBDD) of an n variable Boolean function is presented that offers a simple and straightforward procedure for optimal OBDD generation along with storage economy. This is achieved by generating n! fold tables and applying node reduction rules to each fold table directly instead of generating all n! OBDDs of the function.

Information Sciences, 2017

Although several methods for estimating the resistance of a random Boolean function against (fast) algebraic attacks were proposed, these methods are usually infeasible in practice for relative large input variables n (for instance n ≥ 30) due to increased computational complexity. An efficient estimation the resistance of Boolean function (with relative large input variables n) against (fast) algebraic attacks appears to be a rather difficult task. In this paper, the concept of partial linear relations decomposition is introduced, which decomposes any given nonlinear Boolean function into many linear (affine) subfunctions by using the disjoint sets of input variables. Based on this result, a general probabilistic decomposition algorithm for nonlinear Boolean functions is presented which gives a new framework for estimating the resistance of Boolean function against (fast) algebraic attacks. It is shown that our new probabilistic method gives very tight estimates (lower and upper bound) and it only requires about O(n 2 2 n) operations for a random Boolean function with n variables, thus having much less time complexity than previously known algorithms.

Automatyka/Automatics, 2018

Most methods for determining the prime implicants of a Boolean function depend on the minterms of the function. Deviating from this philosophy, this paper presents a method that depends on maxterms (the minterms of the complement of the function) for this purpose. Normally, maxterms are used to get prime implicates and not prime implicants. It is shown that all prime implicants of a Boolean function can be obtained by expanding and simplifying any product of sums form of the function appropriately. No special form of the product of the sums is required. What is more, prime implicants can generally be generated from any form of the function by converting it into a POS using well-known techniques. The prime implicants of a product of Boolean functions can be obtained from the prime implicants of individual Boolean functions. This allows us to handle big functions by breaking them into the products of smaller functions. A simple method is presented to obtain one minimal set of prime implicants from all prime implicants without using minterms. Similar statements also hold for prime implicates. In particular, all prime implicates can be obtained from any sum of a product's form. Twelve variable examples are solved to illustrate the methods.

Proceedings of 1997 IEEE International Symposium on Circuits and Systems. Circuits and Systems in the Information Age ISCAS '97

BDD propagation is prioritized by Size and discon 56) References Cited tinued once a given limit is exceeded. The resulting Verifi cation engine is reliably accurate and efficient for a wide U.S. PATENT DOCUMENTS variety of practical hardware designs ranging from identical circuits to designs with very few similarities.

2008

In this paper, we compare two different Boolean function reduction methods in order to justify the analytical model of the Monte Carlo data for Boolean function complexity. We use a binary decision diagram (BDD) complexity model (proposed earlier) and weigh it against the complexity behavior generated by Synopsys Design Compiler (DC). We use this synthesis tool (that utilizes a standard cell library) to generate RTL hardware description of Monte Carlo circuits as gate-level netlists. The two reduction methods (model and DC) transform an arbitrary function into a much-reduced representation of the same function. The comparison confirms that the behavior of Boolean function complexity using the model and the DC is visually and statistically similar; the similarity holds true for BDDs representing functions comprising a wide range of variables and minterms.

1993

Abstract We present a method to compute the bdd for an arbitrary Boolean expression, where the operands are themselves bdds. Such expressions are usually computed by the successive application of binary operators. However, cases exist where this method performs wasteful intermediate computations and creates bdd nodes not used in the nal result. In contrast, our method never creates a bdd node unless it is present in the nal result.

2013

Boolean functions play important role in cryptography, since in convention a symmetric encryption algorithm can be designed by composing Boolean functions satisfying good cryptographic criteria. In this paper; state of the art in mathematical and practical study of the most important cryptographic criteria of Boolean functions and how to implement algorithms that fulfill these criteria are introduced. Also; the most known constructions for generating Boolean functions that satisfy good cryptographic criteria are summarized.

Theory of Cryptography, 2013

We present a protocol for securely computing a Boolean circuit C in presence of a dishonest and malicious majority. The protocol is unconditionally secure, assuming a preprocessing functionality that is not given the inputs. For a large number of players the work for each player is the same as computing the circuit in the clear, up to a constant factor. Our protocol is the first to obtain these properties for Boolean circuits. On the technical side, we develop new homomorphic authentication schemes based on asymptotically good codes with an additional multiplication property. We also show a new algorithm for verifying the product of Boolean matrices in quadratic time with exponentially small error probability, where previous methods only achieved constant error.