Boolean Polynomial Research Articles

Redundant Code-based Masking Revisited

Masking schemes are a popular countermeasure against side-channel attacks. To mask bytes, the two classical options are Boolean masking and polynomial masking. The latter lends itself to redundant masking, where leakage emanates from more shares than are strictly necessary to reconstruct, raising the obvious question how well such “redundant” leakage can be exploited by a side-channel adversary. We revisit the recent work by Chabanne et al. (CHES’18) and show that, contrary to their conclusions, said leakage can—in theory—always be exploited. For the Hamming weight scenario in the low-noise regime, we heuristically determine how security degrades in terms of the number of redundant shares for first and second order secure polynomial masking schemes.Furthermore, we leverage a well-established link between linear secret sharing schemes and coding theory to determine when different masking schemes will end up with essentially equivalent leakage profiles. Surprisingly, we conclude that for typical field sizes and security orders, Boolean masking is a special case of polynomial masking. We also identify quasi-Boolean masking schemes as a special class of redundant polynomial masking and point out that the popular “Frobenius-stable” sets of interpolations points typically lead to such quasi-Boolean masking schemes, with subsequent degraded leakage performance.

Application and comparison of different classification methods based on symptom analysis with traditional classification technique for breast cancer diagnosis

Novel approach for classification technique such as Artificial Neural Network (ANN), Linear Discriminant Analysis (LDA) and Random Forest (RF) using factor or dichotomic variables has been introduced. This study searches for the highly informative finitely linear combinations (symptoms) of variables in the finite field on the based of the Fisher’s exact test and accurately predict the target class for each case in the data. There are several super symptoms have comparable p-values. In this case, it becomes possible to choose as a nominative representative the factor which is more accessible for interpretation. The super symptom means a linear combination of various multiplications of k dichotomous variables over a field of characteristic 2 without repeating. In algebra, such functions are called Zhegalkin polynomials or algebraic normal forms.

The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree 8 primitive irreducible polynomial over ℤ2. In this manuscript, we study affine power affine S-boxes for all the 16 distinct degree 8 primitive irreducible polynomials over ℤ2 to propose 16 different 8×8 substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

Solving Boolean polynomial systems by parallelizing characteristic set method for cyber‐physical systems

SummaryMany cyber‐attach schemes and coding models established by algebra tools are build to address the problem of security of cyber‐pysical systems (CPS). As an important field of algebra computing, Boolean Polynomial System Solving (PoSSo) problem plays a very important role in many algebra applications. In this article, we propose an efficient Parallel Boolean Characteristic Set method (PBCS) under the high‐performance computing environment to improve the efficiency of solving Boolean polynomial systems. The PBCS is implemented based on the state‐of‐the‐art Boolean Characteristic Set method (BCS). It adopts a master‐slave parallel pattern, and distributes tasks based on the polynomial sets after initial zero decomposition. We design a strategy of dynamically reallocating tasks to ameliorate load imbalance, which is caused by dynamical zero decomposition of polynomials. Furthermore, we improve its performance by optimizing the parameter settings of PBCS, including the maximum number of polynomial branches that trigger the dynamic allocation policy and the scheduling time. Experimental results with solving several Boolean polynomial systems confirm that PBCS is efficient and scalable, especially for the equations generating from stream ciphers that have block triangular structure. Moreover, the method also has good scalability. It shows a stable speedup as well even extending to the size of thousands of CPU cores.

Analyzing Boolean Functions via Solving Parametric Polynomial Systems

In this paper, a new method to analyze Boolean functions is proposed. By this method, one can analyze the balancedness, the nonlinearity, and the input-output correlation of vectorial Boolean functions. The basic idea of this method is to compute the refined covers of some parametric Boolean polynomial systems which are equivalent to these problems. By a refined cover, the parameter space is divided into several disjoint components, and on each component, the parametric Boolean polynomial system has a fixed number of solutions. An efficient algorithm based on the characteristic set method to compute refined covers of parametric Boolean polynomial systems is presented. The experimental results about some instances generated from cryptanalysis show that this new method is efficient and can solve some instances which can not be solved in reasonable time by other methods.

