Extended Euclidean Algorithm Research Articles

The extended Euclidean Algorithm made easy

The extended Euclidean Algorithm made easy

Efficient Scalable Digit-Serial Inverter Over GF( $2^{m}$ ) for Ultra-Low Power Devices

This paper proposes a novel scalable digit-serial inverter structure with low space complexity to perform inversion operation in GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) based on a previously modified extended Euclidean algorithm. This structure is suitable for fixed size processor that only reuse the core and does not require to modulate the core size when m modified. This structure is extracted by applying a nonlinear methodology that gives the designer more flexibility to control the processing element workload and also reduces the overhead of communication between processing elements. Implementation results of the proposed scalable design and previously reported efficient designs show that the proposed scalable structure achieves a significant reduction in the area ranging from 83.0% to 88.3% and also achieves a significant saving in energy ranging from 75.0% to 85.0% over them, but it has lower throughput compared to them. This makes the proposed design more suitable for constrained implementations of cryptographic primitives in ultra-low power devices such as wireless sensor nodes and radio frequency identification (RFID) devices.

Approaches to performance increasing of extended Euclidean algorithm for double precision division on single precision large integers

Public key cryptography has significant processing and spatial complexity. In this regard, relevant scientific and technical challenge is to improve the performance of such transformations. In this paper proposed approaches of large integer’s division operation with double precision on single precision based on the Extended Euclidean Algorithm. These approaches include handling non-zero machine words in the most frequently used operations; using an semi-exact comparison of large integer, without for-word comparison; knowledge of the of Bézout equation parameters changing. These approaches are implemented in the modified algorithm, which has been software implemented. In experiments were compared numbers with double binary size dividend and single binary size divider with ratios for various filled and total binary length. Modified algorithm showed higher performance in 1.5-3 times, with dividend and divisor increasing binary length.

Performance increasing methods for binary field inversion

Authors propose several methods for increasing performance of multiplicative inversion algorithm in binary fields based on Extended Euclidean Algorithm. First method is based on Extended Euclidean Algorithm specific: either invariant polynomial is a same or swaps with invariant polynomial u. Thus, it does not necessary to compute degree of polynomial . Next method is based on «next fit element» in polynomial degree computation: on each iteration of Extended Euclidean Algorithm execution, degree of invariant polynomial decreases at list one. On next iteration Extended Euclidean Algorithm degree searches from where it left off the previous time. Proposed methods allow to increase performance of software implementation of inversion in binary field for 32-bit architectures on 15-20%.

Reliable Concurrent Error Detection Architectures for Extended Euclidean-Based Division Over &lt;inline-formula&gt; &lt;tex-math notation="TeX"&gt;${\rm GF}(2^{m})$ &lt;/tex-math&gt;&lt;/inline-formula&gt;

The extended Euclidean algorithm (EEA) is an important scheme for performing the division operation in finite fields. Many sensitive and security-constrained applications such as those using the elliptic curve cryptography for establishing key agreement schemes, augmented encryption approaches, and digital signature algorithms utilize this operation in their structures. Although much study is performed to realize the EEA in hardware efficiently, research on its reliable implementations needs to be done to achieve fault-immune reliable structures. In this regard, this paper presents a new concurrent error detection (CED) scheme to provide reliability for the aforementioned sensitive and constrained applications. Our proposed CED architecture is a step forward toward more reliable architectures for the EEA algorithm architectures. Through simulations and based on the number of parity bits used, the error detection capability of our CED architecture is derived to be 100% for single-bit errors and close to 99% for the experimented multiple-bit errors. In addition, we present the performance degradations of the proposed approach, leading to low-cost and reliable EEA architectures. The proposed reliable architectures are also suitable for constrained and fault-sensitive embedded applications utilizing the EEA hardware implementations.

FPGA Implementation of 16 bit RSA Cryptosystem for Text Message

The rapid growth of the internet and electronic commerce has brought to the forefront the issue of privacy in electronic communication. In order to protect the information from unauthorized parties we need to mask the information before sending it through a communication channel. Currently RSA is one of the algorithms which are not broken by hackers due to its mathematical complexity. This paper presents a design and implementation of 16-bit RSA Cryptosystem. The entire cryptosystem is divided in to three parts: key generation, encryption and decryption. The key generation is carried out by using random number generator LFSR, “Sieve of Eratosthenes” algorithm for prime number detection, Booth multiplier for multiplication and Extended Euclidean algorithm to find GCD of the public key and Euler‟s Totient Function. Encryption and decryption is carried out by Modular Multiplication and Modular Exponentiation by using LR binary method. The encryption and decryption of the text message “NAGARJUNA COLLEGE” is verified using proposed RSA algorithm. The design is simulated using Modelsim10.2b simulator and finally implemented on Sparten-6 FPGA using Xilinx 13.4 software. Area and timing parameters are computed with respect to Spartan-6 FPGA. The results obtained from simulation are validated using MATLAB code.

