Expand-and-Randomize: An Algebraic Approach to Secure Computation

Simple arithmetic processing: Surface form effects in a priming task

In this paper neural network implementation of the modified secret sharing scheme based on a matrix projection is offered. Transition from a finite simple Galois field to a complex field allows to reduce by 16 times memory size, necessary for storage of the precalculated constants. Implementation of the modified secret sharing scheme based on a matrix projection with use of the neural network of a finite ring for execution of modular arithmetical addition and multiplication operations in a finite field allows to save on average 30% of the device area and increases the speed of scheme’s work on average by 17%.

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order of a group, subgroups, along with the associated notions of automorphisms group of the dihedral groups, split extensions of groups and vector spaces arises from the varying properties of real and complex numbers. We must have used the Sylow theorems of this work when it's generalized. Here we discuss possible subgroups of a group in different types of order which will give us a practical knowledge to see the applications of the Sylow theorems. In algebraic structures, we deal with operations of addition and multiplication and in order structures, those of greater than, less than and so on. It is through the study of Sylow theorems that we realize the importance of some definitions as like as the exact sequences and split extensions of groups, Sylow p-subgroup and semi-direct product. Thus it has been found necessary and convenient to study these structures in detail. In situations, where it was found that a given situation satisfies the basic axioms of structure and having already known the properties of that structure. Finally, we find out possible subgroups of a group in different types of order for abelian and non-abelian cases.

Machine-Learning-as-a-Service has become increasingly popular, with Recommendation-as-a-Service as one of the representative examples. In such services, providing privacy protection for the users is an important topic. Reviewing privacy-preserving solutions which were proposed in the past decade, privacy and machine learning are often seen as two competing goals at stake. Though improving cryptographic primitives (e.g., secure multi-party computation (SMC) or homomorphic encryption (HE)) or devising sophisticated secure protocols has made a remarkable achievement, but in conjunction with state-of-the-art recommender systems often yields far-from-practical solutions. We tackle this problem from the direction of machine learning. We aim to design crypto-friendly recommendation algorithms, thus to obtain efficient solutions by directly using existing cryptographic tools. In particular, we propose an HE-friendly recommender system, refer to as CryptoRec, which (1) decouples user features from latent feature space, avoiding training the recommendation model on encrypted data; (2) only relies on addition and multiplication operations, making the model straightforwardly compatible with HE schemes. The properties turn recommendation-computations into a simple matrix-multiplication operation. To further improve efficiency, we introduce a sparse-quantization-reuse method which reduces the recommendation-computation time by $9\times$ (compared to using CryptoRec directly), without compromising the accuracy. We demonstrate the efficiency and accuracy of CryptoRec on three real-world datasets. CryptoRec allows a server to estimate a user's preferences on thousands of items within a few seconds on a single PC, with the user's data homomorphically encrypted, while its prediction accuracy is still competitive with state-of-the-art recommender systems computing over clear data. Our solution enables Recommendation-as-a-Service on large datasets in a nearly real-time (seconds) level.

An image encryption algorithm based on Kronecker inner product matrix over the finite field and adversarial neural network (ANN) is designed. The Kronecker inner product matrix transform and the ANN fulfil the tasks of confusion and diffusion simultaneously. In addition, the look-up table method is used to complete the addition and multiplication operations over GF(2 B ) finite fields, which can effectively improve the finite field computation speed while retaining its performance of non rounding errors. At the same time, relying on the secure hash function SHA-256 of the plain image to control the Logistic-Sine map greatly improves the diffusion and security of the encryption system. The simulation results verify that the proposed algorithm not only has high security and sufficient sensitivity, but also has a good resistance to various common attacks.

Signal processing and image processing applications require floating-point computation in digital circuit designs. The floating-point calculation is recommended in almost all digital design applications to improve the accuracy of results. Different mathematical operations use floating-point mathematical operations and can be used to perform calculations and different logical units. The proposed work sugestes a double-MAF unit that uses radix-8 for multiplication and uses a common addition block for addition and multiplication operations. Firstly, floating-point numbers are transformed into IEEE 754 format, next to both addition and multiplication calculations are achieved. Extracts the fractional part of a number and performs a calculation based on the exponential difference between the numbers. The proposed curriculum is designed with a parallel structure that first extracts the decimal and exponential values of the first unit and then performs multiplication and addition operations in the second block. At last, the output is made in the third block where normalization and zero detection are performed. The proposed approach is then compared to the baseline approach, which shows improvements in energy consumption and maximum common path delay. The results showed that the delay is reduced by about 57%.