Symptom analysis of multidimensional categorical data with applications

The linear combinations of dichotomous variables over the field F_2, which are called symptoms, form the projective space from which it is possible select the more informative subspaces for reducing the dimensionality of binary data. In this article, the symptom space expands to the super-symptom space. The super-symptom means a linear combination of various multiplications of k dichotomous variables over a field of characteristic 2 without repeating. In algebra, such functions are called Zhegalkin polynomials or algebraic normal forms. It is known that each logical function can be represented in the form of a Zhegalkin polynomial in a unique way, therefore using them to iterate one can find a logical function to best describe a risk group. The search algorithm of a more informative super-symptom for classification is based on the superposition of impulse sequences with different types of operations: first multiplication and then addition over the F_2 field. Also the super-symptom analysis is a convenient method for a study of the correlation between two sets of categorical variables. This method was applied to identify the most severe forms of the disease by combining hormonal, immunological and genetic tests in patients with breast cancer (data from Cancer Oncology Hospital in Medicine City in Baghdad) and to identify genetic risk factors by patients with alcohol dependence syndrome, receiving alcohol dependence therapy (St.Petersburg V.M. Bekhterev Psychoneurological Research Institute).

Triangle method for constructing Zhegalkin polynomial: connection with Pascal's triangle

Introduced by Soviet scientist I. Zhegalkin in 1927, Zhegalkin polynomial is a way to represent a Boolean function as an exclusive or of conjunctions of variables. One of the known algorithms for constructing Zhegalkin polynomial is so called ‘triangle method’ proposed in 1985–1987 by Soviet mathematician V.P. Suprun. Applying of the triangle method mainly coincides with step-by-step constructing Pascal triangle rows using the equality Cn+1k+1 = Cnk + Cnk+1. Therefore, it would be natural to expect for the relation between the calculation of Zhegalkin polynomial by the triangle method and an arrangement of binomial coefficients in Pascal triangle. In this paper, the connection between the triangle method and constructing of Pascal triangle is studied. Besides that, a rather simple proof of the triangle method correctness is proposed. This proof is based on juxtaposition of the triangle method steps and a step-by-step construction of Pascal triangle.

Construction of a polynomial invariant annihilation attack of degree 7 for T-310

Cryptographic attacks are typically constructed by black-box methods and combinations of simpler properties, for example in [Generalised] Linear Cryptanalysis. In this article, we work with a more recent white-box algebraic-constructive methodology. Polynomial invariant attacks on a block cipher are constructed explicitly through the study of the space of Boolean polynomials which does not have a unique factorisation and solving the so-called Fundamental Equation (FE). Some recent invariant attacks are quite symmetric and exhibit some sort of clear structure, or work only when the Boolean function is degenerate. As a proof of concept, we construct an attack where a highly irregular product of seven polynomials is an invariant for any number of rounds for T-310 under certain conditions on the long term key and for any key and any IV. A key feature of our attack is that it works for any Boolean function which satisfies a specific annihilation property. We evaluate very precisely the probability that our attack works when the Boolean function is chosen uniformly at random.

On the annihilators of Boolean polynomials

On the annihilators of Boolean polynomials

A Signature Based Border Basis Algorithm

The Border Basis Algorithm (BBA) still suffers from the lack of analogues of Buchberger’s criteria for avoiding unnecessary reductions. In this paper we develop a signature based technique which provides a first remedial step: signature bounds allow us to recognize multiple reductions of the same ancestor polynomial. The new signature based algorithm is then combined with the Boolean BBA for ideals of Boolean polynomials. Experiments show that it is at least 5 times faster than the standard Boolean BBA.

On the Annihilators of Boolean Polynomials

Boolean functions in general and Boolean polynomials (Zhegalkin polynomials or algebraic normal forms (ANF)) in particular are the subject of theoretical and applied studies in various fields of computer science. This article addresses the linear operators of the space of Boolean polynomials in n variables, which leads to the results on the problem of finding the minimum annihilator degree for a given Boolean polynomial. This problem is topical in various analytical and algorithmic aspects of cryptography. Boolean polynomials and their combinatorial properties are under study in discrete analysis. The theoretical foundations of information security include the study of the properties of Boolean polynomials in connection with cryptography. In this article, we prove a theorem on the minimum annihilator degree. The class of Boolean polynomials is described for which the degree of an annihilator is at most 1. We give a few combinatorial characteristics related to the properties of the space of Boolean polynomials. Some estimates of the minimum degree of an annihilator are given. We also consider the case of symmetric polynomials.

