Multiplicative complexity of vector valued Boolean functions

The authors theoretically demonstrate the implementation of a low-power 4-state universal logic gate (NOR) using a linear array of three dipole-coupled magnetostrictive-piezoelectric multiferroic nanomagnets (e.g. Ni/PZT) with biaxial magnetocrystalline anisotropy. The two peripheral nanomagnets in the array encode the 4-state input bits in their magnetization orientations and the central nanomagnet's magnetization orientation represents the output bit. Numerical simulations are performed to confirm that the 4-state output bit is the Boolean NOR function of the two 4-state inputs bits when the array reaches its ground state. A voltage pulse alternating between −0.2 and +0.2 V, applied to the piezoelectric layer of the central nanomagnet, generates alternating tensile and compressive stress in its magnetostrictive layer. This drives the array to the correct ground state where dipole interaction between the magnets ensures that the output is the NOR function of the input. For the system considered, the gate operation is executed while dissipating only ∼33 000 kT (0.138 fJ) of energy.

The strict avalanche criterion was introduced by Webster and Tavares [] in order to combine the ideas of completeness and the avalanche effect. A cryptographic transformation is complete if each output bit depends on all the input bits, and it exhibits the avalanche effect if an average of one half of the output bits change whenever a single input bit is complemented. To fulfil the strict avalanche criterion, each output bit should change with probability one half whenever a single input bit is complemented. This means, in particular, that there is no good lower order (fewer bits) approximation to the function. This is clearly a desirable cryptographic property since such an approximation would enable a corresponding reduction in the amount of work needed for an exhaustive search.

4-bit Linear Relations play an important role in Cryptanalysis of 4-bit Bijective S-Boxes. Count of existence of all 4-bit Linear Relations, for all of 16 input and output 4-bit bit patterns of 4-bit Bijective S-Boxes said as S-Boxes has been reported in Linear Cryptanalysis of 4-bit S-Boxes. In this paper a brief review of this cryptanalytic method for 4-bit S-Boxes has been introduced in a very lucid and conceptual manner. A new Analysis to search for the existing Linear Approximations among the input Boolean Functions (BFs) and output BFs of a particular 4-bit S-Box has also been introduced in this paper. The search is limited to find the existing Linear Relations or Approximations in the contrary to count the number existence among all 16 4-bit input and output bit patterns for all possible linear approximations.

Quantum technology application is a rapidly developing discipline in future-generation research. It has grown its appeal due to several unique aspects which cannot be encountered in classical domains. Quantum properties can be used to create powerful and effective technology. The processing capability of a quantum computer is triggered by managing quantum particles in a quantum computer, which improves transformation in many domains. Quantum logic operations have the significant benefit of being reversible. Unlike classical irreversible processing, quantum logic gates are reversible and unitary, allowing input bits to be recovered via a one-to-one connection between input and output bits. In QCADesigner, as well as in the Quantum environment, different reversible logic gates may be constructed. Some other promising features include Superposition (Quantum Parallelism), Entanglement, etc. This study uses IBM quantum to create the QCA-based Majority voter gate.

This paper considers the basic $\mathcal{PULL}$ model of communication, in which in each round, each agent extracts information from few randomly chosen agents. We seek to identify the smallest amount of information revealed in each interaction (message size) that nevertheless allows for efficient and robust computations of fundamental information dissemination tasks. We focus on the Majority Bit Dissemination problem that considers a population of $n$ agents, with a designated subset of source agents. Each source agent holds an input bit and each agent holds an output bit. The goal is to let all agents converge their output bits on the most frequent input bit of the sources (the majority bit). Note that the particular case of a single source agent corresponds to the classical problem of Broadcast. We concentrate on the severe fault-tolerant context of self-stabilization, in which a correct configuration must be reached eventually, despite all agents starting the execution with arbitrary initial states. We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property that uses $\ell$-bits messages to one that uses only $\log \ell$-bits messages, while paying only a small penalty in the running time. By applying this compiler recursively we then obtain a self-stabilizing Clock Synchronization protocol, in which agents synchronize their clocks modulo some given integer $T$, within $\tilde O(\log n\log T)$ rounds w.h.p., and using messages that contain $3$ bits only. We then employ the new Clock Synchronization tool to obtain a self-stabilizing Majority Bit Dissemination protocol which converges in $\tilde O(\log n)$ time, w.h.p., on every initial configuration, provided that the ratio of sources supporting the minority opinion is bounded away from half. Moreover, this protocol also uses only 3 bits per interaction.

