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Elliptic (ECC) and hyperelliptic curve cryptosystems (HECC) have emerged as cryptosystems for small hand-held and mobile devices. Extensive research has been carried out for their secure and efficient implementation on these devices. These devices come with very low amount of resources, efficient memory management is an important issue in all such implementations. HECC arithmetic is now generally performed using so called explicit formulas. The main goal of these formulas is to reduce the number of finite field operations (multiplications and squarings). On the other hand, reducing the memory requirement is also important. To the best of our knowledge, the literature on HECC implementation does not seriously consider this aspect. This is the first work to obtain memory efficient versions of various explicit formulas appearing in the literature. In certain cases, we are also able to determine the minimum memory requirement and obtain a memory optimal implementation. We believe that these formulas will be extremely useful to designers of HECC. Our basic technique is essentially an exhaustive search with heuristic strategies for improving the run-time. [ABSTRACT FROM AUTHOR]

Double base number systems (DBNS) provide an elegant way to represent numbers. These representations also have many interesting and useful properties, which have been exploited to find many applications in Cryptography and Signal Processing. In the current article we present a scheme to represent numbers in double (and multi-) base format by combinatorial objects like graphs and diagraphs. The combinatorial representation leads to proof of some interesting results about the double and multibase representation of integers. These proofs are based on simple combinatorial arguments. In this article we have provided a graph theoretic proof of the recurrence relation satisfied by the number of double base representations of a given integer. The result has been further generalized to more than 2 bases. Also, we have uncovered some interesting properties of the sequence representing the number of double base representation of a positive integer n. It is expected that the combinatorial representation can serve as a tool for a better understanding of the double (and multi-) base number systems. [ABSTRACT FROM AUTHOR]

In the current work we propose two efficient formulas for computing the 5-fold (5P) of an elliptic curve point P. One formula is for curves over finite fields of even characteristic and the other is for curves over prime fields. Double base number systems (DBNS) have been gainfully exploited to compute scalar multiplication efficiently in ECC. Using the proposed point quintupling formulas one can use 2, 5 and 3, 5 (besides 2, 3) as bases of the double base number system. In the current work we propose a scalar multiplication algorithm, which uses a representation of the scalar using three bases 2, 3 and 5 and computes the scalar multiplication very efficiently. The proposed scheme is faster than all sequential scalar multiplication algorithms reported in literature. [ABSTRACT FROM AUTHOR]

Pipelining is a well-known performance enhancing technique in computer science. Point multiplication is the computationally dominant operation in curve based cryptography. It is generally computed by repeatedly invoking some curve (group) operation like doubling, tripling, halving, addition of group elements. Such a computational procedure may be efficiently computed in a pipeline. More generally, let Π be a computational procedure, which computes its output by repeatedly invoking processes from a set of similar processes. Employing pipelining technique may speed up the running time of the computational procedure. To find pipeline sequence by trial and error method is a nontrivial task. In the current work, we present a general methodology, which given any such computational procedure Π can find a pipelined version with improved computational speed. To our knowledge, this is the first such attempt in curve based cryptography, where it can be used to speed up the point multiplication methods using inversion-free explicit formula for curves over prime fields. As an example, we employ the proposed general methodology to derive a pipelined version of the hyperelliptic curve binary algorithm for point multiplication and obtain a performance gain of 32% against the ideal theoretical value of 50%. [ABSTRACT FROM AUTHOR]

Hyperelliptic curve cryptosystems (HECC) can be implemented on a variety of computing devices, starting from smart cards to high end workstations. Side-channel attacks are one of the most potential threats against low genus HECC. Thus efficient algorithms resistant against side channel attacks are the need of the hour. In the current work we provide implementation ready formulae for addition and doubling on curves of genus 2 which are shielded against simple side-channel analysis by having a uniform performance. This is achieved by applying the concept of side-channel atomicity - introducing cheap dummy operations to make all traces look identical. So far a detailed study of countermeasures against side-channel attacks exists only for differential attacks. There one assumes that the performance is made predictable by other means. But apart from the double-and-alway-add approach only generalizations of the Montgomery form were suggested and only for odd characteristic. They are less efficient and do not combine well with some of the countermeasures against differential attacks. Hence, our contribution closes the gap to achieve secured implementations of HECC on devices exposed to side-channel attacks. To increase the performance even further we show how our formulae can be implemented in parallel on two multipliers using a low number of registers. It is also possible to combine our method with precomputations. Keywords: Side-channel attacks, SPA countermeasures, hyperelliptic curve cryptosystems, scalar multiplication, side-channel atomicity, parallel algorithms. [ABSTRACT FROM AUTHOR]

Inversion is the costliest of all finite field operations. Some algorithms require computation of several finite field elements simultaneously (elliptic curve factorization for example). Montgomery's trick is a well known technique for performing the same in a sequential set up with little scope for parallelization. In the current work we propose an algorithm which needs almost same computational resources as Montgomery's trick, but can be easily parallelized. Our algorithm uses binary tree structures for computation and using 2r-1 multipliers, it can simultaneously invert 2r elements in 2r multiplication rounds and one inversion round. We also describe how the algorithm can be used when 2, 4... number of multipliers are available. To exhibit the utility of the method, we apply it to obtain a parallel algorithm for elliptic curve point multiplication. The proposed method is immune to side-channel attacks and compares favourably to many parallel algorithms existing in literature. Keywords: Elliptic Curve Cryptosystems, Scalar Multiplication, parallel algorithm, Montgomery ladder, simultaneous inversion. [ABSTRACT FROM AUTHOR]

In this paper, we propose a efficient and secure point multiplication algorithm, based on double-base chains. This is achieved by taking advantage of the sparseness and the ternary nature of the so-called double-base number system (DBNS). The speed-ups are the results of fewer point additions and improved formulæ for point triplings and quadruplings in both even and odd characteristic. Our algorithms can be protected against simple and differential side-channel analysis by using side-channel atomicity and classical randomization techniques. Our numerical experiments show that our approach leads to speed-ups compared to windowing methods, even with window size equal to 4, and other SCA resistant algorithms. [ABSTRACT FROM AUTHOR]