Improved low-degree testing and its applications

Architects of the information society: Thirty-five years of the laboratory for computer science at MIT: By Simson L. Garfinkel, edited by Hal Abelson. MIT Press, Cambridge, MA. (1999). 72 pages. $20.00

A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof, and in return is allowed to err with some bounded probability. The probability that the verifier accepts a false proof is called the soundness error, and is an important parameter of a PCP system that one seeks to minimize. Constructing PCPs with sub-constant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability. In this work we construct such PCP verifiers, i.e., PCPs that make only two queries and have sub-constant soundness error. Our construction can be viewed as a combinatorial alternative to the {}``manifold vs. point'' construction, which is the only construction in the literature for this parameter range. The {}``manifold vs. point'' PCP is based on a low degree test, while our construction is based on a direct product test. Our construction of a PCP is based on extending the derandomized direct product test of Impagliazzo, Kabanets and Wigderson (STOC 09) to a derandomized parallel repetition theorem. More accurately, our PCP construction is obtained in two steps. We first prove a derandomized parallel repetition theorem for specially structured PCPs. Then, we show that any PCP can be transformed into one that has the required structure, by embedding it on a de-Bruijn graph.

This paper presents a new probabilistie primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.

Robert L. Rees, B.S., M.S., is senior systems analyst at the Laboratory of Computer Science, Massachusetts General Hospital, Boston, Mass

Physically-based modeling: past, present, and future

Polynomial-time safe and unsafe approximations for intractable sets were introduced by Meyer and Paterson [Technical Report TM-126, Laboratory for Computer Science, MIT, Cambridge, MA, 1979] and Yesha [SIAM J. Comput., 12 (1983), pp. 411--425], respectively. The question of which sets have optimal safe and unsafe approximations has been investigated extensively. Duris and Rolim [Lecture Notes in Comput. Sci. 841, Springer-Verlag, Berlin, New York, 1994, pp. 38--51] and Ambos-Spies [Proc. 22nd ICALP, Springer-Verlag, Berlin, New York, 1995, pp. 384--392] showed that the existence of optimal polynomial-time approximations for the safe and unsafe cases is independent. Using the law of the iterated logarithm for p-random sequences (which has been recently proven in [Proc. 11th Conf. Computational Complexity, IEEE Computer Society Press, Piscataway, NJ, 1996, pp. 180--189]), we extend this observation by showing that both the class of polynomial-time $\Delta$-levelable sets and the class of sets which have optimal polynomial-time unsafe approximations have p-measure 0. Hence typical sets in E (in the sense of p-measure) do not have optimal polynomial-time unsafe approximations. We will also establish the relationship between resource bounded genericity concepts and the polynomial-time safe and unsafe approximation concepts.

Hegedűs’s lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field F of characteristic p > 0 and for q a power of p, the lemma says that any multilinear polynomial P∈ F[x 1,…,x n ] of degree less than q that vanishes at all points in {0,1} n of Hamming weight k∈ [q,n−q] must also vanish at all points in {0,1} n of weight k + q. This lemma was used by Hegedűs (2009) to give a solution to Galvin’s problem, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrubes, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-2 threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Hegedűs’s lemma. Informally, this version says that if a polynomial of degree o(q) vanishes at most points of weight k, then it vanishes at many points of weight k+q. We prove this lemma and give the following three different applications. 1. Degree lower bounds for the coin problem: The δ-Coin Problem is the problem of distinguishing between a coin that is heads with probability ((1/2) + δ) and a coin that is heads with probability 1/2. We show that over a field of positive (fixed) characteristic, any polynomial that solves the δ-coin problem with error e must have degree Ω(1/δlog(1/e)), which is tight up to constant factors. 2. Probabilistic degree lower bounds: The Probabilistic degree of a Boolean function is the minimum d such that there is a random polynomial of degree d that agrees with the function at each point with high probability. We give tight lower bounds on the probabilistic degree of every symmetric Boolean function over positive (fixed) characteristic. As far as we know, this was not known even for some very simple functions such as unweighted Exact Threshold functions, and constant error. 3. A robust version of the combinatorial result of Hegedűs (2009) mentioned above.

