Multiplicative Complexity of Autosymmetric Functions: Theory and Applications to Security

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.

Secure multi-party computation (SMC) is a cryptographic protocol that allows participants to compute the desired output without revealing their inputs. A variety of results related to increasing the efficiency of SMC protocol have been reported, and thus, SMC can be used in various applications. With the SMC protocol in smart grids, it becomes possible to obtain information for load balancing and various statistics, without revealing sensitive user information. To prevent malicious users from tampering with input values, SMC requires cheater detection. Several studies have been conducted on SMC with cheater detection, but none of these has been able to guarantee the fairness of the protocol. In such cases, only a malicious user can obtain a correct output prior to detection. This can be a critical problem if the result of the computation is real-time information of considerable economic value. In this paper, we propose a fair and secure multi-party computation protocol, which detects malicious parties participating in the protocol before computing the final output and prevents them from obtaining it. The security of our protocol is proven in the universal composability framework. Furthermore, we develop an enhanced version of the protocol that is more efficient when computing an average after detecting cheaters. We apply the proposed protocols to a smart grid as an application and analyze their efficiency in terms of computational cost.

It is established that secure computation can be achieved by using a deck of physical cards. Almost all existing card-based protocols are based on a specific deck of cards. In this study, we design card-based protocols that are executable using any deck of cards (e.g., playing cards, UNO, and trading cards). Specifically, we construct a card-based protocol for any Boolean function based on any deck of cards. As corollaries of our result, a standard deck of playing cards (having 52 cards) enables secure computation of any 22-variable Boolean function, and UNO (having 112 cards) enables secure computation of any 53-variable Boolean function.

Secure multi-party computations for privacy-preserving machine learning

Secure Computation (SC) enables secure distributed computation of arbitrary functions of private inputs. It has many useful applications, e.g. benchmarking or auctions. Several general protocols for SC have been proposed and recently been implemented in a number of compilers and frameworks. These compilers or frameworks implement one general SC protocol and then require the programmer to implement the function he wants the protocol to compute. Performance remains a challenge for this approach and it has been realized early on that special protocols for important problems can deliver superior performance. In this paper we propose a new intermediate language (L1) for optimizing SC compilers which enables efficient implementation of special protocols potentially mixing several general SC protocols. We show by three case studies - one for computation of the median, one for weighted average, one for division - that special protocols and mixed-protocol implementations in our language L1 can lead to superior performance. Moreover, we show that only a combined view on algorithm and cryptographic protocol can discover SCs with best run-time performance.

Secure multi-party computation (MPC) allows two or more distrusting parties to jointly evaluate a function over private inputs. For a long time considered to be a purely theoretical concept, MPC transitioned into a practical and powerful tool to build privacy-enhancing technologies. However, the practicality of MPC is hindered by the difficulty to implement applications on top of the underlying cryptographic protocols. This is because the manual construction of efficient applications, which need to be represented as Boolean or arithmetic circuits, is a complex, error-prone, and time-consuming task. To facilitate the development of further privacy-enhancing technology, multiple compilers have been proposed that create circuits for MPC. Yet, almost all presented compilers only support domain specific languages or provide very limited optimization methods. In this work (this is an extended and revised version of the paper ‘Secure Two-party Computations in ANSI C’ (Holzer et al., in: ACM CCS, 2012) that reflects the progress in secure computation and describes the current optimization tool chain of CBMC-GC) we describe our compiler CBMC-GC that implements a complete tool chain from ANSI C to circuit. Moreover, we give a comprehensive overview of circuit minimization techniques, which we have identified and adapted for the creation of efficient circuits for MPC. With the help of these techniques, our compilation approach allows for a high level of abstraction from the cryptographic primitives used in MPC protocols, as well as the complex design of digital circuits. By using the model checker CBMC as a compiler frontend, we illustrate the link between MPC, formal methods, and digital logic design. Our experimental results illustrate the effectiveness of the implemented optimizations techniques for various example applications. In particular, compared with other state-of-the-art compilers, we show that CBMC-GC compiles circuits from the same source code that are up to four times smaller.

Multiplicative complexity of vector valued Boolean functions

After while the “Human Genome Project” proposes, the people complete measures the foreword plan after the multi-gene genome team, also starts to change to these genes and their reciprocity function understanding research. Varieties of gene regulation Boolean networks algorithms have been proposed of the gene expression profiles, however, the problem of noise could always be found in creating a Boolean network. Due to gene expression data are always noisy. In this paper, it show that after the Boolean networks logic function are learned from noisy data, some noise in the Boolean function could be restructure Karnaugh Maps. It could find logic relationships between protein and protein and restructure protein logic networks. It find logic relationship among proteins as well as COGs (clusters of orthogous groups) and build the logic network of protein.

