Exponentiation Operations Research Articles

Exponentiating a Bundle of Linear Operators

This paper deals with the concept of exponentiability for a special class of multivalued maps. To be more precise, we discuss the exponentiability of a multivalued map F: X⇉X expressible in the form F(x) = {Ax:A ∈ Ξ}, with Ξ denoting a collection of linear continuous operators defined on a Banach space X. Among other results, we prove that, under suitable assumptions on Ξ, the Painleve–Kuratowski limit $${\left[ {\exp F} \right]}{\left( x \right)} = {\mathop {\lim }\limits_{N \to \infty } }{\left( {I + \frac{1}{N}F} \right)}^{N} {\left( x \right)}$$ exists for all x ∈ X, and it admits the representation [expF](x) = {eAx:A ∈ clco(Ξ)}. The operation of exponentiation has therefore a convexification effect on Ξ. By exploiting the above-mentioned representation formula, we derive general properties for the semigroup {SF(t)}t⩾0 defined by $$S_{F} {\left( t \right)}x = {\mathop {\lim }\limits_{N \to \infty } }{\left( {I + \frac{t}{N}F} \right)}^{N} {\left( x \right)}.$$

Efficient and secure cryptographic systems based on addition chains: Hardware design vs. software/hardware co-design

Most of cryptographic systems are based on modular exponentiation, wherein the operands are considerably large. Generally, modular exponentiation is implemented using a chain of modular multiplications. One way of improving the throughput of a cryptographic system implementation is reducing the number of the required modular multiplications. Given the exponent, finding the minimal number of multiplications that are necessary to reach the exponentiation results is a hard problem. We take advantage of an evolutionary stochastic process, commonly called genetic algorithms, to compute minimal addition chains, allow us to reduce drastically the number of the required modular multiplications. In this paper, we investigate two different ways to implement modular exponentiation based on the evolved addition chain: the first approach consists of implementing all the exponentiation operation on hardware and the second approach combines software and hardware. While the first approach attempts to optimise the encryption/decryption throughput regardless of the cost of system implementation, the second attempts to reach a balance between the throughput and the cost of the implementation.

An improved authenticated key agreement protocol

In 1999, Seo and Sweeney proposed a simple authenticated key agreement protocol that was designed to act as a Diffie-Hellman key agreement protocol with user authentication. Various attacks on this protocol are described and enhanced in the literature. Recently, Ku and Wang proposed an improved authenticated key agreement protocol, where they asserted the protocol could withstand the existing attacks. This paper shows that Ku and Wang’s protocol is still vulnerable to the modification attack and presents an improved authenticated key agreement protocol to enhance the security of Ku and Wang’s protocol. The protocol has more efficient performance by replacing exponentiation operations with message authentication code operations.

Systolic Galois field exponentiation in a multiple-valued logic technique

In this paper, we present two new parallel-in parallel-out systolic array architectures to compute the exponentiation operation for the field GF(2 k ) in a multiple-valued logic (MVL) approach, using the composite field GF((2 2) m ). We compare both circuits with the circuit that use GF(2 k ) as its basis. The proposed circuits require much less amount of chip area, less clock cycles to generate the final output, and are highly regular, thus they are well suited for VLSI.

Robust non-homomorphic approach for speckle reduction in medical ultrasound images

Most existing wavelet-based image denoising techniques are developed for additive white Gaussian noise. In applications to speckle reduction in medical ultrasound (US) images, the traditional approach is first to perform the logarithmic transform (homomorphic processing) to convert the multiplicative speckle noise model to an additive one, and then the wavelet filtering is performed on the log-transformed image, followed by an exponential operation. However, this non-linear operation leads to biased estimation of the signal and increases the computational complexity of the filtering method. To overcome these drawbacks, an efficient, non-homomorphic technique for speckle reduction in medical US images is proposed. The method relies on the true characterisation of the marginal statistics of the signal and speckle wavelet coefficients. The speckle component was modelled using the generalised Nakagami distribution, which is versatile enough to model the speckle statistics under various scattering conditions of interest in medical US images. By combining this speckle model with the generalised Gaussian signal first, the Bayesian shrinkage functions were derived using the maximum a posteriori (MAP) criterion. The resulting Bayesian processor used the local image statistics to achieve soft-adaptation from homogeneous to highly heterogeneous areas. Finally, the results showed that the proposed method, named GNDShrink, yielded a signal-to-noise ratio (SNR) gain of 0.42dB over the best state-of-the-art despeckling method reported in the literature, 1.73dB over the Lee filter and 1.31dB over the Kaun filter at an input SNR of 12.0dB, when tested on a US image. Further, the visual comparison of despeckled US images indicated that the new method suppressed the speckle noise well, while preserving the texture and organ surfaces.

