Double-base Chains Research Articles

Memory-Efficient Algorithm for Scalar Multiplications on Twisted Edwards Curves for Isogeny-Based Cryptosystems

Scalar multiplications are considered an essential aspect of implementations of isogeny-based cryptography. The efficiency of scalar multiplication depends on the equation of the underlying elliptic curves and the addition chain employed. Bos and Friedberger recently stated that, for larger scalar multiplication, addition-subtraction chains will become more useful for twisted Edwards curves because of the differential restriction on Montgomery curves in the setting of isogeny-based cryptosystem. Motivated by these comments, we attempt to increase the efficiency of scalar multiplication in twisted Edwards curves in terms of the memory of algorithms. In this paper, we present a double-base addition-subtraction chain algorithm with memory efficiency for scalar multiplication. The memory usage of this part is O log n 2 / log log n , which is better than the result of the tree-based approach, which is O log n 2 .

Efficient Scalar Multiplication over Elliptic Curve

Elliptic Curve Scalar multiplication is the method of adding a point on the curve to itself every time[1]. In recent years, research in Scalar Multiplication over Elliptic curves (EC) over a finite fields attracted many researchers, working in field of cryptography, to find out how elliptic curves cryptography (ECC) can be implemented and how to reduce its complexity [4]. The efficient techniques used in Elliptic curve cryptography are Elliptic curve scalar multiplication using pointhalving algorithm [2], then double-base (DB) chain algorithm, and after that step multi-base representation (SMBR), but these techniques have their drawbacks. So, it become imperative to find out a new approach which can be efficiently used for implementation of ECC and further reducing its complexity. The paper proposes a new algorithm Treble algorithm for affine coordinates. We continued doing work using the binary concept or double and add operation with the help of treble approach to make it more efficient which relates to the use of all input values in producing any type of output, including how much time and energy are required The results show that our contribution can significantly enhance EC scalar multiplication. Index Term—Elliptic curve cryptography, double and add operation, affine coordinates.

Multibase scalar multiplications in cryptographic pairings

The efficient computation of the Tate pairing is crucial for various cryptographic applications. In the computation the Tate pairing, two types of costs should be considered: that of scalar multiplication and the evaluations of Miller's line functions for elliptic curves. In this paper we optimize the calculation of $$(f_{2j\pm 1}(Q),[2j\pm 1]P)$$(f2j±1(Q),[2j±1]P), $$(f_{3j}(Q),[3]P)$$(f3j(Q),[3]P), $$(f_{3j\pm 1}(Q),[3j\pm 1]P)$$(f3j±1(Q),[3j±1]P) given the points P and Q in an elliptic curve, to improve the efficiency of the Tate pairing, when using the representation of the scalar n in NAF, in signed ternary base, and in double-base chain. Finally we compare their computational costs. In the case of a double-base chain, a general comparison is not simple, so we consider a few examples.

Efficient Double Bases for Scalar Multiplication

In this paper we present efficient algorithms to take advantage of the double-base number system in the context of elliptic curve scalar multiplication. We propose a generalized version of Yao's exponentiation algorithm allowing the use of general double-base expansions instead of the popular double base chains. We introduce a class of constrained double base expansions and prove that the average density of non-zero terms in such expansions is O( log k/ log log k) for any large integer k. We also propose an efficient algorithm for computing constrained expansions and finally provide a comprehensive comparison to double-base chain expansions, including a large variety of curve shapes and various key sizes.

Two Lower Bounds for Shortest Double-Base Number System

In this letter, we derive two lower bounds for the number of terms in a double-base number system (DBNS), when the digit set is {1}. For a positive integer n, we show that the number of terms obtained from the greedy algorithm proposed by Dimitrov, Imbert, and Mishra [1] is $\Theta\left(\frac{\log n}{\log \log n}\right)$. Also, we show that the number of terms in the shortest double-base chain is Θ(log n).

A new algorithm for computing triple-base number system

We introduce here a generalized method a new Algorithm to find Triple-Base number system and Triple-Base chain and hence in turn Single Digit Triple-Base number system(SDTBNS). The proposed method is not only simpler and faster than the Algorithms to find Double-Base number system or Double-Base chain, experimentally it also returns a shorter length of Triple-Base chain which in turn reduces the size of the look-up-table to find out SDTBNS. The complexity analysis and experimental results shows the novelty of the proposed Algorithm. Moreover when the proposed method is applied to find scalar multiplication in case of Elliptic Curve Cryptography and coefficient multiplication in case of designing digital filter, its efficiency also proves its novelty. Here we have used third base as 5 because when it is multiplied by 2 gives 10 which can be efficiently used for decimal shifting, i.e. if an integer 'n' can be represented in SDTBNS form, then n/10 x or nx10 x can also represented in SDTBNS only by diving or multiplying 'n' by 10.