Algebraic immunity is an important cryptographic property for Boolean functions against algebraic attacks. Constructions of Boolean functions with the maximum algebraic immunity (MAI Boolean functions) by using univariate polynomial representation of Boolean functions over finite fields have received more and more attention. In this paper, how to obtain more MAI Boolean functions from a known MAI Boolean function under univariate polynomial representation is further investigated. The sufficient condition of Boolean functions having the maximum algebraic immunity obtained by changing a known MAI Boolean function under univariate polynomial representation is given. With this condition, more balanced MAI Boolean functions under univariate polynomial representation can be obtained. The algebraic degree and the nonlinearity of these Boolean functions are analyzed.

Random Boolean networks (RBNs) have been widely explored as models for understanding complex systems, in particular gene regulatory networks (GRNs). Stability (order) and instability (chaos), critical characteristics of any dynamic system, have naturally been a central focus in RBN research. Several measures have previously been introduced to assess stability, e.g., via damage spreading or via statistical properties of attractors such as their number, lengths or basin sizes. Undoubtedly, network topology plays an important role in shaping the dynamics of RBN. Another key factor influencing the dynamics is the Boolean functions (BFs) assigned to each node. In this work, we conduct a systematic investigation to examine the influence of five different classes of BFs on the dynamics of RBNs. By employing various dynamical stability measures, we show that biologically meaningful BFs consistently drive the dynamics toward the stable regime compared to random BFs. This observation holds across networks with varying size, connectivity, and degree distributions. Additionally, we show that the change in the values of our stability measures with increasing connectivity or size of RBNs is different across the BF classes. For most observables, random BFs change the values significantly, pushing the dynamics toward a more chaotic regime with increasing connectivity or network size. In contrast, biologically meaningful BFs show very minimal fluctuations for most stability measures. These findings emphasize the advantage of restricting to biologically meaningful classes in the reconstruction and modeling of biological systems within Boolean framework.

The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n−Hℕ(n), where Hℕ(n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexity of symmetric functions and fractals derived from the parity of binomial coefficients.

Observability of Boolean networks uniquely determines an initial condition of Boolean networks with a given output sequence. The objective of this study is to simplify the observability determination of Boolean networks and to offer a strategy to reconstruct states of Boolean networks with partial states' measurement. For a given Boolean dynamics and its measurement function, we extracted implicit useful information and deleted the redundant information based on its boolean dynamics and measurement function. Then, the extracted information was transferred to an undirected graph. We derived the necessary and sufficient condition of observability determination based on the corresponding undirected graph. Such necessary and sufficient condition significantly reduced the computational cost comparing with current methods. Further, we proposed a design method to determine the measurement function and reconstruct the unmeasured states. The effectiveness of the method was illustrated with an example. In conclusion, our result established an effective tool to determine the observability for Boolean networks with arbitrary number of nodes and Boolean functions, which further provided a novel way to allocate sensors for partial state measurement and to reconstruct the entire states.

Secure computation (SC) are cryptographic protocols that enable multiple parties to perform a joint computation while retaining the privacy of their inputs. It is current practice to evaluate the performance of SC protocols using complexity approximations of computation and communication. Due to the disparate complexity measures and constants this approach fails at reliably predicting the performance. We contribute a performance model (PM) for forecasting run-times of secure two-party computations. We show the correctness of our PM by an empirical study on the problem of secure division which is relevant for many real world SCs, e.g., k-means clustering or supply chain optimization. We show that our PM can be used to make an optimal selection of an algorithm and cryptographic protocol combination, as well as to determine the implicit security tradeoffs. The predictions of our PM can be used to design or select more efficient or more secure protocols.

Secure Multi-party Computation (SMC) protocols are exploited to perform collaborative computation of a function between two or more parties while keeping the privacy of the private inputs and sharing the computed result only. The Garbled Circuit (GC) protocol, proposed by Yao, is one of the possible approaches to solve the SMC problem, based on the evaluation of the Boolean Circuit representing the given function. Recently, the question to improve efficiency in secure multi-party computation has gained much interest. One of the proposed techniques to increase the efficiency of the GC protocol is based on the reduction of the number of non-XOR gates in the Boolean circuit, since the evaluation of XOR gates have no cost for the execution of the whole protocol. The aim of this work is to define a post-processing procedure that, given an optimized GC, decreases the number of non-XOR gates by transforming some parts of the circuit. The strategy is based on the fact that some gates behave as XORs apart from one output and then, if that input never occurs, those gates can be replaced by a XOR without changing the output of the overall network. The technique we propose is based on the analysis of the GC by using Ordered Binary Decision Diagrams (OBDD) representation. We present the application of our technique to some standard circuits to show the effectiveness of our proposal.

In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks’ classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed 3max(a,b) in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation.