arXiv subjects

Bruce M. Kapron

Publications and source records attributed to Bruce M. Kapron.

16 recordsLinked to original sources

arXiv2025

Complete and tractable machine-independent characterizations of second-order polytime

The class of Basic Feasible Functionals BFF is the second-order counterpart of the class of first-order functions computable in polynomial time. We present several implicit characterizations of BFF based on a typed programming language of terms. These terms may perform calls to non-recursive imperative procedures. The type discipline has two layers: the terms follow a standard simply-typed discipline and the procedures follow a standard tier-based type discipline. BFF consists exactly of the second-order functionals that are computed by typable and terminating programs. The completeness of this characterization surprisingly still holds in the absence of lambda-abstraction. Moreover, the termination requirement can be specified as a completeness-preserving instance, which can be decided in time quadratic in the size of the program. As typing is decidable in polynomial time, we obtain the first tractable (i.e., decidable in polynomial time), sound, complete, and implicit characterization of BFF, thus solving a problem opened for more than 20 years.

cs.LO↗

arXiv2024

The Computational Complexity of Variational Inequalities and Applications in Game Theory

We present a computational formulation for the approximate version of several variational inequality problems, investigating their computational complexity and establishing PPAD-completeness. Examining applications in computational game theory, we specifically focus on two key concepts: resilient Nash equilibrium, and multi-leader-follower games -- domains traditionally known for the absence of general solutions. In the presence of standard assumptions and relaxation techniques, we formulate problem versions for such games that are expressible in terms of variational inequalities, ultimately leading to proofs of PPAD-completeness.

cs.CC↗

arXiv2024

On Separation Logic, Computational Independence, and Pseudorandomness (Extended Version)

Separation logic is a substructural logic which has proved to have numerous and fruitful applications to the verification of programs working on dynamic data structures. Recently, Barthe, Hsu and Liao have proposed a new way of giving semantics to separation logic formulas in which separating conjunction is interpreted in terms of probabilistic independence. The latter is taken in its exact form, i.e., two events are independent if and only if the joint probability is the product of the probabilities of the two events. There is indeed a literature on weaker notions of independence which are computational in nature, i.e. independence holds only against efficient adversaries and modulo a negligible probability of success. The aim of this work is to explore the nature of computational independence in a cryptographic scenario, in view of the aforementioned advances in separation logic. We show on the one hand that the semantics of separation logic can be adapted so as to account for complexity bounded adversaries, and on the other hand that the obtained logical system is useful for writing simple and compact proofs of standard cryptographic results in which the adversary remains hidden. Remarkably, this allows for a fruitful interplay between independence and pseudorandomness, itself a crucial notion in cryptography.

cs.CR↗

arXiv2024

Declassification Policy for Program Complexity Analysis

In automated complexity analysis, noninterference-based type systems statically guarantee, via soundness, the property that well-typed programs compute functions of a given complexity class, e.g., the class FP of functions computable in polynomial time. These characterizations are also extensionally complete -- they capture all functions -- but are not intensionally complete as some polytime algorithms are rejected. This impact on expressive power is an unavoidable cost of achieving a tractable characterization. To overcome this issue, an avenue arising from security applications is to find a relaxation of noninterference based on a declassification mechanism that allows critical data to be released in a safe and controlled manner. Following this path, we present a new and intuitive declassification policy preserving FP-soundness and capturing strictly more programs than existing noninterference-based systems. We show the versatility of the approach: it also provides a new characterization of the class BFF of second-order polynomial time computable functions in a second-order imperative language, with first-order procedure calls. Type inference is tractable: it can be done in polynomial time.

cs.LO↗

arXiv2023

The Computational Complexity of Equilibria with Strategic Constraints

Computational aspects of solution notions such as Nash equilibrium have been extensively studied, including settings where the ultimate goal is to find an equilibrium that possesses some additional properties. Furthermore, in order to address issues of tractability, attention has been given to approximate versions of these problems. Our work extends this direction by considering games with constraints in which players are subject to some form of restrictions on their strategic choices. We also consider the relationship between Nash equilibria and so-called constrained or social equilibria in this context, with particular attention to how they are related with respect to totality and complexity. Our results demonstrate that the computational complexity of finding an equilibrium varies significantly between games with slightly different strategic constraints. In addition to examining the computational aspects of such strategic constraints, we also demonstrate that these constraints are useful for modeling problems involving strategic resource allocation and also are of interest from the perspective of behavioral game theory.

cs.CC↗

arXiv2022

A tier-based typed programming language characterizing Feasible Functionals

The class of Basic Feasible Functionals BFF$_2$ is the type-2 counterpart of the class FP of type-1 functions computable in polynomial time. Several characterizations have been suggested in the literature, but none of these present a programming language with a type system guaranteeing this complexity bound. We give a characterization of BFF$_2$ based on an imperative language with oracle calls using a tier-based type system whose inference is decidable. Such a characterization should make it possible to link higher-order complexity with programming theory. The low complexity (cubic in the size of the program) of the type inference algorithm contrasts with the intractability of the aforementioned methods and does not overly constrain the expressive power of the language.