This work presents the coder and decoder of block mBnB of the type 1B2B or Manchester. In the coder 1B2B each block/word of 1 input bit is coded in the block / word of 2 output bits. In the decoder 1B2B happen the inverse, each block of 2 input bits is newly converted in the original block of 1 output bit. The coder injects in the transmission line a number of 1’s exactly equal to the number of 0’s, what guarantees a DC constant component and maximizes the transitions number. The objective is to implement the pair coder and decoder 1B2B so that it improves the transmission quality and increases the information security. Keywords: Block codes, Digital systems, Transmission lines

With the growing availability of experimental loophole-free Bell tests1–5, it has become possible to implement a new class of device-independent random number generators whose output can be certified6,7 to be uniformly random without requiring a detailed model of the quantum devices used8–10. However, all these experiments require many input bits to certify a small number of output bits, and it is an outstanding challenge to develop a system that generates more randomness than is consumed. Here we devise a device-independent spot-checking protocol that consumes only uniform bits without requiring any additional bits with a specific bias. Implemented with a photonic loophole-free Bell test, we can produce 24% more certified output bits (1,181,264,237) than consumed input bits (953,301,640). The experiment ran for 91.0 h, creating randomness at an average rate of 3,606 bits s–1 with a soundness error bounded by 5.7 × 10−7 in the presence of classical side information. Our system allows for greater trust in public sources of randomness, such as randomness beacons11, and may one day enable high-quality private sources of randomness as the device footprint shrinks. Device-independent randomness expansion is demonstrated in an experiment that is secure in the presence of a classical eavesdropper who does not share any entanglement with the setup.

Non-local games are widely studied as a model to investigate the properties of quantum mechanics as opposed to classical mechanics. In this paper, we consider a subset of non-local games: symmetric XOR games of n players with 0-1 valued questions. For this class of games, each player receives an input bit and responds with an output bit without communicating to the other players. The winning condition only depends on XOR of output bits and is constant w.r.t. permutation of players. We prove that for almost any n-player symmetric XOR game the entangled value of the game is Theta((sqrt(ln(n)))/(n^{1/4})) adapting an old result by Salem and Zygmund on the asymptotics of random trigonometric polynomials. Consequently, we show that the classical-quantum gap is Theta(sqrt(ln(n))) for almost any symmetric XOR game.

We obtain the first deterministic extractors for sources generated (or sampled) by small circuits of bounded depth. Our main results are: (1) We extract k(k/nd) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1)</sup> bits with exponentially small error from n-bit sources of min-entropy k that are generated by functions f : {0,1} → {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> where each output bit depends on ≤ d input bits. In particular, we extract from NC sources, corresponding to d = O(1). (2) We extract k(k/n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+γ</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1)</sup> bits with super-polynomially small error from ri-bit sources of min-entropy k that are generated by poly(n)-size AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</sup> circuits, for any γ >; 0. As our starting point, we revisit the connection by Trevisan and Vadhan (FOCS 2000) between circuit lower bounds and extractors for sources generated by circuits. We note that such extractors (with very weak parameters) are equivalent to lower bounds for generating distributions (FOCS 2010; with Lovett, CCC 2011). Building on those bounds, we prove that the sources in (1) and (2) are (close to) a convex combination of high-entropy "bit-block" sources. Introduced here, such sources are a special case of affine ones. As extractors for (1) and (2) one can use the extractor for low-weight affine sources by Rao (CCC 2009). Along the way, we exhibit an explicit boolean function b : {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> → {0,1} such that poly(n)-size AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</sup> circuits cannot generate the distribution (Y, b(Y)), solving a problem about the complexity of distributions. Independently, De and Watson (RANDOM 2011) obtain a result similar to (1) in the special case d = o(lg n).

Multiplicative complexity is a complexity measure defined as the minimum number of AND gates required to implement a given primitive by a circuit over the basis (AND, XOR, NOT). Implementations of cyphers with a small number of AND gates are preferred in protocols for fully homomorphic encryption, multiparty computation and zero-knowledge proofs. Fischer and Peralta (2002) computed the number of n-variable Boolean functions with multiplicative complexity 1. In this paper, we study Boolean functions that can be constructed with two AND gates. By characterising the structure of these functions in terms of affine equivalence relations, we provide a closed-form formula for the number of Boolean functions with multiplicative complexity 2.