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.

We study the question of protecting arithmetic circuits against additive attacks, which can add an arbitrary fixed value to each wire in the circuit. This extends the notion of algebraic manipulation detection (AMD) codes, which protect information against additive attacks, to that of AMD circuits which protect computation. We present a construction of such AMD circuits: any arithmetic circuit C over a finite field F can be converted into a functionally-equivalent randomized arithmetic circuit • C of size O(|C|) that is fault-tolerant in the following sense. For any additive attack on the wires of C, its effect on the output of C can be simulated, up to O(|C|/|F|) statistical distance, by an additive attack on just the input and output. Given a small tamper-proof encoder/decoder for AMD codes, the input and output can be protected as well. We also give an alternative construction, applicable to small fields (for example, to protect Boolean circuits against wire-toggling attacks). It uses a small tamper-proof decoder to ensure that, except with negligible failure probability, either the output is correct or tampering is detected. Our study of AMD circuits is motivated by simplifying and improving protocols for secure multiparty computation (MPC). Typically, securing MPC protocols against active adversaries is much more difficult than securing them against passive adversaries. We observe that in simple passive-secure MPC protocols for circuit evaluation, the effect of any active adversary corresponds precisely to an additive attack on the original circuit's wires. Thus, to securely evaluate a circuit C in the presence of active adversaries, it suffices to apply the passive-secure protocol to C. We use this methodology to simplify feasibility results and attain efficiency improvements in several standard MPC models.

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [ 7 ], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [ 25 ] and Kedlaya & Umans [ 16 ] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most \(d^{o(1)}\) where the degree of the input polynomial in every variable is less than d . They also stated the question of designing fast algorithms for the large variable case (i.e., \(n \notin d^{o(1)}\) ) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field \(\mathbb {F}_{q}\) of characteristic p , which evaluates an n -variate polynomial of degree less than d in each variable on N inputs in time \(\begin{equation*} \left((N + d^n)^{1 + o(1)}\text{poly}(\log q, d, n, p)\right), \end{equation*}\) provided that p is at most d o (1) , and q is at most (exp (exp (exp (...(exp ( d ))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n ∉ d o (1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: — We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [ 21 ] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. — We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program [ 26 ]. More precisely, for every fixed prime p , we show that for every constant ɛ > 0, and large enough n , the rank of any \(n \times n\) Vandermonde matrix V over the field \(\mathbb {F}_{p^a}\) can be reduced to ( n /exp (Ω (poly(ɛ)log 0.53 n ))) by changing at most n Θ (ɛ) entries in every row of V , provided a ≤ poly(log n ). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [ 9 ].

Secure multiparty computation refers to the problem where a number of users wish to securely compute a function on their inputs without revealing any unnecessary information. This dissertation focuses on the fundamental limits of secure summation under different constraints. We first focus on the minimal model of secure computation, in which two users each hold an input and wish to securely compute a function of their inputs at the server. We propose a novel scheme base on the algebraic structure of finite field and modulo ring of integers. Then we extend the minimal model of secure computation, in which K users wish to securely compute the sum of their inputs at the server. We prove a folklore result on the limits of communication cost and randomness cost. Then we characterized the optimal communication cost with user dropouts constraint, when some users may lose connection to the server and the server wishes to compute the sum of remaining inputs. Next, we characterize the optimal communication and randomness cost for symmetric groupwise keys and find the feasibility condition for arbitrary groupwise keys. Last, we study the secure summation with user selection, such that the server may select any subset of users to compute the sum of their inputs. This leads us to the MDS variable generation problem. We characterize the optimal individual key rate and the result is interestingly the harmonic number.