On a similarity measure between LR-type fuzzy numbers and its application to database acquisition

This article presents a new similarity measure for LR-type fuzzy numbers. The proposed similarity measure is based on a defined metric between LR-type fuzzy numbers. It is known that an exponential operation is highly useful in dealing with the classical Shannon entropy and cluster analysis. We adopted, therefore, the exponential operation on this metric. Furthermore, we analyze its properties and make numerical comparisons to several similarity measures. The results show that the proposed similarity measure can overcome the drawbacks of the existing similarity measures. We then apply it to compound attributes for handling null queries to database systems. These applications can also be widely used in fuzzy queries to databases. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1001–1016, 2005.

A hardware version of the RSA using the Montgomery's algorithm with systolic arrays

Rivest–Shamir–Adleman (RSA) is one of the most widely preferred algorithms used in public-key cryptography systems. RSA has a very slow ciphering rate if used in software. The use of a specific hardware is the only reasonable solution in applications where performance is the key factor. To speed up the modular multiplication and squaring, bit level systolic arrays are used with the Montgomery's modular multiplication algorithm to constitute the core of modular exponentiation operation. The squaring systolic structure is also performed in parallel with the systolic multiplication in the modular exponentiation. The novel idea in this paper is to use the systolic array cells with increased performance of up to 20% and use them in a single row organization. The final RSA design is configurable and can operate both for encryption and decryption. 1024-bit RSA algorithm is designed for the Xilinx Virtex FPGA and 0.7 μ ASIC.

A new efficient (t, n) verifiable multi-secret sharing (VMSS) based on YCH scheme

A verifiable multi-secret sharing scheme (VMSS) is a multi-secret sharing scheme with the verifiable property. This paper presents a new efficient verifiable multi-secret sharing based on YCH [C.-C. Yang, T.-Y. Chang, M.-S. Hwang, A (t,n) multi-secret sharing scheme, Appl. Math. Comput. 151 (2004) 483–490] scheme and the intractability of the discrete logarithm (DL). And it just needs few public values and low modular exponentiation operation.

A hardware version of the RSA using the Montgomery's algorithm with systolic arrays

Rivest–Shamir–Adleman (RSA) is one of the most widely preferred algorithms used in public-key cryptography systems. RSA has a very slow ciphering rate if used in software. The use of a specific hardware is the only reasonable solution in applications where performance is the key factor. To speed up the modular multiplication and squaring, bit level systolic arrays are used with the Montgomery's modular multiplication algorithm to constitute the core of modular exponentiation operation. The squaring systolic structure is also performed in parallel with the systolic multiplication in the modular exponentiation. The novel idea in this paper is to use the systolic array cells with increased performance of up to 20% and use them in a single row organization. The final RSA design is configurable and can operate both for encryption and decryption. 1024-bit RSA algorithm is designed for the Xilinx Virtex FPGA and 0.7 μ ASIC.

Algorithm and architecture for logarithm, exponential, and powering computation

An architecture for the computation of logarithm, exponential, and powering operations is presented in this paper, based on a high-radix composite algorithm for the computation of the powering function (X/sup Y/). The algorithm consists of a sequence of overlapped operations: 1) digit-recurrence logarithm, 2) left-to-right carry-free (LRCF) multiplication, and 3) online exponential. A redundant number system is used and the selection in 1) and 3) is done by rounding except from the first iteration, when selection by table look-up is necessary to guarantee the convergence of the recurrences. A sequential implementation of the algorithm, with a control unit which allows the independent computation of logarithm and exponential, is proposed and the execution times and hardware requirements are estimated for single and double-precision floating-point computations. These estimates are obtained for radices from r=8 to r=1,024, according to an approximate model for the delay and area of the main logic blocks and help determining the radix values which lead to the most efficient implementations: r=32 and r=128.