In this paper we study the second preimage resistance of Hamsi-256, a second round SHA-3 candidate. We show that it is possible to find affine equations between some input bits and some output bits on the 3-round compression function. This property enables an attacker to find pseudo preimages for the Hamsi-256 compression function. The pseudo preimage algorithm can be used to find second preimages of the digests of messages M with complexity 2251.3, which is lower than the best generic attacks when M is short.

We obtain the first deterministic extractors for sources generated (or sampled) by small circuits of bounded depth. Our main results are: (1) We extract k(k/nd) O(1) bits with exponentially small error from n-bit sources of min-entropy k that are generated by functions f: {0, 1} ℓ → {0, 1} n where each output bit depends on ≤ d input bits. In particular, we extract from NC 0 sources, corresponding to d = O(1). (2) We extract k(k/n 1+γ) O(1) bits with super-polynomially small error from n-bit sources of min-entropy k that are generated by poly(n)-size AC 0 circuits, for any γ&amp;gt; 0. As our starting point, we revisit the connection by Trevisan and Vadhan (FOCS 2000) between circuit lower bounds and extractors for sources generated by circuits. We note that such extractors (with very weak parameters) are equivalent to lower bounds for generating distributions (FOCS 2010; with Lovett, CCC 2011). Building on those bounds, we prove that the sources in (1) and (2) are (close to) a convex combination of high-entropy “bit-block ” sources. Introduced here, such sources are a special case of affine ones. As extractors for (1) and (2) one can use the extractor for low-weight affine sources by Rao (CCC 2009). Along the way, we exhibit an explicit boolean function b: {0, 1} n → {0, 1} such that poly(n)-size AC 0 circuits cannot generate the distribution (Y, b(Y)), solving a problem about the complexity of distributions. Independently, De and Watson (RANDOM 2011) obtain a result similar to (1) in the special case d = o(lg n). Supported by NSF grant CCF-0845003.

In this study the authors propose a new multivariate hash function with HAsh Iterative FrAmework framework which we call the hash function quadratic polynomials multiplying linear polynomials (QML). The new hash function is made of cubic polynomials which are the products of quadratic polynomials and linear polynomials. The authors design the quadratic-polynomial part of the compression function based on the centre map of the multivariate public key cryptosystem Matsumoto-Imai cryptosystem (MI). The hash function QML can keep the three cryptography properties and be immune to the pre-image attack, second pre-image attack, collision attack, differential attack and algebraic attack. The required memory storage is about 50% of the one which is built of the cubic polynomials and their coefficients are random. On the avalanche effect, by experiments the authors get the result that about one half of the output bits are different when one input bit is changed randomly. The one-round diffusion of the hash function QML is twice of that of Blake. Also the authors simplify the matrixes of the new hash function, analyse the rationality and show the comparable data. Finally, the authors give the advice to the parameters of the new hash function and summarise the paper.

We establish new hardness amplification results for one-way functions in which each input bit influences only a small number of output bits (a.k.a. input-local functions). Our transformations differ from previous ones in that they approximately preserve input locality and at the same time retain the input size of the original function.Let f : {0, 1}n → {0, 1}m be a one-way function with input locality d, and suppose that f cannot be inverted in time \(\exp(\tilde{O}(\sqrt{n}\cdot d))\) on an ε-fraction of inputs. Our main results can be summarized as follows: If f is injective then it is equally hard to invert f on a (1 − ε)-fraction of inputs. If f is regular then there is a function g: {0, 1}n → {0, 1}m + O(n) that is d + O(log3 n) input local and is equally hard to invert on a (1 − ε)-fraction of inputs. A natural candidate for a function with small input locality and for which no sub-exponential time attacks are known is Goldreich’s one-way function. To make our results applicable to this function, we prove that when its input locality is set to be d = O(logn) certain variants of the function are (almost) regular with high probability.In some cases, our techniques are applicable even when the input locality is not small. We demonstrate this by extending our first main result to one-way functions of the “parity with noise” type.Keywordsone-way functioninput localityhardness amplificationparity with noise

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.