Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes

We present polynomial families complete for the well-studied algebraic complexity classes VF, VBP, VP, and VNP. The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. (2018). We consider three different variants of graph homomorphisms, namely injective homomorphisms , directed homomorphisms , and injective directed homomorphisms , and obtain polynomial families complete for VF, VBP, VP, and VNP under each one of these. The polynomial families have the following properties: • The polynomial families complete for VF, VBP, and VP are model independent, i.e., they do not use a particular instance of a formula, algebraic branching programs, or circuit for characterising VF, VBP, or VP, respectively. • All the polynomial families are hard under p -projections.

Designing an Efficient and Highly Dynamic Substitution-Box Generator for Block Ciphers Based on Finite Elliptic Curves

Developing a substitution-box (S-box) generator that can efficiently generate a highly dynamic S-box with good cryptographic properties is a hot topic in the field of cryptography. Recently, elliptic curve (EC)-based S-box generators have shown promising results. However, these generators use large ECs to generate highly dynamic S-boxes and thus may not be suitable for lightweight cryptography, where the computational power is limited. The aim of this paper is to develop and implement such an S-box generator that can be used in lightweight cryptography and perform better in terms of computation time and security resistance than recently designed S-box generators. To achieve this goal, we use ordered ECs of small size and binary sequences to generate certain sequences of integers which are then used to generate S-boxes. We performed several standard analyses to test the efficiency of the proposed generator. On an average, the proposed generator can generate an S-box in 0.003 seconds, and from 20,000 S-boxes generated by the proposed generator, 93 % S-boxes have at least the nonlinearity 96. The linear approximation probability of 1000 S-boxes that have the best nonlinearity is in the range [0.117, 0.172] and more than 99% S-boxes have algebraic complexity at least 251. All these S-boxes have the differential approximation probability value in the interval [0.039, 0.063]. Computational results and comparisons suggest that our newly developed generator takes less running time and has high security against modern attacks as compared to several existing well-known generators, and hence, our generator is suitable for lightweight cryptography. Furthermore, the usage of binary sequences in our generator allows generating plaintext-dependent S-boxes which is crucial to resist chosen-plaintext attacks.

Fast Quadratic Shape From Shading With Unknown Light Direction

An algorithm is proposed that extracts 3D shape from shading information in a digital image. The algorithm assumes that there is only a single source of light producing the image, that the surface of the shape giving rise to the image is Lambertian (matte) and that its shape can be locally approximated by a quadratic function. Previous work shows that under these assumptions, robust shape from shading is possible, though slow for large images because a non-linear optimization method is applied in order to estimate local quadratic surface patches from image intensities. The work presented here shows that local quadratic surface patch estimates can be computed, without prior knowledge of the light source direction, via a linear least squares optimization, thus greatly improving the algebraic complexity and run-time of the existing algorithms.

Fast Quadratic Shape From Shading With Unknown Light Direction

An algorithm is proposed that extracts 3D shape from shading information in a digital image. The algorithm assumes that there is only a single source of light producing the image, that the surface of the shape giving rise to the image is Lambertian (matte) and that its shape can be locally approximated by a quadratic function. Previous work shows that under these assumptions, robust shape from shading is possible, though slow for large images because a non-linear optimization method is applied in order to estimate local quadratic surface patches from image intensities. The work presented here shows that local quadratic surface patch estimates can be computed, without prior knowledge of the light source direction, via a linear least squares optimization, thus greatly improving the algebraic complexity and run-time of this existing algorithms.

Fast Quadratic Shape From Shading With Unknown Light Direction

An algorithm is proposed that extracts 3D shape from shading information in a digital image. The algorithm assumes that there is only a single source of light producing the image, that the surface of the shape giving rise to the image is Lambertian (matte) and that its shape can be locally approximated by a quadratic function. Previous work shows that under these assumptions, robust shape from shading is possible, though slow for large images because a non-linear optimization method is applied in order to estimate local quadratic surface patches from image intensities. The work presented here shows that local quadratic surface patch estimates can be computed, without prior knowledge of the light source direction, via a linear least squares optimization, thus greatly improving the algebraic complexity and run-time of this existing algorithms.

Evidence for and against Zauner's MUB conjecture in C^6

The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case,~$d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~$d=6$ by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~$q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

The effect of strong shear on internal solitary-like waves

Large amplitude internal waves in the ocean propagate in a dynamic, highly variable environment with changes in background current, local depth, and stratification. The Dubreil-Jacotin-Long, or DJL, theory of exact internal solitary waves can account for a background shear, doing so at a cost of algebraic complexity and a lack of a mathematical proof of algorithm convergence. Waves in the presence of shear that is strong enough to preclude theoretical calculations have been reported in observations. We report on high resolution simulations of stratified adjustment in the presence of strong shear currents. We find instances of large amplitude solitary-like waves with recirculating cores in parameter regimes for which DJL theory fails, and of wave types that are completely different in shape from classical internal solitary waves. Both are spontaneously generated from general initial conditions. Some of the waves observed are associated with critical layers, but others exhibit a propagation speed that is very near the background current maximum. As such they are not freely propagating solitary waves, and a DJL theory would not apply. We thus provide a partial reconciliation between observations and theory.

The Computational Complexity of Plethysm Coefficients

In two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity.In this paper, we show that deciding positivity of plethysm coefficients is -hard and that computing plethysm coefficients is #-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.

Rational approximation of P-wave kinematics — Part 1: Transversely isotropic media

In seismic data processing and several wave propagation modeling algorithms, the phase velocity, group velocity, and traveltime equations are essential. To have these equations in explicit form, or to reduce algebraic complexity, approximation methods are used. For the approximation of P-wave kinematics in acoustic transversely isotropic media, we have developed a new flexible 2D functional equation in a continued fraction form. Using different orders of the continued fraction, we obtain different approximations for (1) phase velocity as a function of phase direction, (2) group velocity as a function of group direction, and (3) traveltime as a function of offset. Then, we use them in the approximation of the group direction as a function of phase direction, and phase direction as a function of group direction. The proposed approximations have a rational form, which is considered algebraically simple and computationally efficient. The used continued fraction form rapidly converges to exact kinematics. By introducing the optimal ray into our approximations and using it for parameter definition, the convergence becomes faster, so the accuracy of the existing most accurate approximations is available by the third order, and new most accurate approximations are obtained by the fourth order of the proposed general form. The error of the most accurate version of the proposed approximations is below 0.001% for moderate anisotropic models with an anellipticity parameter up to 0.3. This high accuracy is considered to be attractive in practical implementations that use the kinematic equations and their derivatives.

The Asymptotic Induced Matching Number of Hypergraphs: Balanced Binary Strings

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith–Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science.Our proof relies on a new combinatorial inequality that may be of independent interest. This inequality concerns how many pairs of Boolean vectors of fixed Hamming weight can have their sum in a fixed subspace.