On the Relationship between Boolean Algebra and Quantum Informatics

The fundamental relationship between quantum physics and discrete mathematics is examined. A method for representing Boolean functions in the form of unitary transformations is described. The question of the connection of Zhegalkin polynomials defining the algebraic normal form of a Boolean function with quantum circuits is considered. It is shown that the quantum information language provides a simple algorithm for constructing the Zhegalkin polynomial based on the truth table. The developed methods and algorithms are generalized to the case of an arbitrary Boolean function with a multibit domain of definition and a multibit set of values, as well as to the case of multivalued ($$k$$-value) logic when $$k = p$$ is a prime number. The developed approach is important for the implementation of quantum computer technologies and is the foundation for the transition from classical computer logic to quantum hardware.

Deterministic Algorithms for Solving Boolean Polynomial Equations Based on Channel Coding Theory

Solving the satisfiability problems of Boolean polynomial equations is still an open challenge in the fields of mathematics and computer science. In this paper, our goal is to propose a non-algebraic method for solving maximal Boolean polynomial equations (Max-PoSSo problem). By leveraging channel coding theory and dynamic programming, we propose three deterministic and robust algorithms for solving the satisfiability problems of Boolean polynomial equations. Comparisons are made among the three proposed algorithms and Genetic and Grobner algorithms. Simulation results show that the proposed algorithms exhibit better performance in terms of the largest number of Boolean polynomials equal to 0 compared to the benchmark schemes in the literature.

Efficient On-Chip Randomness Testing Utilizing Machine Learning Techniques

Randomness testing is an important procedure that bit streams, produced by critical cryptographic primitives such as encryption functions and hash functions, have to undergo. In this paper, a new hardware platform for the randomness testing is proposed. The platform exploits the principles of genetic programming, which is a machine learning technique developed for the automated program and circuit design. The platform is capable of evolving efficient randomness distinguishers directly on a chip. Each distinguisher is represented as a Boolean polynomial in the algebraic normal form. The randomness testing is conducted for bit streams that are either stored in an on-chip memory or generated by a circuit placed on the chip. The platform is developed with a Xilinx Zynq-7000 All Programmable System on Chip that integrates a field programmable gate array with on-chip ARM processors. The platform is evaluated in terms of the quality of randomness testing, performance, and resources utilization. With power budget less than 3 W, the platform provides comparable randomness testing capabilities with the standard testing batteries running on a personal computer.

On the efficiency of solving Boolean polynomial systems with the characteristic set method

An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain pseudo-remainders. Three important techniques are applied in the algorithm. The first one is eliminating variables by new generated linear polynomials. The second one is optimizing the strategy of choosing polynomial for zero decomposition. The third one is to compute add-remainders to eliminate the leading variable of new generated monic polynomials. By analyzing the depth of the zero decomposition tree, we present some complexity bounds of this algorithm, which are lower than the complexity bounds of previous characteristic set algorithms. Extensive experimental results show that this new algorithm is more efficient than previous characteristic set algorithms for solving Boolean polynomial systems.

The complexity of AND—decomposition of Boolean functions

We consider the complexity of decomposing a Boolean function into a conjunction of components, which may share a (possibly empty) given set of variables Δ. Boolean functions are given as expressions in CNF, DNF, full DNF, and ANF and it is assumed that decomposition components must be in the same normal form as the input expression. We show that deciding decomposability is in general intractable for CNF and DNF already for the empty set of shared variables between the components. On the other hand, we demonstrate that for positive CNF and DNF and full DNF, the finest decomposition components wrt a given Δ can be computed in polynomial time (in the size of the input expression given as a string). The tractability results employ an efficient transformation of input expressions into series of expressions in ANFs aka Boolean polynomials, for which we provide a polynomial time factorization algorithm. Since the reduction leads to solving series of factorization tasks for Boolean polynomials, we discuss a number of ideas on how to optimize massive factorization and report preliminary experimental results.