Huffman based conditional access system for key distribution in digital TV multicast

The advance of modern network technologies has made Digital TV systems available throughout the world. To provide secure media delivery in Digital TV systems, a large number of messages are exchanged for key updates in the conventional key distributed schemes of conditional access systems (CAS). In this paper, proposed a new multicast key distribution scheme based on Huffman grouping scheme of access control for conditional access system (CAS) in digital television multicast. The proposed key distribution scheme can greatly reduce the computation using fast Fourier transform and acquire higher efficiency and security using extended Euclidean algorithm. With this scheme, only authorized subscribers can watch the subscriber programs correctly. Unauthorized subscribers have no information to retrieve the correct programs over the networks. Moreover, the proposed scheme is more flexible in processing subscribers joining and leaving which is achieved by using Huffman based grouping scheme and is very important for service provider to dynamic manage the subscriber.

A Unified View on Known Algebraic Decoding Algorithms and New Decoding Concepts

Known properties of cyclic codes are used to give a unified description of many classical decoding algorithms for Reed-Solomon codes up to half the minimum distance. This description allows also simplified proofs for these decoders. Further, a novel decoding algorithm is derived using these properties directly and variants of a new error/erasure decoding algorithm are given. For decoding beyond half the minimum distance, a basis of all solutions for decoding is derived. This basis allows to use side information in order to decode beyond half the minimum distance. Other methods where this basis can be used are power decoding, also known as virtual syndrome extension, where additional equations are generated by taking powers of the received symbols, and interleaved Reed-Solomon codes. The extended Euclidean algorithm, which calculates the greatest common divisor, plays an essential role in many presented methods.

Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$

A significant amount of effort has been devoted to improving divisor arithmetic on low-genus hyperelliptic curves via explicit versions of generic algorithms. Moderate and high genus curves also arise in cryptographic applications, for example, via the Weil descent attack on the elliptic curve discrete logarithm problem, but for these curves, the generic algorithms are to date the most efficient available. Nagao [22] described how some of the techniques used in deriving efficient explicit formulas can be used to speed up divisor arithmetic using Cantor's algorithm on curves of arbitrary genus. In this paper, we describe how Nagao's methods, together with a sub-quadratic complexity partial extended Euclidean algorithm using the half-gcd algorithm can be applied to improve arithmetic in the degree zero divisor class group. We present numerical results showing which combination of techniques is more efficient for hyperelliptic curves over $\mathbb{F}_{2^n}$ of various genera.

Extended Euclidean Algorithm and CRT Algorithm

Summary In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by Tokyo Metropolitan University [8].

Computation of Private Key Based on Divide-By-Prime for Luc Cryptosystems

Problem statement: One of the public key cryptosystem is Luc cryptosystems. This system used Lucas Function for encryption and decryption process. Lucas Function is a special form of second-order linear recurrence relation. An encyption process is used to encrypt an original message to ciphertext by using public key. A decryption process is the process to decrypt a ciphertext into original message using private key. The existing algorithm on computing private key computation involved some redundant computations. Approach: In this study, an efficient algorithm to compute private key for Luc cryptosystem is developed. The Extended Euclidean Algorithm will be enhanced by implementing Divide-By-Prime in its computations. The comparison is focused on the computation time by the existing and new algorithms. The more efficient algorithm means the better computation time. The shorter computation time the better algorithm. Results: A new algorithm shows better computation time. In all experiments, the computation time by new algorithm is always better than the existing algorithm. Conclusion: The new computation algorithm that based on Divide-By-Prime provided better efficiency of decryption process compared to the existing algorithm.

A new bound on the minimum distance of cyclic codes using small-minimum-distance cyclic codes

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann–Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes. We define syndromes and formulate a Key Equation that allows an efficient decoding up to our bound with the Extended Euclidean Algorithm. It turns out that lowest-code-rate cyclic codes with small minimum distances are useful for our approach. Therefore, we give a sufficient condition for binary cyclic codes of arbitrary length to have minimum distance two or three and lowest code-rate.

Analysis of Modular Inverse and Improvement of Extended Euclid Algorithm

Analysis of Modular Inverse and Improvement of Extended Euclid Algorithm

Synthesis of first-order dynamic programming algorithms

To solve a problem with a dynamic programming algorithm, one must reformulate the problem such that its solution can be formed from solutions to overlapping subproblems. Because overlapping subproblems may not be apparent in the specification, it is desirable to obtain the algorithm directly from the specification. We describe a semi-automatic synthesizer of linear-time dynamic programming algorithms. The programmer supplies a declarative specification of the problem and the operators that might appear in the solution. The synthesizer obtains the algorithm by searching a space of candidate algorithms; internally, the search is implemented with constraint solving. The space of candidate algorithms is defined with a program template reusable across all linear-time dynamic programming algorithms, which we characterize as first-order recurrences. This paper focuses on how to write the template so that the constraint solving process scales to real-world linear-time dynamic programming algorithms. We show how to reduce the space with (i)~symmetry reduction and (ii)~domain knowledge of dynamic programming algorithms. We have synthesized algorithms for variants of maximal substring matching, an assembly-line optimization, and the extended Euclid algorithm. We have also synthesized a problem outside the class of first-order recurrences, by composing three instances of the algorithm template.