The purpose of the research in the NJIKI’s fundamental THEOREM-DEFINITION on fractions in the mathematical set ℚ and by extension in ℝ and ℂ and in order to construct some algebraic structures is about the proved EXISTENCE and the DEFINITION by NJIKI of two INNOVATIVE, IMPORTANT and TEACHABLE operations of addition or additive operations, in ℚ , marked ⊕ and + α,β , and taken as VECTORIAL, TRIANGULAR, of THREE or PROPORTIONAL operations and in order to make THEM not be different from the RATIONAL ONE, +, but to bring much more and new information on fractions, and, by extension in ℝ and ℂ . And the very NJIKI’s fundamental THEOREM-DEFINITION having many APPLICATIONS in the everyday life of the HUMAN BEINGS and without talking about computer sciences, henceforth being supplied with very interesting new ALGORITHMS. And as for the work done in the research, it will be waiting for its extension to be done after publication and along with the research results concerned.

Problems of secure communication and computation have been studied extensively in network models. Goldreich, Goldwasser, and Linial, Franklin and Yung, and Franklin and Wright have initiated the study of secure communication and secure computation in multi-recipient (broadcast) models. A “broadcast channel” (such as ethernet) enables one processor to send the same message—simultaneously and privately—to a fixed subset of processors. In their Eurocrypt ’98 paper, Franklin and Wright have shown that if there are n broadcast lines between a sender and a receiver and there are at most t malicious (Byzantine style) processors, then the condition n > t is necessary and sufficient for achieving efficient probabilisticly reliable and probabilisticly private communication. They also showed that if n > [3t=2] then there is an efficient protocol to achieve probabilisticly reliable and perfectly private communication. And they left open the question whether there exists an efficient protocol to achieve probabilisticly reliable and perfectly private communication when [3t=2] ≥ n > t. In this paper, by using a different authentication scheme, we will answer this question affirmatively and study related problems.KeywordsNetwork securityPrivacyPerfect secrecyReliability

Algebraic structures play a vital role in securing important data. These structures are utilized to construct the non-linear components of block ciphers. Since constructing non-linear components through algebraic structures is crucial for the confusion aspects of encryption schemes, relying solely on these structures can result in limited key spaces. To address this issue, a complex non-chain ring with combination of Henon map based algorithm is presented. This proposed work aims to enhance both security and key space by utilizing complex algebraic structures. In this study, we investigate the use of unit elements from non-chain rings, establishing a mapping from the group of units to Galois fields to enhance the algebraic robustness of the scheme. A substitution box is constructed from Galois field and is applied in image processing. For pixel mixing, the Henon map is employed, and the exclusive OR operation is performed using a key generated from image pixels and the substitution box. The algorithm’s performance is evaluated using standard parameters such as histograms, Structural Similarity Index Measure (SSIM), correlation analysis, and National Institute of Standards and Technology (NIST) tests. For S-box analysis, we apply the strict avalanche criterion (SAC), bit independence criterion (BIC), linear approximation probability (LAP), and differential approximation probability (DAP). This algebraic approach utilizes a robust and state-of-the-art image encryption system that is highly effective against potential attackers.

Secure multi-party computations for privacy-preserving machine learning

The main point of this paper is a further development of some aspects of the recent theory of recursively enumerable (r.e.) algebraic structures. Initial work in this area is due to Frölich and Shepherdson [4] and Rabin [10]. Here we are only concerned with vector space structure. The previous work on r.e. vector spaces is due to Dekker [2], [3], Metakides and Nerode [8], Remmel [11], Retzlaff [13], and the author [5].Our object of study is V∞ a countably infinite dimensional fully effective vector space over a countable recursive field . By fully effective we mean that V∞. under a fixed Godel numbering has the following properties:(i) Operations of vector addition and scalar multiplication on V∞ are presented by partial recursive functions on the Gödel numbers of elements of V∞.(ii) V∞ has a dependence algorithm, i.e., there is a uniform effective procedure which applied to any n vectors of V∞ determines whether or not they are linearly independent.We also study , the lattice of r.e. subspaces of V∞ (under the operations of intersection, ⋂ and (weak) sum, +). We note that if is not distributive and is merely modular (see [1]). This fact indicates the essential difference between the lattice of r.e. sets and .