Triple-Base Hybrid Joint Sparse Form and its Applications

Multi-scalar multiplication and multi-exponentiation are the major problems in digital signal processing(DSP) and in public-key cryptography. In DSP, multiplication of the filter coefficients, which is used in different signal processing algorithms, is a time consuming operation. Also in elliptic curve cryptography(ECC), this multiplication process takes a lot of time. Though there are several algorithms to speed up this multiplication process, but they are not up to the satisfactory limit. The algorithms are based on the minimization of the multipliers or adders required, i.e. the minimization of the numbers of „one‟ appear in the binary representation of the signal and the coefficients in DSP or of the number of general multiplications and points addition in ECC. Different existing algorithms in this context are Joint Sparse Form(JSF), w-NAF, Double Base Chain(DBC), Hybrid Binery-Ternary Joint Sparse Form(HBTJSF) etc. In this paper, a novel algorithm has been proposed which is the modified HBTJSF, known as Triple-Base Hybrid Joint Sparse Form(TBHJSF). The proposed method is based on the decomposition of an integer or fraction that mixes the power of the base 2,3 and 5. The experimental results show that it requires less numbers of multiplier and adder and hence show its novelty over other algorithms.

An Efficient Algorithm for Elliptic Curve Scalar Multiplication Using Extended Double-Base Chain and Point Halving

We propose to represent a scalar in the form of d (1/2)(superscript a)3(superscript b), where a and b are integers and d is an odd integer belonging to a given set. This representation is a combination of the extended double-base number system (DBNS) and the double-base chain representation using powers of 1/2 and 3. Experimental results show that our approach leads to a shorter DBNS expansion and a lower complexity in elliptic curve scalar multiplication, with the cost of only a few pre-computations and storages. This contributes to the efficient implementation of elliptic curve cryptography (ECC).

Bounds and trade-offs for Double-Base Number Systems

During the past twenty years, elliptic curves have attracted more and more attention from the cryptography community. One of the core operations of elliptic curve cryptography protocols is the scalar multiplication. Any algorithm reducing the complexity of such multiplications will speed up cryptographic primitives based on these algebraic structures. In this paper, we study two recently introduced techniques for scalar multiplication: Double-Base Number System (DBNS) and Double-Base Chain (DBC). Our results are twofold. First, we demonstrate a theoretical bound Ω ( log n ) on the length of any DBC used to decompose some ( log n ) -bit integers. Second, we present a new algorithm to obtain a { 2 , 3 } -integer expansion of n. The bound previously computed will imply the optimality of this algorithm. Our scheme represents a trade-off method between the DBNS approach and the DBC technique. Experimentally, our algorithm constructs shorter chains on average than both the binary/ternary method and the greedy algorithm approach.

Fast ideal cubing in imaginary quadratic number and function fields

We present algorithms for computing the cube of an ideal in an imaginary quadratic number field or function field. In addition to a version that computes a non-reduced output, we present a variation based on Shanks' NUCOMP algorithm that computes a reduced output and keeps the sizes of the intermediate operands small. Extensive numerical results are included demonstrating that in many cases our formulas, when combined with double base chains using binary and ternary exponents, lead to faster exponentiation.

Efficient Tate pairing computation using double-base chains

Pairing-based cryptosystems have developed very fast in the last few years. The efficiencies of these cryptosystems depend on the computation of the bilinear pairings. In this paper, a new efficient algorithm based on double-base chains for computing the Tate pairing is proposed for odd characteristicp > 3. The inherent sparseness of double-base number system reduces the computational cost for computing the Tate pairing evidently. The new algorithm is 9% faster than the previous fastest method for the embedding degree k = 6.

Fast elliptic scalar multiplication using new double-base chain and point halving

The fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication. Based on the double-base chain representation of scalar using powers of 2 and 3, we propose a new representation with powers of 1/2 and 3 instead. Thus the efficient point halving operation can be incorporated in the new double-base chain to achieve fast scalar multiplication. Experimental results show that our approach leads to a lower complexity which contributes to the efficient implementation of elliptic curve cryptosystems.