cs.LO↗

arXiv2020

Type-two polynomial-time and restricted lookahead

This paper provides an alternate characterization of type-two polynomial-time computability, with the goal of making second-order complexity theory more approachable. We rely on the usual oracle machines to model programs with subroutine calls. In contrast to previous results, the use of higher-order objects as running times is avoided, either explicitly or implicitly. Instead, regular polynomials are used. This is achieved by refining the notion of oracle-polynomial-time introduced by Cook. We impose a further restriction on the oracle interactions to force feasibility. Both the restriction as well as its purpose are very simple: it is well-known that Cook's model allows polynomial depth iteration of functional inputs with no restrictions on size, and thus does not guarantee that polynomial-time computability is preserved. To mend this we restrict the number of lookahead revisions, that is the number of times a query can be asked that is bigger than any of the previous queries. We prove that this leads to a class of feasible functionals and that all feasible problems can be solved within this class if one is allowed to separate a task into efficiently solvable subtasks. Formally put: the closure of our class under lambda-abstraction and application includes all feasible operations. We also revisit the very similar class of strongly polynomial-time computable operators previously introduced by Kawamura and Steinberg. We prove it to be strictly included in our class and, somewhat surprisingly, to have the same closure property. This can be attributed to properties of the limited recursion operator: It is not strongly polynomial-time computable but decomposes into two such operations and lies in our class.

cs.CC↗

arXiv2020

Unweighted linear congruences with distinct coordinates and the Varshamov--Tenengolts codes

In this paper, we first give explicit formulas for the number of solutions of unweighted linear congruences with distinct coordinates. Our main tools are properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions. Then, as an application, we derive an explicit formula for the number of codewords in the Varshamov--Tenengolts code $VT_b(n)$ with Hamming weight $k$, that is, with exactly $k$ $1$'s. The Varshamov--Tenengolts codes are an important class of codes that are capable of correcting asymmetric errors on a $Z$-channel. As another application, we derive Ginzburg's formula for the number of codewords in $VT_b(n)$, that is, $|VT_b(n)|$. We even go further and discuss connections to several other combinatorial problems, some of which have appeared in seemingly unrelated contexts. This provides a general framework and gives new insight into all these problems which might lead to further work.

cs.IT↗

arXiv2020

MMH* with arbitrary modulus is always almost-universal

Universal hash functions, discovered by Carter and Wegman in 1979, are of great importance in computer science with many applications. MMH$^*$ is a well-known $\triangle$-universal hash function family, based on the evaluation of a dot product modulo a prime. In this paper, we introduce a generalization of MMH$^*$, that we call GMMH$^*$, using the same construction as MMH$^*$ but with an arbitrary integer modulus $n>1$, and show that GMMH$^*$ is $\frac{1}{p}$-almost-$\triangle$-universal, where $p$ is the smallest prime divisor of $n$. This bound is tight.

cs.CR↗

arXiv2020

The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs

Let $\Z_n[i]$ be the ring of Gaussian integers modulo a positive integer $n$. Very recently, Camarero and Martínez [IEEE Trans. Inform. Theory, {\bf 62} (2016), 1183--1192], showed that for every prime number $p>5$ such that $p\equiv \pm 5 \pmod{12}$, the Cayley graph $\mathcal{G}_p=\textnormal{Cay}(\Z_p[i], S_2)$, where $S_2$ is the set of units of $\Z_p[i]$, induces a 2-quasi-perfect Lee code over $\Z_p^m$, where $m=2\lfloor \frac{p}{4}\rfloor$. They also conjectured that $\mathcal{G}_p$ is a Ramanujan graph for every prime $p$ such that $p\equiv 3 \pmod{4}$. In this paper, we solve this conjecture. Our main tools are Deligne's bound from 1977 for estimating a particular kind of trigonometric sum and a result of Lovász from 1975 (or of Babai from 1979) which gives the eigenvalues of Cayley graphs of finite Abelian groups. Our proof techniques may motivate more work in the interactions between spectral graph theory, character theory, and coding theory, and may provide new ideas towards the famous Golomb--Welch conjecture on the existence of perfect Lee codes.

cs.IT↗

arXiv2019

Type-two Iteration with Bounded Query Revision

Motivated by recent results of Kapron and Steinberg (LICS 2018) we introduce new forms of iteration on length in the setting of applied lambda-calculi for higher-type poly-time computability. In particular, in a type-two setting, we consider functionals which capture iteration on input length which bound interaction with the type-one input parameter, by restricting to a constant either the number of times the function parameter may return a value of increasing size, or the number of times the function parameter may be applied to an argument of increasing size. We prove that for any constant bound, the iterators obtained are equivalent, with respect to lambda-definability over type-one poly-time functions, to the recursor of Cook and Urquhart which captures Cobham's notion of limited recursion on notation in this setting.

