The Hidden Subgroup Problem and Non-interactive Perfect Zero-Knowledge Proofs

In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum cryptography. We review the relationship between HSP and other computational problems, discuss an optimal solution method, and review results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in the proposed infinite group platforms are not yet known.

It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was...

Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity $O(n^{35/27})$ that is better than $O(n^{13/10})$ of the best previously known algorithm by Magniez et al.

In this chapter, we introduce some mathematical tools used throughout the rest of the textbook, to provide readers from different backgrounds with a good understanding of our notation and terminology. These mathematical tools include the binary system, Boolean functions, and sets of logic gates universal for classical computation. We then introduce the notion of a decision problem, a mathematical problem whose solution is either yes or no and discuss how a computation solves it by exploiting two physical resources: time and space. We conclude by presenting several examples of decision problems, including satisfiability, MAXCUT, traveling salesman, and the hidden subgroup problem

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

In this brief chapter we present a quantum algorithm, which can be seen as a generalization of Shor’s algorithm. We can here explain, in a more structural manner, why quantum computers can speed up some computational problems. The so-called Simon’s hidden subgroup problem can be stated in a general form as follows [18]: Input: A finite abelian group G and function ρ: G → R, where R is a finite set. Promise: There exists a nontrivial subgroup H ≤ G such that ρ is constant and distinct on each coset of H. Output: A generating set for H.

We study perfect zero-knowledge proofs ( Open image in new window ). Unlike statistical zero-knowledge, where many fundamental questions have been answered, virtually nothing is known about these proofs.We consider reductions that yield hard and complete problems in the statistical setting. The issue with these reductions is that they introduce errors into the simulation, and therefore they do not yield analogous problems in the perfect setting. We overcome this issue using an error shifting technique. This technique allows us to remove the error from the simulation. Consequently, we obtain the first complete problem for the class of problems possessing non-interactive perfect zero-knowledge proofs ( Open image in new window ), and the first hard problem for the class of problems possessing public-coin Open image in new window proofs.We get the following applications. Using the error shifting technique, we show that the notion of zero-knowledge where the simulator is allowed to fail is equivalent to the one where it is not allowed to fail. Using our complete problem, we show that under certain restrictions Open image in new window is closed under the OR operator. Using our hard problem, we show how a constant-round, perfectly hiding instance-dependent commitment may be obtained (this would collapse the round complexity of public-coin Open image in new window proofs to a constant).

We prove a number of general theorems about CZK, the class of problems possessing computational zero knowledge proofs. Our results are unconditional, in contrast to most previous works on CZK which rely on the assumption that one-way functions exist. We establish several new characterizations of CZK, and use these characterizations to prove results such as: 1) Honest-verifier CZK equals general CZK. 2) Public-coin CZK equals private-coin CZK. 3) CZK is closed under union (and more generally, monotone formula closure). 4) CZK with imperfect completeness equals CZK with perfect completeness. 5) Any problem in CZK /spl cap/ NP can be proven in computational zero knowledge by a BPP/sup NP/ prover. 6) CZK with black-box simulators equals CZK with general, non-black-box simulators. The above equalities refer to the resulting class of problems (and do not necessarily preserve other efficiency measures such as round complexity). Our approach is to combine the conditional techniques previously used in the study of CZK with the unconditional techniques developed in the study of SZK, the class of problems possessing statistical zero knowledge proofs. To enable this combination, we prove that every problem in CZK can be decomposed into a problem in SZK together with a set of instances from which a one-way function can be constructed.

This paper introduces quantum analogues of non-interactive perfect and statistical zero-knowledge proof systems. Similar to the classical cases, it is shown that sharing randomness or entanglement is necessary for non-trivial protocols of non-interactive quantum perfect and statistical zero-knowledge. It is also shown that, with sharing EPR pairs a priori, the complexity class resulting from non-interactive quantum perfect zero-knowledge proof systems of perfect completeness has a natural complete promise problem. Using our complete promise problem, the Graph Non-Automorphism problem is shown to have a non-interactive quantum perfect zero-knowledge proof system.

This paper proves that the computational power of quantum interactive proof systems, with a double-exponentially small gap in acceptance probability between the completeness and soundness cases, is precisely characterized by EXP, the class of problems solvable in exponential time by deterministic Turing machines. This fact, and our proof of it, has implications concerning quantum and classical interactive proof systems in the setting of unbounded error that include the following: * Quantum interactive proof systems are strictly more powerful than their classical counterparts in the unbounded-error setting unless PSPACE=EXP, as even unbounded error classical interactive proof systems can be simulated in PSPACE. * The recent proof of Jain, Ji, Upadhyay, and Watrous (STOC 2010) establishing QIP=PSPACE relies heavily on the fact that the quantum interactive proof systems defining the class QIP have bounded error. Our result implies that some nontrivial assumption on the error bounds for quantum interactive proofs is unavoidable to establish this result (unless PSPACE=EXP). * To prove our result, we give a quantum interactive proof system for EXP with perfect completeness and soundness error 1-2^{-2^poly}, for which the soundness error bound is provably tight. This establishes another respect in which quantum and classical interactive proof systems differ, because such a bound cannot hold for any classical interactive proof system: distinct acceptance probabilities for classical interactive proof systems must be separated by a gap that is at least (single-)exponentially small. We also study the computational power of a few other related unbounded-error complexity classes.

