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Finally, we present the function fsqrt64, which performs double-, single-, and half-precision square root extraction. This function, which is listed in Appendix E, is based on an implementation of the minimally redundant radix-4 case of SRT square root extraction characterized by Lemma 10.15 of Sect. 10.5. As noted in Chap. 18, it is derived from the same RTL module as the function fdiv64. The design shares hardware between the two operations for post-processing; therefore, the auxiliary functions computeQ, rounder, and final are shared by the two models.

The deepest point of the nose (where the upper nose joins the forehead) is the radix, which is both the root and the starting point of the nose. “Radix” is a clinical definition. Other structures of the nasal root, such as the nasofrontal suture and sellion, are anatomical definitions that do not correspond to the radix; we do not need to know these structures. When analysing the nose, it is necessary to commence at the radix; this is the starting point of the nasal structure. Radix position and height determine the nasal profile. If this is to be ideal (well-proportioned), the radix must be in the right place. Meticulous analysis is necessary using two descriptive parameters.

Modification of radix-2 algorithms to their corresponding high-radix algorithms can be performed, in order to complete the task in fewer clock cycles. In a high-radix algorithm, an equal amount of computation can be performed in less number of iterations. There is a hidden latency for determining the complex quotient and modifying the partial results. This latency is present in every iteration and increases the overall running time of the algorithm when implemented in hardware.

In order to evaluate the performance of any cryptosystem certain parameters are taken into consideration. Parameters such as the amount of power consumption and running-time. Cryptosystem having low power consumption and minimum running-time is the most desirable one. The performance of the cryptosystem varies from system to system. Some cryptosystem consumes less power in a particular system whereas, some take less running time in another system. Based on priority one needs to choose the best cryptosystem. This chapter discusses the various Montgomery multiplication with Radix-2, which are having less power consumption and minimum running time. Hardware and software both areas tend to search for new ways of efficient implementation of Montgomery multiplication. Hardware approaches to speed up Montgomery multiplication are way more superior than software approaches. For real-world scenarios, the hardware approach is not so fast, but the other advantages like flexibility and customization according to the need of the algorithms are achieved.

This chapter discusses the implementation of modular Exponentiation in Dual-core with High Radix Multiplication (DCHRM), for further reduction of clock cycles and better frequency. The main mathematical operation in maximum public-key cryptography (PKC) (William 2006) is a Modular Exponentiation (ME). The ME evaluation is done by performing a series of modular multiplications (MMs). As ME is a core operation thus, the competent implementation of PKC is directly proportional to the efficient performance of ME. similarly efficient implementation of ME is directly proportional to the better performing MM. In order to obtain efficient computation, high-radix implementation is preferable. As increment in radix is done, the number of digits to be executed in each iteration is decreased. Thus, reduces the required number of the clock cycles but, increases the frequency and additional space. This chapter also explores the conventional right-to-left ME approach for DCRSAP

We introduce a dynamic precision floating-point arithmetic based on the Infinity Computer. This latter is a computational platform which can handle both infinite and infinitesimal quantities epitomized by the positive and negative finite powers of the symbol Open image in new window, which acts as a radix in a corresponding positional numeral system. The idea is to use the positional numeral system from the Infinity Computer to devise a variable precision representation of finite floating-point numbers and to execute arithmetical operations between them using the Infinity Computer Arithmetics. Here, numerals with negative finite powers of Open image in new window will act as infinitesimal-like quantities whose aim is to dynamically improve the accuracy of representation only when needed during the execution of a computation. An application to the iterative refinement technique to solve linear systems is also presented.

This article presents three algebraic proofs of Steiner’s 1-Cycle Theorem [14] within the context of the (accelerated) \(3x+1\) dynamical system. Furthermore, under an assumption of an exponential upper-bound on the iterates, the article demonstrates that the only 1-cycles in the (accelerated) \(3x-1\) dynamical system are (1) and (5, 7).

This paper proposes efficient split-radix and radix-4 Discrete Cosine Transform (DCT) of types II/III algorithms. The proposed fast split-radix and radix-4 algorithms extend the previous work on the lowest multiplication complexity, self-recursive, radix-2 DCT II/III algorithms. The paper also addresses the self-recursive and stable aspects of split-radix and radix-4 DCT II/III algorithms having simple, sparse, and scaled orthogonal factors. Moreover, the proposed split-radix and radix-4 algorithms attain the lowest theoretical multiplication complexity and arithmetic complexity for 8-point DCT II/III matrices. The factorization corresponding to the proposed DCT algorithms contains sparse and scaled orthogonal matrices. Numerical results are presented for the arithmetic complexity comparison of the proposed algorithms with the known fast and stable DCT algorithms. Execution time of the proposed algorithms is presented while verifying the connection to the order of the arithmetic complexity. Moreover, we will show that the execution time of the proposed split-radix and radix-4 algorithms are more efficient than the radix-2 DCT algorithms. Finally, the implementations of the proposed DCT algorithms are stated using signal-flow graphs.

The simplest and most common approach to computer division is digit recurrence, an iterative process whereby at each step, a multiple of the divisor is subtracted from the current remainder and the quotient is updated accordingly by appending a fixed number of bits k, determined by the underlying radix, r = 2k. Thus, quotient convergence is linear, resulting in fairly high latencies of high-precision operations for the most common radices, r = 2, 4, and 8.

Radix pseudostellaria is widely cultivated because of its good pharmacology of traditional Chinese medicine. However, due to the lack of excavation device that suitable for digging operation in rolling foothills country, which leads to the low efficiency of harvesting of radix pseudostellaria in harvest season. Based on the characteristics, in this paper the specific design scheme of automatic rotary excavator is carried on and the design idea of the structure of disengaging device and excavation device is analyzed. In the end of article, the performance of excavation device is analyzed by using Finite Element Software. The experimental results show that the design of the excavation equipment of radix pseudostellariae meets the actual requirements.

