Single Monomial Research Articles

Using easy coefficients conjecture for rotation symmetric Boolean functions

A Boolean function in n variables is rotation symmetric (RS) if it is invariant under powers of ρ(x1,…,xn)=(x2,…,xn,x1). An RS function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ to a single monomial. Completing earlier research on special cases, the author showed in 2018 that for any RS function fn in n variables, the sequence of Hamming weights wt(fn) for all values of n satisfies a linear recurrence relation. It was also proved that the associated recursion polynomial could be explicitly calculated as the minimal polynomial of a rules matrix and an algorithm for computing the rules matrix was explained. Examples showed that the usual formula which gives the values of wt(fn)−2n−1 as a linear combination of powers of the irrational roots of the minimal polynomial has simple coefficients which are all 1/2 if the multiset made up of the roots of the characteristic polynomial of the rules matrix is used instead, no matter what the degree of the Boolean function is. The conjecture that this is always true is called the Easy Coefficients Conjecture (ECC). The special case of MRS quadratics was proved in an earlier paper. The present paper gives some ECC applications valid for any function for which the ECC is true, even though there is a proof only for the quadratic MRS case so far.

The weight recursions for the 2-rotation symmetric quartic Boolean functions

A Boolean function in $ n $ variables is 2-rotation symmetric if it is invariant under even powers of $ \rho(x_1, \ldots, x_n) = (x_2, \ldots, x_n, x_1) $, but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of $ \rho^2 $ to a single monomial. If the quartic MRS 2-function in $ 2n $ variables has a monomial $ x_1 x_q x_r x_s $, then we use the notation $ {2-}(1,q,r,s)_{2n} $ for the function. A detailed theory of equivalence of quartic MRS 2-functions in $ 2n $ variables was given in a $ 2020 $ paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called $ mf1 $ and $ mf2 $ in the paper. After describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions $ {2-}(1,q,r,s)_{2n} $ (with $ q < r < s, $ say), $ n = s, s+1, \ldots $ can be shown to satisfy. This problem was solved for the $ mf1 $ case only in the $ 2020 $ paper. Using new ideas about 'short' functions, Cusick and Cheon found formulas for the $ mf2 $ weights in a $ 2021 $ sequel to the $ 2020 $ paper. In this paper the actual recursions for the weights in the $ mf2 $ case are determined.

Characters, Weil sums and c-differential uniformity with an application to the perturbed Gold function

Building upon the observation that the newly defined Ellingsen, et al. (2020) concept of c-differential uniformity is not invariant under EA or CCZ-equivalence Hasan et al. (2021), we showed in Stanica and Geary (2021) that adding some appropriate linearized monomials increases the c-differential uniformity of the inverse function, significantly, for some c. We continue that investigation here. First, by analyzing the involved equations, we find bounds for the uniformity of the Gold function perturbed by a single monomial, exhibiting the discrepancy we previously observed on the inverse function. Secondly, to treat the general case of perturbations via any linearized polynomial, we use characters in the finite field to express all entries in the c-Differential Distribution Table (DDT) of an (n, n)-function on the finite field ${\mathbb {F}}_{p^{n}}$ , and further, we use that method to find explicit expressions for all entries of the c-DDT of the perturbed Gold function (via an arbitrary linearized polynomial).

Regularity of [formula omitted]-invariant monomial ideals

For a polynomial ring S in n variables, we consider the natural action of the symmetric group Sn on S by permuting the variables. For an Sn-invariant monomial ideal I⊆S and j≥0, we give an explicit recipe for computing the modules ExtSj(S/I,S), and use this to describe the projective dimension and regularity of I. We classify the Sn-invariant monomial ideals I that have a linear free resolution, and also characterize those which are Cohen–Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and another where we vary the dimension of the ambient polynomial ring and examine the invariant monomial ideals induced by I. In the first case we determine the asymptotic regularity for those ideals I that are generated by the Sn-orbit of a single monomial by solving an integer linear optimization problem. In the second case we describe the behavior of regularity for any I, recovering a recent result of Murai.

Weights for short quartic Boolean functions

A Boolean function in n variables is 2-rotation symmetric if it is invariant under even powers of ρ(x1,…,xn)=(x2,…,xn,x1), but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ2 to a single monomial. If the quartic MRS 2-function in 2n variables has a monomial x1xqxrxs, then we use the notation 2-(1,q,r,s)2n for the function. A detailed theory of equivalence of quartic MRS 2-functions in 2n variables was given in a 2020 paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called mf1 and mf2 in the paper. Next to describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions 2-(1,q,r,s)2n (with q<r<s, say), n=s,s+1,… can be shown to satisfy. This problem was solved for the mf1 case only in the 2020 paper. In this paper the problem for the mf2 case is solved, using new ideas about short functions, as defined in this paper. These short functions are also of independent interest, and further results about them are in the paper.