This paper is devoted to the study of complexity of finding a special covering for a set, as well as to obtaining some important applications of special decomposition. We formulate the problem of existence of a special covering as a decision problem. To determine the complexity class in which this problem is located, we study the relationship between this problem and the Boolean satisfiability problem, treating them as formal languages. We prove that these problems are polynomially equivalent, which means that the problem of existence of a special covering for a set is an -complete problem. In this article we also introduce a new concept ‘Replaceable Subsets’. The properties of such subsets are used to fill in the missing elements needed to obtain a special set covering. It is proved that when searching for missing elements to fill a special covering, the order in which these elements are considered does not matter. This result is of great importance in the search for satisfiability of Boolean functions.

Unification is a basic operation in theorem proving, in type inference algorithms, and in logic programming languages such as Prolog. Prolog will play a major role in software development for the Fifth Generation project, and thus developing fast algorithms for unification is an important goal. This paper shows that the running time for a linear unification algorithm can often be improved substantially by use of parallel processing. The same is true for algorithms for some other complete problems in P, namely, the monotone circuit value problem and the path system accessibility problem. Previous theoretical work in computational complexity has suggested that these problems are not parallelizable; in practice this is not the case. To resolve this paradox, we introduce new complexity classes PC and PC* that capture the practical notion of parallelizability we discuss in this paper. We pose several open questions concerning PC and PC*.

A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A k-clique-colouring of a graph is a colouring of the vertices with at most k colours such that no clique is monochromatic. Defossez proved that the 2-clique-colouring of perfect graphs is a \(\Sigma_2^P\)-complete problem [J. Graph Theory 62 (2009) 139–156]. We strengthen this result by showing that it is still \(\Sigma_2^P\)-complete for weakly chordal graphs. We then determine a hierarchy of nested subclasses of weakly chordal graphs whereby each graph class is in a distinct complexity class, namely \(\Sigma_2^P\)-complete, \(\mathcal{NP}\)-complete, and \(\mathcal{P}\). We solve an open problem posed by Kratochvil and Tuza to determine the complexity of 2-clique-colouring of perfect graphs with all cliques having size at least 3 [J. Algorithms 45 (2002), 40–54], proving that it is a \(\Sigma_2^P\)-complete problem. We then determine a hierarchy of nested subclasses of perfect graphs with all cliques having size at least 3 whereby each graph class is in a distinct complexity class, namely \(\Sigma_2^P\)-complete, \(\mathcal{NP}\)-complete, and \(\mathcal{P}\).

The solutions of certain natural decision problems such as the halting problem and the boolean satisfiability problem contain large amounts of useful information about computation that is highly organized and readily available to efficient computational processes. Such problems are computationally useful. This dissertation investigates the complexity and distribution of these computationally useful problems. The main results of this dissertation are of the following three general types. (1) Useful problems contain highly organized information. (2) Very useful problems are so highly organized that they are unusually simple and hence rare. (3) Useful problems are, as a whole, not rare and thus are not necessarily simple. A result of type (1) is proven in Chapter 3. Bennett recently extended algorithmic information theory to include a notion of computational depth that appears to quantify the level of organization in binary strings and sequences. The main result of Chapter 3 states that every weakly useful sequence is strongly deep. (A sequence x is weakly useful if a non-negligible set of recursive problems are decidable within a fixed recursive time bound when given access to x.) Results of type (2) are presented in Chapters 4 and 5. These results say that the $\le\sbsp{\rm m}{\rm P}$-complete problems for E = DTIME(2$\sp{\rm linear}$) and the $\le\sbsp{\rm m}{\rm p/poly}$-complete problems for ESPACE = DSPACE(2$\sp{\rm linear}$) are unusually simple and hence rare. Complete problems are very useful because every problem in E or ESPACE is efficiently decidable when given access to one of these problems. Chapter 6 develops a result of type (3). This result says that the weakly $\le\sbsp{\rm m}{\rm P}$-complete problems for E and ESPACE are not rare and hence are not necessarily simple. Weakly complete problems are useful because every problem in a non-negligible subset of E or ESPACE is efficiently decidable when given access to one of these problems. The above results (and others along the way) are obtained through a systematic investigation of the measure-theoretic structure of complexity classes.

We describe parameterized complexity classes by means of classical complexity theory and descriptive complexity theory. For every classical complexity class we introduce a parameterized analogue in a natural way. In particular, the analogue of polynomial time is the class of all fixed-parameter tractable problems. We develop a basic complexity theory for the parameterized analogues of classical complexity classes and give, among other things, complete problems and logical descriptions. We then show that most of the well-known intractable parameterized complexity classes are not analogues of classical classes. Nevertheless, for all these classes we can provide natural logical descriptions.