BLIND SEPARATION OF MIXED KURTOSIS SIGNED SIGNALS USING PARTIAL OBSERVATIONS AND LOW COMPLEXITY ACTIVATION FUNCTIONS

Although several highly accurate blind source separation algorithms have already been proposed in the literature, these algorithms must store and process the whole data set which may be tremendous in some situations. This makes the blind source separation infeasible and not realisable on VLSI level, due to a large memory requirement and costly computation. This paper concerns the algorithms for solving the problem of tremendous data sets and high computational complexity, so that the algorithms could be run on-line and implementable on VLSI level with acceptable accuracy. Our approach is to partition the observed signals into several parts and to extract the partitioned observations with a simple activation function performing only the "shift-and-add" micro-operation. No division, multiplication and exponential operations are needed. Moreover, obtaining an optimal initial de-mixing weight matrix for speeding up the separating time will be also presented. The proposed algorithm is tested on some benchmarks available online. The experimental results show that our solution provides comparable efficiency with other approaches, but lower space and time complexity.

A fast modular square computing method based on the generalized Chinese remainder theorem for prime moduli

This paper introduces a very efficient way to compute modular exponentiations modulo to prime numbers. A prime modular exponentiation operation is replaced by two modular operations employed with decomposable moduli. Each substitute modular operation can be performed with the generalized Chinese remainder theorem (GCRT) for computing efficiency. Due to the independent computing property of the GCRT, these two substitute operations can be computed concurrently. Assuming the parallel computation, the computation complexity is better than the conventional modular exponentiation operations. The computational costs is reduced approximately to 22% if these factors of decomposable moduli are on average in quarter bit lengths of decomposable moduli and the smallest factor is in half average bit lengths.

Divisible Blind Signatures Based on Hash Chains

This article presents an efficient divisible blind signature scheme that makes it possible to divide a blind signature with a specified value into a sequence of subsignatures with smaller values for a sequence of different designated verifications. In the proposed scheme every signature can be divided into subsignatures with arbitrary positive values as long as the total value of them is equal to the specified value of that signature. Furthermore, no time-consuming operations such as modular exponentiation operations and inverse computations are required for a user to obtain and verify a divisible blind signature in the protocol. In particular, the proposed scheme can be applied to many interesting communication services, such as anonymous scoring services, divisible electronic cash systems, and multi-recastable voting protocols.

Divisible Blind Signatures based on Hash Chains

AbstractThis article presents an efficient divisible blind signature scheme that makes it possible to divide a blind signature with a specified value into a sequence of subsignatures with smaller values for a sequence of different designated verifications. In the proposed scheme every signature can be divided into subsignatures with arbitrary positive values as long as the total value of them is equal to the specified value of that signature. Furthermore, no time-consuming operations such as modular exponentiation operations and inverse computations are required for a user to obtain and verify a divisible blind signature in the protocol. In particular, the proposed scheme can be applied to many interesting communication services, such as anonymous scoring services, divisible electronic cash systems, and multi-recastable voting protocols.

Anonymous message communications with user hierarchy in a multicast system

In this paper, an anonymous communication protocol is proposed. The proposed scheme is then further expanded to support the user hierarchy. Higher security class users have the right to monitor a private communication involving lower security class users in the hierarchy. In our scheme, the identities of the sender and the destinations can be concealed. Every participant of the multicast connection can determine whether it is on the list of recipients of a packet, but can not derive any further information about who else is also able to receive that packet. The main mechanism we adopted in our scheme is kind of knapsack-like system in which we define the volume-increasing knapsack to encrypt a bit vector, and reveal the specified bit in the decryption manner. In our scheme, less multiplication and addition operations on modular computation are required to conceal the addresses, as long as all the encryption keys are set up in a table in advance. As a result we obtain better benefits in the encryption/decryption operations when compared to the exponential operations in general crypto manner. Consequently, the anonymous message communications with an addressed packet are efficiently implemented in our proposed scheme.

