Maximum and Minimum Norms for τ-NAF Expansion on Koblitz Curve

Background/Objectives: The scalar multiplication in Elliptic Curve Cryptosystem (ECC) is the dominant operation of computing integer multiple for an integer n and a point P on elliptic curve. In 1997, Solinas4 introduced the τ-adic non-adjacent form (τ-NAF) expansion of an element n of ring Z(τ) on Koblitz Curve. However in 2000, Solinas estimated the length of τ-NAF expansion by using maximum and minimum norms that obtained by direct evaluation method. In 2014, Yunos et al.9 introduced the formula of norm for every τ-NAF to improve this method. However, a lot of combination of norm should be considered when length of expansion is more than 15. So, the objective of this paper is to built the formulas to calculate the number of maximum and minimum norms for τ-NAF occurring among of all elements in Z(τ). Application/Improvement: With these formulas, we can estimate the length of τ-NAF expansion more accurately.


Introduction
ECC was proposed by Miller 1 and Koblitz 2 . Koblitz 2 found that the Koblitz curves are a special type of curves for which the Forbenius endomorphism can be used to improve the performance of computing a scalar multiplication. Scalar multiplication of elliptic curve is the operation of successively adding a point along an elliptic curve defined as nP where P is the point on the curve and n is an integer called scalar. Koblitz curve defined over 2 m F as follows. Other than that, there are several other research have been carried out with the same aim which is to improve or upgrade the performances of scalar multiplication on ECC. Solinas 4 introduced Non-Adjacent Form (NAF) expansion followed by Solinas 5 proposed τ-adic Non-Adjacent Form (τ-NAF) expansion for an integer ( ) τ ∈ n Z to compute scalar multiplication on Koblitz curve more efficiently than the previous version. Solinas 5 also claimed that reduced τ-NAF can be used to replace τ-NAF for scalar multiplication on Koblitz   Before we give the property of norms of element in ( ) τ Z , we would like to give the following table of short analysis of norms for τ-NAF with length-3 that was stated in Yunos et al. 9 . c c c C 2 C 1 C 0 t r= C 0 -2 C 2 s= C 1 + C 2 t N(r+s τ) Based on the analysis of Table 1, we found that the maximum norm of τ -NAF expansion for length-3 is equal to 8 and the minimum norm of τ -NAF expansion for length-3 is equal to 2. Table 1 gives us the basic idea on how to obtained maximum and minimum norms for τ -NAF expansion those have certain length by analysing each norm for every integer in that length.
Based on the above theorem, Solinas 5 By listing the value of all the norms ( ) τ + N r s consist in a certain length, Solinas 5 can determine the value of maximum and minimum norms for that length. A few years later, Yunos et al. 9 gave an alternative to the direct evaluation method by introducing the formula of norm for every length-l that is . However, the direct evaluation method by Solinas 5 and the alternative formula produced by Yunos et al. 9 are not practically applicable for a large length where there are more than 43692 combination of ( ) τ + N r s that must be considered for length more than 15. Therefore, forming a formula to calculate maximum and minimum norms directly are important because we can estimate both maximum and minimum norms for any length. As mentioned in Solinas 5 , the practical size of τ -NAF was at least 83.
That is why we need to choose at least 42 d = to estimate the length of τ -NAF expansion. But the question now, is relation (1) still practical to be used to estimate the length for the case 83 l > or not? So in the next chapter, we will discuss in detail about this matter.

2.Result and Discussion
In this section, we discuss on how to obtain the formulas of maximum and minimum norms for τ-NAF occurring among all length element of ( ).  Based on Table 2, we can see that the values of maximum and minimum norms of τ-NAF are increase as the length increase. Other than that, we also found that the increment of maximum and minimum norms for every 1 l + are about 2 . l By making an analysis on the sequence of maximum norms as shown in Table 2 can be written as 2 2 (2954) + 0, 2 2 (5915) + 2, 2 2 (11831) + 0, 2 2 (23655) + 2, +2 2 (47229) + 0, 2 2 (94585) + 0, 2 2 (189151) + 0, 2 2 (378275) + 2.
Based on the above sequence, we obtain the following theorem.            By comparing relation (4) that has been produced with the relation (1), we found that the boundary of τ-NAF expansion is not same. As relation (4)