Equivalence of 2-rotation symmetric quartic Boolean functions

A Boolean function in n variables is 2-rotation symmetric if it is invariant under even powers of ρ(x1,……,xn)=(x2,…,xn,x1), but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ2 to a single monomial. If the quartic MRS 2-function in 2n variables has a monomial x1xqxrxs, then we use the notation 2-(1, q, r, s)2n for the function. This paper gives a detailed theory of equivalence of quartic MRS 2-functions in 2n variables. Such a theory was provided for the cubic MRS 2-functions in two 2015 papers of Cusick and Johns. As in the earlier papers, the two main topics in the theory are describing the affine equivalence classes of the functions under certain groups of permutations; and giving details of the linear recursions that the Hamming weights of any sequence of functions 2-(1, q, r, s)2n (with q < r < s, say), n=s,s+1,… can be shown to satisfy. The discussion for both of these topics uses new ideas because the quartic theory naturally divides into two cases.

Affine equivalence classes of 2-rotation symmetric cubic Boolean functions

ABSTRACTA Boolean function in variables that is generated by applying all even powers of the cyclic permutation of the variables to a single cubic monomial is called a cubic monomial rotation symmetric 2-function (or cubic 2-MRS function for short). In 2015, Cusick and Johns gave a complete description of the affine equivalence classes for such functions with generating monomial with 1<r<s and r and s not both odd. In this paper we develop the theory further by determining the smallest sets that act on the set of all these cubic 2-MRS functions to give the affine equivalence classes. Also, suppose f and g are two of these functions such that there exists a permutation σ of the variables where We give a complete description of all such permutations σ and an exact count of their number.

Divisibility and groups in one-generated semirings

Let [Formula: see text] be a semiring generated by one element. Let us denote this element by [Formula: see text] and let [Formula: see text] be a polynomial. It has been proved that if [Formula: see text] contains at least two different monomials, then the elements of the form [Formula: see text] may possibly be contained in any countable commutative semigroup. In particular, divisibility of such elements does not imply their torsion. Let, on the other hand, [Formula: see text] consist of a single monomial (i.e. [Formula: see text], where [Formula: see text]). We show that in this case, the divisibility of [Formula: see text] by infinitely many primes implies that [Formula: see text] generates a group within [Formula: see text]. Further, an element [Formula: see text] is called an [Formula: see text]-fraction of an element [Formula: see text] if [Formula: see text] and [Formula: see text]. We prove that “almost every” [Formula: see text]-fraction of [Formula: see text] can be expressed as [Formula: see text] for some polynomial [Formula: see text] of degree at most [Formula: see text].

Using enchanted features of Constancy Adding Space Extension (CASE) to reduce the dimension of evolver dynamics: Single Monomial Probabilistic Evolution Theory

From physics point of view, this paper focuses on evolver dynamics which is in fact somehow another interpretation of quantum dynamics’ Heisenberg picture combined with the Probabilistic Evolution Theory (PREVTH) quite recently developed in Group for Science and Methods of Computing (Demiralp’s group). Even though PREVTH has been based on conicality, during CMMSE 2017 a new point of view and a novel extension have been built for PREVTH, Single Monomial Probabilistic Evolution Theory has been made available by the second author. This work focuses on a one-degree-of-freedom quantum system whose potential may be come very steep (anharmonic symmetric exponential potential) for constructing, first, evolver dynamics equations, and then, single monomial PREVTH by using degree escalating procedure together with commutativity relation based addition to the single monomiality coefficient matrix. Both procedures introduce certain arbitrary parameters into the structure of that matrix by using Constancy Adding Space Extension. Parameters are determined to suppress the norm square of single monomiality coefficient matrix. A convergence analysis for this new PREVTH has also been presented.

Promenading in the enchanted realm of Kronecker powers: single monomial probabilistic evolution theory (PREVTH) in evolver dynamics

This paper takes a CMMSE 2107 proceedings paper about “Highest Monomiality Based Probabilistic Evolution Theory (PREVTH)” as starting point. The focus is again set of ODEs but, this time, not on function unknowns. Instead, we deal with the temporally evolving operators, quantum mechanics’ Heisenberg picture entities we call evolvers. Our work here uses the degree changing possibilities in Kronecker powers of vectors by using so-called constancy adding space extension. Beyond the degree escalation, vanishing expression addition is also used to introduce certain arbitrary parameters and therefore to increase the flexibility in the single monomial coefficient matrix. These two issues have been deeply investigated to learn how to get single monomiality in highest degree additive term. So, the PREVTH previous-to-this paper, somehow arrested by the conicality, has now been emancipated to single monomiality which is certainly much more relaxed than before.

Somehow emancipating Probabilistic Evolution Theory (PREVTH) from singularities via getting single monomial PREVTH

This paper deals with the combination of the evolver dynamics of a hydrogen-like quantum system with the conical PREVTH (Probabilistic Evolution Theory) and excessively with the single monomial PREVTH. The new original applications are degree escalations in each monomial of a set of Poisson Bracket equations with multinomial right hand side and arbitrary, optimisable, parameters insertion to the resulting single monomial via an approach based on commutativity relations amongst the system basis operators. What we have shown that the additions based on commutativities with the Constancy Adding Space Extension (CASE) operator which is in fact proportional to identity operator, does not contribute to the suppression of the norm square of the single monomial coefficient matrix. This is just an observation for a specific family of systems but may be signaling a more general reality. If so it needs rather a rigorous proof.