Efficient proxy multisignature schemes based on the elliptic curve cryptosystem

For improving proxy-signature research, Sun [5] attempted to resolve problems related to defective security in the scheme of Yi [3]. However, both Yi and Sun’s schemes involve a significant number of exponential operations to verify the proxy signature. Accordingly, an improvement is proposed here to change the exponential operations into elliptic curve multiplicative ones. As proposed by both Koblitz [6-7] and Miller [8] in 1985, the elliptic curve is used in developing the cryptosystems. The elliptic curve cryptosystem can achieve a level of security equal to that of RSA or DSA but has a lower computational overhead and a smaller key size than both of these. Therefore, it is used in Sun’s schemes to improve their efficiency.

XML 문서 보안을 위한 새로운 XML-Signcryption scheme 설계 및 구현

As the XML is approved standard language by the UN, the progress which complemented the XML security has being processed rapidly. In this paper, we design and implement the "XML-Signcryption" as a security mechanism to protect the XML document that can operate between other platforms. The signature and encryption which is the standard specification in W3C needs to be able to proceed them separately. Generally the signature and encryption require four times modular exponential operation, however the signcryption only needed three times modular exponential operation. This will benefit overall system effectiveness in terms of cost. And this scheme offers to convenient the user, because the signature and encryption implement as a single XML format. This tool can save the parsing time as a number of tags is few within a document. And also, in this paper, based on a research of Web Services security, we can apply XML-Signcryption to the SOAP message to provide the security services. Based on the XML-Signcryption scheme which provides confidentiality, integrity, authentication and non-repudiation to the XML document and Web Service security simultaneously.

Architectural tradeoff in implementing RSA processors

An investigation of a suite of RSA processors using different exponentiation and modular arithmetic algorithms is the main theme of this paper. The execution time and the amount of hardware required of different algorithms used to implement the RSA processor are compared. The modular algorithms examined in this paper are classical modular algorithm, Barrett's modular algorithm, Hensel's odd division and Montgomery's modular algorithm. The exponentiation algorithms implemented are the left-to-right binary method, the right-to-left binary method, the Chinese remainder theorem. This work finds that the fast RSA processor is the one using the Chinese remainder theorem with right to left scan for exponentiation operations and Barrett's algorithm for modular arithmetic operations. The RSA processor using least amount of hardware is the one using the left-o-right binary method for exponentiation operations and Montgomery's algorithm for modular operations.

A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

This paper is concerned with reliable multistation series queueing networks. Items arrive at the first station according to a Poisson distribution and an operation is performed on each item by a server at each station. Every station is allowed to have more than one server with the same characteristics. The processing times at each station are exponentially distributed. Buffers of nonidentical finite capacities are allowed between successive stations. The structure of the transition matrices of these specific type of queueing networks is examined and a recursive algorithm is developed for generating them. The transition matrices are block-structured and very sparse. By applying the proposed algorithm the transition matrix of a K-station network can be created for any K. This process allows one to obtain the exact solution of the large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures can be calculated. Scope and purpose The exact analysis of queueing networks with multiple servers at each workstation and finite capacities of the intermediate queues is extremely difficult as for even the case of exponential operation (service or processing) times the Markovian chain that models the system consists of a huge number of states which grows exponentially with the number of stations, the number of servers at each station and the queue capacity of each intermediate queue of the resulting system. The scope and purpose of the present paper is to analyze and provide a recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks. By applying the proposed algorithm one may create the transition matrix of a K-station queueing network for any K. This process allows one to obtain the exact solution of the resulting large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures of the system can be obtained.

Algorithms for optoelectronic implementation of modified signed-digit division, square-root, logarithmic, and exponential functions

Algorithms for computing complex elementary functions based on the modified signed-digit (MSD) number system representations are proposed. An arithmetic unit that performs parallel one-step addition (subtraction), multiplication, and division is proposed to achieve the computation of complex functions such as the square-root, logarithm, exponential, and other related operations. An optoelectronic correlator-based architecture is suggested for implementing the proposed MSD algorithms to compute the above-mentioned elementary functions. We utilized the symbolic substitution technique to reduce the number of computation rules involved.