Erkay Savas

A Scalable and Unified Multiplier Architecture for Finite Fields GF(p) and GF(2m)

We describe a scalable and unified architecture for a Montgomery multiplication module which operates in both types of finite fields GF(p) and GF(2m). The unified architecture requires only slightly more area than that of the multiplier architecture for the field GF(p). The multiplier is scalable, which means that a fixed-area multiplication module can handle operands of any size, and also, the wordsize can be selected based on the area and performance requirements. We utilize the concurrency in the Montgomery multiplication operation by employing a pipelining design methodology. The upper limit on the precision of the scalable and unified Montgomery multiplier is dictated only by the available memory to store the operands and internal results, and the module is capable of performing infinite-precision Montgomery multiplication in both types of finite fields.

The Montgomery modular inverse-revisited

We modify an algorithm given by Kaliski to compute the Montgomery inverse of an integer modulo a prime number. We also give a new definition of the Montgomery inverse, and introduce efficient algorithms for computing the classical modular inverse, the Kaliski-Montgomery inverse, and the new Montgomery inverse. The proposed algorithms are suitable for software implementations on general-purpose microprocessors.

Privacy preserving clustering on horizontally partitioned data

Data mining has been a popular research area for more than a decade due to its vast spectrum of applications. However, the popularity and wide availability of data mining tools also raised concerns about the privacy of individuals. The aim of privacy preserving data mining researchers is to develop data mining techniques that could be applied on databases without violating the privacy of individuals. Privacy preserving techniques for various data mining models have been proposed, initially for classification on centralized data then for association rules in distributed environments. In this work, we propose methods for constructing the dissimilarity matrix of objects from different sites in a privacy preserving manner which can be used for privacy preserving clustering as well as database joins, record linkage and other operations that require pair-wise comparison of individual private data objects horizontally distributed to multiple sites. We show communication and computation complexity of our protocol by conducting experiments over synthetically generated and real datasets. Each experiment is also performed for a baseline protocol, which has no privacy concern to show that the overhead comes with security and privacy by comparing the baseline protocol and our protocol.

Low-Power Elliptic Curve Cryptography Using Scaled Modular Arithmetic

We introduce new modulus scaling techniques for transforming a class of primes into special forms which enables efficient arithmetic. The scaling technique may be used to improve multiplication and inversion in finite fields. We present an efficient inversion algorithm that utilizes the structure of scaled modulus. Our inversion algorithm exhibits superior performance to the Euclidean algorithm and lends itself to efficient hardware implementation due to its simplicity. Using the scaled modulus technique and our specialized inversion algorithm we develop an elliptic curve processor architecture. The resulting architecture successfully utilizes redundant representation of elements in GF(p) and provides a low-power, high speed, and small footprint specialized elliptic curve implementation.

Instruction Set Extensions for Fast Arithmetic in Finite Fields GF(p) and GF(2m)

Instruction set extensions are a small number of custom instructions specifically designed to accelerate the processing of a given kind of workload such as multimedia or cryptography. Enhancing a general-purpose RISC processor with a few application-specific instructions to facilitate the inner loop operations of public-key cryptosystems can result in a significant performance gain. In this paper we introduce a set of five custom instructions to accelerate arithmetic operations in finite fields GF(p) and GF(2 m ). The custom instructions can be easily integrated into a standard RISC architecture like MIPS32 and require only little extra hardware. Our experimental results show that an extended MIPS32 core is able to perform an elliptic curve scalar multiplication over a 192-bit prime field in 36 msec, assuming a clock speed of 33 MHz. An elliptic curve scalar multiplication over the binary field GF(2191) takes only 21 msec, which is approximately six times faster than a software implementation on a standard MIPS32 processor.

Computer and information sciences-ISCIS 2006

This book constitutes the refereed proceedings of the 21st International Symposium on Computer and Information Sciences, ISCIS 2006, held in Istanbul, Turkey in October 2006. The 106 revised full papers presented together with 5 invited lectures were carefully reviewed and selected from 606 submissions. The papers are organized in topical sections on algorithms and theory, bioinformatics, computational intelligence, computer architecture, computer graphics, computer networks, computer vision, data mining, databases, embedded systems, information retrieval, mobile computing, parallel and distributed computing, performance evaluation, security and cryptography, as well as software engineering.

Energy-efficient software implementation of long integer modular arithmetic

This paper investigates performance and energy characteristics of software algorithms for long integer arithmetic. We analyze and compare the number of RISC-like processor instructions (e.g. single-precision multiplication, addition, load, and store instructions) required for the execution of different algorithms such as Schoolbook multiplication, Karatsuba and Comba multiplication, as well as Montgomery reduction. Our analysis shows that a combination of Karatsuba-Comba multiplication and Montgomery reduction (the so-called KCM method) allows to achieve better performance than other algorithms for modular multiplication. Furthermore, we present a simple model to compare the energy-efficiency of arithmetic algorithms. This model considers the clock cycles and average current consumption of the base instructions to estimate the overall amount of energy consumed during the execution of an algorithm. Our experiments, conducted on a StrongARM SA-1100 processor, indicate that a 1024-bit KCM multiplication consumes about 22% less energy than other modular multiplication techniques.

Privacy Preserving Clustering on Horizontally Partitioned Data

Data mining has been a popular research area for more than a decade due to its vast spectrum of applications. The power of data mining tools to extract hidden information that cannot be otherwise seen by simple querying proved to be useful. However, the popularity and wide availability of data mining tools also raised concerns about the privacy of individuals. The aim of privacy preserving data mining researchers is to develop data mining techniques that could be applied on databases without violating the privacy of individuals. Privacy preserving techniques for various data mining models have been proposed, initially for classification on centralized data then for association rules in distributed environments. In this work, we propose methods for constructing the dissimilarity matrix of objects from different sites in a privacy preserving manner which can be used for privacy preserving clustering as well as database joins, record linkage and other operations that require pair-wise comparison of individual private data objects horizontally distributed to multiple sites.

Public key cryptography based privacy preserving multi-context RFID infrastructure

In this paper, we propose a novel radio frequency identification (RFID) infrastructure enabling multi-purpose RFID tags realized by the use of privacy preserving public key cryptography (PKC) architecture. The infrastructure ensures that the access rights of the tags are preserved based on the spatial and temporal information collected from the RFID readers. We demonstrate that the proposed scheme is secure with respect to cryptanalytic, impersonation, tracking, replay, and relay attacks. We also analyze the feasibility of PKC implementation on passive class 2 RFID tags, and show that the requirements for PKC are comparable to those of other cryptographic implementations based on symmetric ciphers. Our numerical results indicate PKC based systems can outperform symmetric cipher based systems, since the back end servers can identify RFID tags with PKC based systems approximately 57 times faster than the best symmetric cipher based systems.

Scalable and Unified Hardware to Compute Montgomery Inverse in GF(p) and GF(2)

Computing the inverse of a number in finite fields GF(p) or GF(2n) is equally important for cryptographic applications. This paper proposes a novel scalable and unified architecture for a Montgomery inverse hardware that operates in both GF(p) and GF(2n) fields. We adjust and modify a GF(2n) Montgomery inverse algorithm to accommodate multi-bit shifting hardware, making it very similar to a previously proposed GF(p) algorithm. The architecture is intended to be scalable, which allows the hardware to compute the inverse of long precision numbers in a repetitive way. After implementing this unified design it was compared with other designs. The unified hardware was found to be eight times smaller than another reconfigurable design, with comparable performance. Even though the unified design consumes slightly more area and it is slightly slower than the scalable inverter implementations for GF(p) only, it is a practical solution whenever arithmetic in the two finite fields is needed.