Polynomial Carleson Operators Along Monomial Curves in the Plane

We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem.

Affine equivalence of monomial rotation symmetric Boolean functions: A Pólya’s theorem approach

AbstractTwo Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

On a class of repeated-root monomial-like abelian codes

In this paper we study polycyclic codes of length $p^{s_1} \times \cdots \times p^{s_n}$ over $\F_{p^a}$ generated by a single monomial. These codes form a special class of abelian codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. Finally we extend the results of Massey et. al. in \cite{MASSEY_1973} on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables.

Recursion orders for weights of Boolean cubic rotation symmetric functions

Rotation symmetric (RS) Boolean functions have been extensively studied in recent years because of their applications in cryptography. In cryptographic applications, it is usually important to know the weight of the functions, so much research has been done on the problem of determining such weights. Recently it was proved that for cubic RS functions in n variables generated by a single monomial, the weights of the functions as n increases satisfy a linear recursion. Furthermore, explicit methods were found for generating these recursions and the initial values needed to use the recursions. It is important to be able to compute the order of these recursions without needing to determine all of the coefficients. This paper gives a technique for doing that in many cases, based on a new notion of towers of RS Boolean functions.

Permutation equivalence of cubic rotation symmetric Boolean functions

Rotation symmetric Boolean functions have been extensively studied for about 15 years because of their applications in cryptography and coding theory. Until recently little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. For the much more complicated case of cubic rotation symmetric functions generated by a single monomial, the affine equivalence classes under permutations which preserve rotation symmetry were determined in 2011. It was conjectured then that the cubic equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. This conjecture is probably difficult, but here we take a step towards it by proving that the cubic affine equivalence classes found in 2011 are the same if all permutations, not just those preserving rotation symmetry, are allowed. The needed new idea uses the theory of circulant matrices.

Theory of 2-rotation symmetric cubic Boolean functions

A Boolean function in $$n$$ n variables is $$2$$ 2 -rotation symmetric if it is invariant under even powers of the cyclic permutation $$\rho (x_1,\ldots ,x_n)=(x_2,\ldots ,x_n,x_1)$$ ? ( x 1 , ? , x n ) = ( x 2 , ? , x n , x 1 ) of the variables, but not under the first power (ordinary rotation symmetry); for brevity, we call such a function a $$2$$ 2 -function. A $$2$$ 2 -function is said to be monomial rotation symmetric (MRS) if it is generated by applying powers of $$\rho ^2$$ ? 2 to a single monomial. This paper develops the theory of cubic MRS $$2$$ 2 -functions in $$2n$$ 2 n variables generated by a monomial $$x_1 x_r x_s$$ x 1 x r x s with $$1 $$ > 2. An exact count of the equivalence classes is given and their number is proved to be very small, in fact $$O(n^{\epsilon })$$ O ( n ∈ ) for any $$\epsilon > 0.$$ ∈ > 0 . It is proved that the sequence of Hamming weights $$\{wt(2 \text {-}(1,r,s)_{2n}):~2n \ge s\}$$ { w t ( 2 - ( 1 , r , s ) 2 n ) : 2 n ? s } satisfies a linear recursion with integer coefficients. A similar result for ordinary cubic MRS functions was proved recently (papers by Bileschi, Cusick and Padgett, and by Brown and Cusick), but this paper uses a new method for the $$2$$ 2 -functions proof. Unlike the ordinary MRS function case, both the orders of the recursions for the $$2$$ 2 -functions and the precise values of the roots of the corresponding recursion polynomials can be given explicitly. Finally, a precise value for the weights of the $$2$$ 2 -functions is proved, using a 2011 formula of Cusick and Padgett. These weights are connected to powers of members of the well known Lucas sequence, and so the weights can be found without computing initial values for the recursions.

Affine equivalence for cubic rotation symmetric Boolean functions with [formula omitted] variables

Rotation symmetric Boolean functions have been extensively studied in the last fifteen years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in a 2009 paper of Kim, Park and Hahn. The much more complicated analogous problem for cubic functions was solved for permutations using a new concept of patterns in a 2011 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions generated by a single monomial and having pq variables, where p and q are primes, is examined in detail, and in particular, a formula for the number of classes is proved. This is significant because it is the first time that a complete enumeration of the number of classes has been found when the number of variables is divisible by two distinct primes.

Affine equivalence of quartic homogeneous rotation symmetric Boolean functions

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions in n variables are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009, and the first substantial progress on the much more complicated cubic case came in 2010. In this paper, we show that much of the work on the cubic case can be extended to the quartic case. We also prove an exact formula for the number and sizes of the affine equivalence classes when n is a prime.

Affine equivalence for rotation symmetric Boolean functions with [formula omitted] variables

Rotation symmetric Boolean functions have been extensively studied in the last dozen years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in a 2009 paper of Kim, Park and Hahn. The much more complicated analogous problem for cubic functions was solved for permutations using a new concept of patterns in a 2010 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions generated by a single monomial and having pk variables, where p>3 is prime, is examined in detail, and in particular, a formula for the number of classes is proved.