cs.CC↗

arXiv2019

A generalization of Schönemann's theorem via a graph theoretic method

Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, where $a_1,\ldots,a_k,b,n$ ($n\geq 1$) are arbitrary integers, has a solution $\langle x_1,\ldots,x_k \rangle \in \Z_{n}^k$ with all $x_i$ distinct. So, it would be an interesting problem to give an explicit formula for the number of such solutions. Quite surprisingly, this problem was first considered, in a special case, by Schönemann almost two centuries ago(!) but his result seems to have been forgotten. Schönemann [{\it J. Reine Angew. Math.} {\bf 1839} (1839), 231--243] proved an explicit formula for the number of such solutions when $b=0$, $n=p$ a prime, and $\sum_{i=1}^k a_i \equiv 0 \pmod{p}$ but $\sum_{i \in I} a_i \not\equiv 0 \pmod{p}$ for all $\emptyset \not= I\varsubsetneq \lbrace 1, \ldots, k\rbrace$. In this paper, we generalize Schönemann's theorem using a result on the number of solutions of linear congruences due to D. N. Lehmer and also a result on graph enumeration. This seems to be a rather uncommon method in the area; besides, our proof technique or its modifications may be useful for dealing with other cases of this problem (or even the general case) or other relevant problems.

cs.DM↗

arXiv2017

On an almost-universal hash function family with applications to authentication and secrecy codes

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH$^*$, which was shown to be $Δ$-universal by Halevi and Krawczyk in 1997, is a well-known universal hash function family. We introduce a variant of MMH$^*$, that we call GRDH, where we use an arbitrary integer $n>1$ instead of prime $p$ and let the keys $\mathbf{x}=\langle x_1, \ldots, x_k \rangle \in \mathbb{Z}_n^k$ satisfy the conditions $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $t_1,\ldots,t_k$ are given positive divisors of $n$. Then via connecting the universal hashing problem to the number of solutions of restricted linear congruences, we prove that the family GRDH is an $\varepsilon$-almost-$Δ$-universal family of hash functions for some $\varepsilon<1$ if and only if $n$ is odd and $\gcd(x_i,n)=t_i=1$ $(1\leq i\leq k)$. Furthermore, if these conditions are satisfied then GRDH is $\frac{1}{p-1}$-almost-$Δ$-universal, where $p$ is the smallest prime divisor of $n$. Finally, as an application of our results, we propose an authentication code with secrecy scheme which strongly generalizes the scheme studied by Alomair et al. [{\it J. Math. Cryptol.} {\bf 4} (2010), 121--148], and [{\it J.UCS} {\bf 15} (2009), 2937--2956].

cs.CR↗

arXiv2016

On a restricted linear congruence

Let $b,n\in \mathbb{Z}$, $n\geq 1$, and ${\cal D}_1, \ldots, {\cal D}_{τ(n)}$ be all positive divisors of $n$. For $1\leq l \leq τ(n)$, define ${\cal C}_l:=\lbrace 1 \leqslant x\leqslant n \; : \; (x,n)={\cal D}_l\rbrace$. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence $x_1+\cdots +x_k\equiv b \pmod{n}$, with $κ_{l}=|\lbrace x_1, \ldots, x_k \rbrace \cap {\cal C}_l|$, $1\leq l \leq τ(n)$, is \begin{align*} \frac{1}{n}\mathlarger{\sum}_{d\, \mid \, n}c_{d}(b)\mathlarger{\prod}_{l=1}^{τ(n)}\left(c_{\frac{n}{{\cal D}_l}}(d)\right)^{κ_{l}}, \end{align*} where $c_{d}(b)$ is a Ramanujan sum. Some special cases and other forms of this problem have been already studied by several authors. The problem has recently found very interesting applications in number theory, combinatorics, computer science, and cryptography. The above explicit formula generalizes the main results of several papers, for example, the main result of the paper by Sander and Sander [J. Number Theory {\bf 133} (2013), 705--718], one of the main results of the paper by Sander [J. Number Theory {\bf 129} (2009), 2260--2266], and also gives an equivalent formula for the main result of the paper by Sun and Yang [Int. J. Number Theory {\bf 10} (2014), 1355--1363].

math.NT↗

arXiv2016

Restricted linear congruences

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, with $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $a_1,t_1,\ldots,a_k,t_k, b,n$ ($n\geq 1$) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of $a_i=t_i=1$ $(1\leq i \leq k)$. Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [{\it Duke Math. J.} {\bf 39} (1972), 521--527] gave a nice explicit formula for the number of such solutions when $(a_1,\ldots,a_k)=t_i=1$ $(1\leq i \leq k)$. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.

math.NT↗

arXiv2016

Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch, Ramgoolam, and Wen [Nuclear Phys.\,B {\bf 870} (2013), 530--581] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an `equivalent' form of the famous Harvey's theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions.

math.CO↗