A proof system for a language L is a function f such that Range( f ) is exactly L . In this article, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is proof systems that can be computed by bounded fanin circuits of constant depth (NC 0 ) or of O (log log n ) depth but with O (1) alternations (poly log AC 0 ). Each output bit depends on very few input bits; thus such proof systems correspond to a kind of local error correction on a theorem-proof pair. We identify exactly how much power we need for proof systems to capture all regular languages. We show that all regular languages have poly log AC 0 proof systems, and from a previous result (Beyersdorff et al. [2011a], where NC 0 proof systems were first introduced), this is tight. Our technique also shows that M aj has poly log AC 0 proof system. We explore the question of whether T aut has NC 0 proof systems. Addressing this question about 2TAUT, and since 2TAUT is closely related to reachability in graphs, we ask the same question about Reachability. We show that if Directed reachability has NC 0 proof systems, then so does 2TAUT. We then show that both Undirected Reachability and Directed UnReachability have NC 0 proof systems, but Directed Reachability is still open. In the context of how much power is needed for proof systems for languages in NP, we observe that proof systems for a good fraction of languages in NP do not need the full power of AC 0 ; they have SAC 0 or coSAC 0 proof systems.

Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if f ( x ) f(x) and g ( x ) g(x) are integral polynomials of degree n ≥ 2 n \geq 2 and p is a prime exceeding ( n 2 − 3 n + 4 ) 2 {({n^2} - 3n + 4)^2} for which f and g are both permutation polynomials of the finite field F p {F_p} , then their difference h = f − g h = f - g cannot be such that h ( x ) = c x h(x) = cx for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in F p {F_p} and t is the degree of h, then t ≥ 3 n / 5 t \geq 3n/5 and, provided t ≤ n − 3 t \leq n - 3 , t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.

Differential uniformity of permutation polynomials has been studied intensively in recent years due to the differential cryptanalysis of S-boxes. The boomerang attack is a variant of differential cryptanalysis which combines two differentials for the upper part and the lower part of the block cipher. The boomerang uniformity measures the resistance of block ciphers to the boomerang attack. In this paper, by using the resultant elimination method, we study the boomerang uniformity of normalized permutation polynomials of the low degree over finite fields. As a result, we determine the boomerang uniformity of all normalized permutation polynomials of degree up to six over the finite field $${\mathbb {F}}_{q}$$.

We study gossip algorithms for the rumor spreading problem which asks one node to deliver a rumor to all nodes in an unknown network. We present the first protocol for any expander graph $G$ with $n$ nodes such that, the protocol informs every node in $O(\log n)$ rounds with high probability, and uses $\tilde{O}(\log n)$ random bits in total. The runtime of our protocol is tight, and the randomness requirement of $\tilde{O}(\log n)$ random bits almost matches the lower bound of $\Omega(\log n)$ random bits for dense graphs. We further show that, for many graph families, polylogarithmic number of random bits in total suffice to spread the rumor in $O(\mathrm{poly}\log n)$ rounds. These results together give us an almost complete understanding of the randomness requirement of this fundamental gossip process. Our analysis relies on unexpectedly tight connections among gossip processes, Markov chains, and branching programs. First, we establish a connection between rumor spreading processes and Markov chains, which is used to approximate the rumor spreading time by the mixing time of Markov chains. Second, we show a reduction from rumor spreading processes to branching programs, and this reduction provides a general framework to derandomize gossip processes. In addition to designing rumor spreading protocols, these novel techniques may have applications in studying parallel and multiple random walks, and randomness complexity of distributed algorithms.

In this chapter we study the basic notions from mathematics and computer science that we will be using throughout the book. Its purpose is not to give a comprehensive account of these concepts but, rather, to study them from the point of view of their applications to cryptography, emphasizing the algorithmic aspects. For this task we will be using Maple as a tool to implement the relevant algorithms and to experiment with them. Bearing in mind these premises, we include here the required concepts and results from probability, complexity theory and modular arithmetic, including quadratic residues and modular square roots, and we also give a brief introduction to the relevant aspects of group theory and finite fields. Thus, for readers already familiar with these subjects, this chapter will hopefully serve as a refresher of the topics mentioned in the Preface (with the exception of linear algebra which is not included here) and, at the same time, as an introduction to the use of Maple to explore the algorithms discussed. Even for readers not familiar with these subjects we hope that sufficient information is included here to allow them to profitably read the rest of the book. We include proofs of most of the algorithmic and mathematical results mentioned, with pointers to the literature for the reader who wishes to go deeper on these topics and, at the same time, we will be introducing here a great deal of the notation used throughout the book.

