International Journal of Computer Applications (0975 – 8887) Volume 105 – No. 6, November 2014 19 Evaluating Composite EC Operations and their Applicability to the on-the-fly and Non-Window Multiplication Methods Mohammad Rasmi Zarqa University, Jordan Hani Mimi Alzaytoonah University, Jordan Mohammad Sh. Daoud Al Ain University of Science and Technology, UAE ABSTRACT In order to improve the efficiency of elliptic curve multiplication methods, extended and composite elliptic curve operations such as ࢔ࡼ, ࢓ࡼ + ࡽ , where ࢔ >2 and ࢓≥ , and repeated doublings were proposed. These operations have lower complexity, in terms of field operations, than that for classical methods. Moreover, they are supposed to replace the classical methods. In this paper, repeated doublings and odd point computation are deeply analyzed in order to measure their actual efficiency. According to the gained results, the improvement ratio in the execution time is not the same as the improvement ratio measured in terms of field operations. Moreover, different implementations of Sakai repeated doubling method yield different results. For example, implementing 4P as a separate function gives lower complexity than implementing repeated doublings as a general function. On the other hand, other methods for computing nP, where n is odd, have been analyzed. Dahmen method failed to meet the expected results for computing odd points in elliptic curve multiplication methods that employ the on-the-fly strategy since its time complexity was more than that for classical methods. It was also found that new techniques should be devised to improve the efficiency of window methods for calculating odd points such as: 5P, 7P, and 15P, which have lower cost than that for classical method. Keywords Repeated doublings, extended elliptic curve operations, pre- computations, single scalar multiplications, recoding methods 1. INTRODUCTION Elliptic Curve Cryptography (ECC) was proposed in 80’s of the previous century. The same level of the well-known RSA cryptographic algorithm security can be achieved by smaller key sizes in ECC systems [1]. The performance of ECC schemes is better than that for other public key schemes. Therefore, it is more suitable for devices with limited resources such as personal digital assistants (PDAs) and mobile phones [2]. The hamming weight () , which is the ratio of the nonzero digits to the key length, affects the efficiency of elliptic curve (EC) multiplication methods. Other recoding methods such as signed methods were invented in order to reduce the amount of hamming weight, for example the non-adjacent form (NAF) signed binary method were used to accelerate the EC multiplications [3]. Window methods, such as Mutual opposite form (wMOF) [4] and wNAF, were also used in EC multiplications in order to its efficiency. Other recoding schemes were also introduced to improve the efficiency of computing the point  ,wehre ∈ , ∈ ܧ( ܨ  ) such as multi-base or mixed-base recoding methods, double base number system (DBNS) [5, 6] is an example. Computing kP involves two basic EC operations: doubling (2) and addition ( + ) . Extended and Composite (simply Composite) EC operations are those other than the basic operations, i.e. , ݓℎݎ  >2   + , ݓℎݎ ≥ 2 . Recoding methods, such as window and multi-base, involve both types of operations: basic and composite. Therefore, researchers such as Ciet[7], Sakai [8], Dahmen[9], and Eisentrager[10] explored this area and come up with extended and composite EC operations based on Affine and Jacobian, coordinate systems. The cost, in terms of the number of underlying field operations, of these new invented operations is lower than the cost of classical methods. The EC operations, basic and composite, involve underlying field operation such as inversions (i), multiplications (m), squarings (s), and additions. In large prime fields, additions are neglected and the ratio ߚ= ݏ  is considered 0.8 if the algorithm is not protected against side channel attack (SCA) [11]. If the algorithm is protected against SCA , the same multiplier is used to perform both operations in order to be indistinguishable [12]. The ratio ߙ=   is used to represent the relative cost between field inversion and field multiplication [12, 13]. Up to our knowledge and based on the literature, there is no study that combines the composite EC operations together and takes advantage of these methods or analyzes the actual performance (execution time) of these methods. Thus, the goal of this paper is to measure the actual performance of these methods and find out if the amount of enhancement in the actual performance will meet the expected performance. This paper investigates these methods and analyzes them in order to come up with some recommendations by answering the question: Can these methods really reduce the cost of EC scalar multiplication methods? A feasibility study of exploiting these methods in EC multiplication methods is conducted. The expected performance is measured by calculating the field complexity (FIELD- COMPLEXITY), while the actual performance represents the execution time measured by implementing the algorithm (TIME- COMPLEXITY). The FIELD-COMPLEXITY is defined as the number of underlying field operations required by the EC operations. Inversion operations and squaring are replaced by equivalent multiplication. Thus, FIELED-COMPLEXITY is measured by the total number of required multiplications. 2. BACKGROUND TEHORY This section represents an introduction about the elliptic curve arithmetic, basic elliptic operations, and the composite operations. The main concentration will be given to composite EC methods, repeated doublings and computing the odd point nP in general. Ciet formulas also will be explored.