Boolean Gröbner Basis Reductions on Finite Field Datapath Circuits Using the Unate Cube Set Algebra

Recent developments in formal verification of arithmetic datapaths make efficient use of symbolic computer algebra algorithms. The circuit is modeled as an ideal in polynomial rings, and Grobner basis (GB) reductions are performed over these polynomials to derive a canonical representation. As they model logic gates of the circuit, the ideals comprise largely of Boolean (or pseudo-Boolean) polynomials. This paper considers a logic synthesis analogue of GB reductions over Boolean polynomials, by interpreting symbolic algebra as the unate cube set algebra over characteristic sets. By representing Boolean polynomials as characteristic sets using zero-suppressed binary decision diagrams (ZDDs), implicit algorithms are efficiently designed for GB-reduction for datapath circuits. We show that the imposition of circuit-topology-based monomial orders exposes a special structure on the ZDD representation of the polynomials. The subexpressions employed in the GB-reduction are readily visible as subgraphs on the ZDDs, which are directly used to compose the result. Our division algorithms effectively cancel multiple monomials implicitly in one-step, simplify the search for divisors, and avoid intermediate size explosion. Experiments performed over various finite field arithmetic architectures demonstrate the efficiency of our algorithms and implementations; our approach is orders of magnitude faster as compared to conventional methods.

Incremental column-wise verification of arithmetic circuits using computer algebra

Verifying arithmetic circuits and most prominently multiplier circuits is an important problem which in practice still requires substantial manual effort. The currently most effective approach uses polynomial reasoning over pseudo boolean polynomials. In this approach a word-level specification is reduced by a Gröbner basis which is implied by the gate-level representation of the circuit. This reduction returns zero if and only if the circuit is correct. We give a rigorous formalization of this approach including soundness and completeness arguments. Furthermore we present a novel incremental column-wise technique to verify gate-level multipliers. This approach is further improved by extracting full- and half-adder constraints in the circuit which allows to rewrite and reduce the Gröbner basis. We also present a new technical theorem which allows to rewrite local parts of the Gröbner basis. Optimizing the Gröbner basis reduces computation time substantially. In addition we extend these algebraic techniques to verify the equivalence of bit-level multipliers without using a word-level specification. Our experiments show that regular multipliers can be verified efficiently by using off-the-shelf computer algebra tools, while more complex and optimized multipliers require more sophisticated techniques. We discuss in detail our complete verification approach including all optimizations.

Geometry of Interaction for MALL via Hughes--Van Glabbeek Proof-Nets

This article presents, for the first time, a Geometry of Interaction (GoI) interpretation inspired from Hughes--Van Glabbeek (HvG) proof-nets for multiplicative additive linear logic (MALL). Our GoI dynamically captures HvG’s geometric correctness criterion—the toggling cycle condition—in terms of algebraic operators. Our new ingredient is a scalar extension of the *-algebra in Girard’s *-ring of partial isometries over a Boolean polynomial ring with literals of eigenweights as indeterminates. To capture feedback arising from cuts, we construct a finer-grained execution formula. The expansion of this execution formula is longer than that for collections of slices for multiplicative GoI, hence it is harder to prove termination. Our GoI gives a dynamical, semantical account of Boolean valuations (in particular, pruning sub-proofs), conversion of weights (in particular, α-conversion), and additive (co)contraction, peculiar to additive proof-theory. Termination of our execution formula is shown to correspond to HvG’s toggling criterion. The slice-wise restriction of our execution formula (by collapsing the Boolean structure) yields the well-known correspondence, explicit or implicit in previous works on multiplicative GoI, between the convergence of execution formulas and acyclicity of proof-nets. Feedback arising from the execution formula by restricting to the Boolean polynomial structure yields autonomous definability of eigenweights among cuts from the rest of the eigenweights.

On the Number of Roots of Boolean Polynomials

On the Number of Roots of Boolean Polynomials