A Novel approach for Privacy Preserving in Video using Extended Euclidean algorithm Based on Chinese remainder theorem

The development in the modern technology paved a path in the utilization of surveillance cameras in streets, offices and other areas but this significantly leads a threat to the privacy of visitors, passengers or employees, leakage of information etc.. To overcome this threat, privacy and security needs to be incorporated in the practical surveillance system. It secures the video information which is resided in various video file types. In this process we used an efficient framework to preserve the privacy while distributing secret among ‘N’ number of parties. In this paper we analyzed various techniques of Chinese Remainder Theorem.

A Comparison of Several Greatest Common Divisor 'GCD' Algorithms

paper about Greater Common Divisor GCD, the paper shows that there is a lot of algorithms, some of these algorithm is good in timing and make low number of iteration, the other make a lot of iteration with a lot of time! But as we see in the analysis of the algorithms that some of the algorithms is faster than the others in small numbers (like Brute force is faster than Bishop Algorithm in the small numbers, but in the large numbers the Bishop Algorithm is too fast with comparison with the brute force) so the researchers recommend to develop the Bishop algorithm the make it more efficient in computing the GCD for small numbers. In the other hand the Dijkstra algorithm is too close in timing and number of iteration with the Bishop algorithm. But as we see in the analysis the best algorithm to use in computing the GCD in all type of integers is the Extended Euclidean algorithm which makes few loops with small or large numbers.

A suite of algorithms for key distribution and authentication in centralized secure multicast environments

The Extended Euclidean algorithm provides a fast solution to the problem of finding the greatest common divisor of two numbers. In this paper, we present three applications of the algorithm to the security and privacy field. The first one allows one to privately distribute a secret to a set of recipients with only one multicast communication. It can be used for rekeying purposes in a Secure Multicast scenario. The second one is an authentication mechanism to be used in environments in which a public-key infrastructure is not available. Finally, the third application of the Extended Euclidean algorithm is a zero-knowledge proof that reduces the number of messages between the two parts involved, with the aid of a central server.

A simple power analysis attack on a McEliece cryptoprocessor

The security of McEliece public-key cryptosystem is based on the difficulty of the decoding problem which is NP-hard. In this article, we propose a simple power analysis attack on this cryptosystem. The attack exploits an information leakage, which results from the relation between the error vector weight and the iteration number of the extended Euclidean algorithm used in Patterson Algorithm. Executing the proposed attacks enables the extraction of the secret error vector, and thus the plain text with minimal overhead. A countermeasure is presented which removes the information leakage and prevents the simple power analysis attack. The attack procedure and the countermeasure are applied to a cryptoprocessor implementation of the McEliece cryptosystem running on a FPGA platform.

Computing the Modular Inverse of a Polynomial Function over GF(2 P ) Using Bit Wise Operation

Most public key crypto systems use finite field modulo arithmetic. This modulo arithmetic is applied on real numbers, binary values and polynomial functions. The computation cost is based on how it works with minimum use of scarce resources like processor and memory We have implemented the determination of the multiplicative in-verse of a polynomial over GF(2(superscript p)) with minimum computational cost. The ”Extended Euclidean Algorithm” (EEA) has been demonstrated to work very well manually for integers and polynomials. However polynomial manipulation cannot be computerized directly. We have implemented the same by using simple bit wise shift and XOR operations. In small applications like smart cards, mo-bile devices and other small memory devices, this method works very well. To the best of our knowledge, the pro-posed algorithm seems to be the first, efficient and cost effective implementation of determining the multiplicative inverse of polynomials over GF(2(superscript p)) using computers. As this is a pioneering work, the results could not be compared with that of any previous work.

Recursive formulation of the matrix Padé approximation in packed storage

The Extended Euclidean algorithm for matrix Padé approximants is applied to compute matrix Padé approximants when the coefficient matrices of the input matrix polynomial are triangular. The procedure given by Bjarne S. Anderson et al. for packing a triangular matrix in recursive packed storage is applied to pack a sequence of lower triangular matrices of a matrix polynomial in recursive packed storage. This recursive packed storage for a matrix polynomial is applied to compute matrix Padé approximants of the matrix polynomial using the Matrix Padé Extended Euclidean algorithm in packed form. The CPU time and memory comparison, in computing the matrix Padé approximants of a matrix polynomial, between the packed case and the non-packed case are described in detail.