As said in the Introduction, roughly speaking, a radix-β floating-point number x is a number of the form m · β e , where β is the radix of the floating-point system, m such that |m| < β is the significand of x, and e is its exponent. And yet, portability, accuracy, and the ability to prove interesting and useful properties as well as to design smart algorithms require more rigorous definitions, and much care in the specifications. This is the first purpose of this chapter. The second one is to deal with basic problems: rounding, exceptions, properties of real arithmetic that become wrong in floating-point arithmetic, best choices for the radix, and radix conversions.

We propose some new baby-step giant-step algorithms for computing “low-weight” discrete logarithms; that is, for computing discrete logarithms in which the radix-b representation of the exponent is known to have only a small number of nonzero digits. Prior to this work, such algorithms had been proposed for the case where the exponent is known to have low Hamming weight (i.e., the radix-2 case). Our new algorithms (i) improve the best-known deterministic complexity for the radix-2 case, and then (ii) generalize from radix-2 to arbitrary radixes \(b>1\). We also discuss how our new algorithms can be used to attack several recent Verifier-based Password Authenticated Key Exchange (VPAKE) protocols from the cryptographic literature with the conclusion that the new algorithms render those constructions completely insecure in practice.

In this paper we introduce RADULS2, the fastest parallel sorter based on radix algorithm. It is optimized to process huge amounts of data making use of modern multicore CPUs. The main novelties include: high performance algorithm for handling tiny arrays (up to about a hundred of records) that could appear even billions times as subproblems to handle and improved processing of larger subarrays with better use of non-temporal memory stores.

Radix trees belong to the class of trie data structures, used for storing both sets and dictionaries in a way optimized for space and lookup. In this work, we present an efficient non-blocking implementation of radix tree data structure that can be configured for arbitrary alphabet sizes. Our algorithm implements a linearizable set with contains, insert and remove operations and uses single word compare-and-swap (CAS) instruction for synchronization. We extend the idea of marking the child edges instead of nodes to improve the parallel performance of the data structure. Experimental evaluation indicates that our implementation out-performs other known lock-free implementations of trie and binary search tree data structures using CAS by more than 100% under heavy contention.

We consider the problem of mining for functional dependencies in relational databases. Intermediate data structures, although simple, explode in size and a solution is proposed using radix trees to reduce memory utilization. Parallelism is further applied in a Multi-Core computer to further speedup the process. Because bit-permutations are the basis of the construction of a binary intermediate matrix, radix trees reduce the memory usage 10 times. Multi-Threading the construction and processing of the intermediate data leads to a concurrent computing average-over-time of 63 % on an equivalent speedup of 6.3 on a system with 12 cores, 256 GB of memory and 1 TB SSD.

In this paper, a recent computational methodology is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (patented in USA and EU) working numerically with infinite and infinitesimal numbers that can be written in a positional system with an infinite radix. On a number of examples dealing mainly with infinite sets and Turing machines with different infinite tapes it is shown that it becomes possible to execute a fine analysis of these mathematical objects. The accuracy of the obtained results is continuously compared with results obtained by traditional tools used to work with mathematical objects involving infinity.

Description of the drug: Compressed-cylindrical, somewhat curved, varying in length, 0.5–2 cm in diameter. Externally greyish-yellow or dark grey, with longitudinal wrinkles and transverse cracks, some bark transversely broken and wood exposed. Texture tough, fracture bark thick, purple or yellowish-brown or yellowish-white, 1–5 mm in diameter. Odour, slight; taste, sweetish and slightly astringent.

Description of the drugs: Root of Glycyrrhiza uralensis: Roots cylindrical, 25–100 cm long, 0.6–3.5 cm in diameter. The outer bark loose or tight. Externally reddish-brown or grayish-brown, obviously longitudinally wrinkled, furrowed, lenticel-like protruded and with sparse rootlet scars. Texture compact, structure slightly fibrous, yellowish-white, starchy, cambium ring distinct, rays radiate, some with clefts. Rhizomes cylindrical, externally with bud scars, pith present in the center of fracture. Odour, slight; taste, sweet and characteristic.

The Quantitative Pattern-Activity Relationship (QPAR) approach has been proposed recently by us and applied to the herbal medicine Radix Puerariae Lobatae and a related synthetic mixture system. Two different types of data from the chromatographic fingerprint and related bioactivity capacities of the samples were correlated quantitatively. The method thus developed provided a model for predicting total bioactivity from the chromatographic fingerprints and features in the chromatographic profiles responsible for the bioactivity. In this work, we propose a new methodology called QPAR-F here, to provide another piece of information: recommending the bioactive regions to facilitate bioassay-guided fractionation and related studies. QPAR-F makes use of chromatographic profiles instead of individual data points utilized in our previous work. The chromatograms of the system concerned are firstly divided into different regions or related fractions representing different groups of constituents. Then different combinations of these regions using the exhaustive searching strategy are processed by the partial least squares (PLS) methods to build models. The optimal models give smaller errors between the predicted and measured total bioactivity capacities. The performance of the proposed QPAR-F methodology is first evaluated by a known mixture system with combinations with active ingredients. The results confirmed that QPAR-F works very well in predicting the total antioxidant bioactivity capacities and the active regions could be correctly identified. These findings are very helpful in planning the bioassay-guided fractionation. For this data-mining process, only limited chemical and bioactivity information of the original samples or crude extracts are required. No prior knowledge of activities of the fractions under study is needed. The QPAR-F methodology was also applied to the herbal medicine, Radix Puerariae Lobatae and similar predicted models give smaller errors between the predicted and measured total antioxidant bioactivity capacities could be successfully built.