In this paper we discuss the semantics of a simple block-structured programming language which allows sharing or aliasing. Sharing, which arises naturally in procedural languages which permit certain forms of parameter passing, has typically been regarded as problematical for the semantic treatment of a language. Difficulties have been encountered in both denotational and axiomatic treatments of sharing in the literature. Nevertheless, we find that it is possible to define a clean and elegant formal semantics for sharing. The key to our success is the choice of semantic model; we show that conventional approaches based on locations are less than satisfactory for the purposes of reasoning about partial correctness, and that in a well defined sense locations are unnecessary.We begin by defining a denotational semantics for our programming language. The semantic model is not based on locations, but instead uses an abstract sharing relation on identifiers to represent the notion of aliasing, and uses an abstract state with a stack-like structure to capture the semantics of blocks and accurately model the scope rules. The semantics is shown to be fully abstract with respect to partial correctness properties, in contrast to conventional location-based models. This means that the semantics identifies terms if and only if they induce identical partial correctness behaviour in all program contexts. This property usually fails for location-based semantics because in such models it is possible to distinguish between terms on the basis of their effect on individual locations, which has no bearing on partial correctness.We believe that axiomatic reasoning about program behaviour should be based directly on a semantic model specifically tailored for that purpose; full abstraction with respect to an appropriate behavioural notion is a formal criterion against which to judge the suitability of a semantics. The structure of a semantic model should be used directly to suggest the structure of an assertion language for expressing program properties. With this in mind, we build a Hoare-style (syntax-directed) proof system for partial correctness properties of our programming language, and we prove soundness and relative completeness of this system. The proof system is built up in a hierarchical manner which reflects the syntactic and semantic structure of the programming language. We first design a proof system for declarations, and then use it in building proof rules for commands. We claim that our proof rules are conceptually simpler to understand than other rules proposed in the literature for aliasing, without losing any expressive power. We show, for example, that it is possible to define a “generic” inference rule for blocks which is uniformly applicable to3 blocks headed by different forms of declaration. The important point here is that, unlike most of the proof systems for these constructs in the literature, we do not have to design a separate rule for blocks for each possible form of declaration. This results in greater flexibility and adaptability in our proof system. We demonstrate that some well known rules from the literature for blocks can be derived in our system.

Let \(\mathbb{F}_{q}\) be a finite field, where \(q\) is an odd prime such that \(q>3\). Let \(f\left(t\right) =t^{3}-t\) \(\in \mathbb{F}_{q}\left[ t\right]\) be a polynomial of degree 3. For \(\lambda \neq 0\) in \(\mathbb{F}_{q}\), consider families of elliptic curves \(\left\{ E_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast}}\) and \(\left\{ H_{\lambda }\right\} _{\lambda \in \mathbb{F}_{q}^{\ast }}\) defined respectively by\[v^{2}=\lambda f(u)\text{ and }f\left( v\right) =\lambda f(u).\]In this paper, I investigate the relation between the rational points over finite field on \(\left\{E_{\lambda }\left( \mathbb{F}_{q}\right) \right\}_{\lambda \in \mathbb{F}_{q}^{\ast }}\) and \(\left\{ H_{\lambda }\left(\mathbb{F}_{q}\right) \right\} _{_{\lambda \in \mathbb{F}_{q}^{\ast }}}\), and determine the number of rational points on both of these family of curves.

In this paper we develop new algorithms for factoring polynomials over finite fields by exploring an interesting connection between the algebraic factoring problem and the combinatorial problem of stable coloring of tournaments. We present an algorithm which can be viewed as a recursive refinement scheme through which most cases of polynomials are completely factored in deterministic polynomial time within the first level of refinement, most of the remaining cases are factored completely before the end of the second level refinement, and so on. The algorithm has average polynomial time complexity and (n \(^{\rm log{\it n}}\) logp)O(1) worst case complexity. Under a purely combinatorial conjecture concerning tournaments, the algorithm has worst case complexity (n \(^{\rm loglog{\it n}}\) logp)O(1). Our approach is also useful in reducing the amount of randomness needed to factor a polynomial completely in expected polynomial time. We present a random polynomial time algorithm for factoring polynomials over finite fields which requires only log p random bits. All these results assume the Extended Riemann Hypothesis.KeywordsPolynomial TimeZero DivisorStable ColoringPrimitive IdempotentFactoring PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.