Privacy Preserving Multi-Server k-means Computation over Horizontally Partitioned Data

The k-means clustering is one of the most popular clustering algorithms in data mining. Recently a lot of research has been concentrated on the algorithm when the dataset is divided into multiple parties or when the dataset is too large to be handled by the data owner. In the latter case, usually some servers are hired to perform the task of clustering. The dataset is divided by the data owner among the servers who together perform the k-means and return the cluster labels to the owner. The major challenge in this method is to prevent the servers from gaining substantial information about the actual data of the owner. Several algorithms have been designed in the past that provide cryptographic solutions to perform privacy preserving k-means. We provide a new method to perform k-means over a large set using multiple servers. Our technique avoids heavy cryptographic computations and instead we use a simple randomization technique to preserve the privacy of the data. The k-means computed has exactly the same efficiency and accuracy as the k-means computed over the original dataset without any randomization. We argue that our algorithm is secure against honest but curious and passive adversary.Comment: 19 pages, 4 tables. International Conference on Information Systems Security. Springer, Cham, 201

Algorithms for Finding Inverse of Two Patterned Matrices over Z

Circulant matrix families have become an important tool in network engineering. In this paper, two new patterned matrices over Zp which include row skew first-plus-last right circulant matrix and row first-plus-last left circulant matrix are presented. Their basic properties are discussed. Based on Newton-Hensel lifting and Chinese remaindering, two different algorithms are obtained. Moreover, the cost in terms of bit operations for each algorithm is given

An Implementation of the Chor-Rivest Knapsack Type Public Key Cryptosystem

The Chor-Rivest cryptosystem is a public key cryptosystem first proposed by MIT cryptographers Ben Zion Chor and Ronald Rivest [Chor84]. More recently Chor has imple mented the cryptosystem as part of his doctoral thesis [Chor85]. Derived from the knapsack problem, this cryptosystem differs from earlier knapsack public key systems in that computa tions to create the knapsack are done over finite algebraic fields. An interesting result of Bose and Chowla supplies a method of constructing higher densities than previously attain able [Bose62]. Not only does an increased information rate arise, but the new system so far is immune to the low density attacks levied against its predecessors, notably those of Lagarias- Odlyzko and Radziszowski-Kreher [Laga85, Radz86]. An implementation of this cryptosystem is really an instance of the general scheme, dis tinguished by fixing a pair of parameters, p and h , at the outset. These parameters then remain constant throughout the life of the implementation (which supports a community of users). Chor has implemented one such instance of his cryptosystem, where p =197 and h =24. This thesis aspires to extend Chor\u27s work by admitting p and h as variable inputs at run time. In so doing, a cryptanalyst is afforded the means to mimic the action of arbitrary implementations. A high degree of success has been achieved with respect to this goal. There are only a few restrictions on the choice of parameters that may be selected. Unfortunately this general ity incurs a high cost in efficiency; up to thirty hours of (VAX1 1-780) processor time are needed to generate a single key pair in the desired range (p = 243 and h =18)

An Approach to Reduce Storage for Homomorphic Computations

We introduce a hybrid homomorphic encryption by combining public key encryption (PKE) and somewhat homomorphic encryption (SHE) to reduce storage for most applications of somewhat or fully homomorphic encryption (FHE). In this model, one encrypts messages with a PKE and computes on encrypted data using a SHE or a FHE after homomorphic decryption. To obtain efficient homomorphic decryption, our hybrid schemes is constructed by combining IND-CPA PKE schemes without complicated message paddings with SHE schemes with large integer message space. Furthermore, we remark that if the underlying PKE is multiplicative on a domain closed under addition and multiplication, this scheme has an important advantage that one can evaluate a polynomial of arbitrary degree without recryption. We propose such a scheme by concatenating ElGamal and Goldwasser-Micali scheme over a ring $\Z_N$ for a composite integer $N$ whose message space is $\Z_N^\times$. To be used in practical applications, homomorphic decryption of the base PKE is too expensive. We accelerate the homomorphic evaluation of the decryption by introducing a method to reduce the degree of exponentiation circuit at the cost of additional public keys. Using same technique, we give an efficient solution to the open problem~\cite{KLYC13} partially. As an independent interest, we obtain another generic conversion method from private key SHE to public key SHE. Differently from Rothblum~\cite{RothTCC11}, it is free to choose the message space of SHE

Multilinear Map via Scale-Invariant FHE: Enhancing Security and Efficiency

Cryptographic multilinear map is a useful tool for constructing numerous secure protocols and Graded Encoding System (GES) is an {\em approximate} concept of multilinear map. In multilinear map context, there are several important issues, mainly about security and efficiency. All early stage candidate multilinear maps are recently broken by so-called zeroizing attack, so that it is highly required to develop reliable mechanisms to prevent zeroizing attacks. Moreover, the encoding size in all candidate multilinear maps grows quadratically in terms of multilinearity parameter $\kappa$ and it makes them less attractive for applications requiring large $\kappa$. In this paper, we propose a new integer-based multilinear map that has several advantages over previous schemes. In terms of security, we expect that our construction is resistant to the zeroizing attack. In terms of efficiency, the bit-size of an encoding grows sublinearly with $\kappa$, more precisely $O((\log_2\kappa)^2)$. To this end, we essentially utilize a technique of the multiplication procedure in {\em scale-invariant} fully homomorphic encryption (FHE), which enables to achieve sublinear complexity in terms of multilinearity and at the same time security against the zeroizing attacks (EUROCRYPT 2015, IACR-Eprint 2015/934, IACR-Eprint 2015/941), which totally broke Coron, Lepoint, and Tibouchi\u27s integer-based construction (CRYPTO 2013, CRYPTO2015). We find that the technique of scale-invariant FHE is not very well harmonized with previous approaches of making GES from (non-scale-invariant) FHE. Therefore, we first devise a new approach for approximate multilinear maps, called {\em Ring Encoding System (RES)}, and prove that a multilinear map built via RES is generically secure. Next, we propose a new efficient scale-invariant FHE with special properties, and then construct a candidate RES based on a newly proposed scale-invariant FHE. It is worth noting that, contrary to the CLT multilinear map (CRYPTO 2015), multiplication procedure in our construction does not add hidden constants generated by ladders of zero encodings, but mixes randoms in encodings in non-linear ways without using ladders of zero encodings. This feature is obtained by using the scale-invariant FHE and essential to prevent the Cheon et al.\u27s zeroizing attack