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**2014/371**

**Title:**

On the Enumeration of Double-Base Chains with Applications to Elliptic Curve Cryptography

**Authors:**
Christophe Doche

**Abstract:**

The Double-Base Number System (DBNS) uses two bases, 2 and 3, in order to represent any integer n. A Double-Base Chain (DBC) is a special case of a DBNS expansion. DBCs have been introduced to speed up the scalar multiplication [n]P on certain families of elliptic curves used in cryptography. In this context, our contributions are twofold. First, given integers n, a, and b, we outline a recursive algorithm to compute the number of different DBCs with a leading factor dividing 2^a3^b and representing n. A simple modification of the algorithm allows to determine the number of DBCs with a specified length as well as the actual expansions. In turn, this gives rise to a method to compute an optimal DBC representing n, i.e. an expansion with minimal length. Our implementation is able to return an optimal expansion for most integers up to 2^{60} bits in a few minutes. Second, we introduce an original and potentially more efficient approach to compute a random scalar multiplication [n]P, based on the concept of controlled DBC. Instead of generating a random integer n and then trying to find an optimal, or at least a short DBC to represent it, we propose to directly generate n as a random DBC with a chosen leading factor 2^a3^b and length \ell. To inform the selection of those parameters, in particular \ell, which drives the trade-off between the efficiency and the security of the underlying cryptosystem, we enumerate the total number of DBCs having a given leading factor 2^a3^b and a certain length \ell. The comparison between this total number of DBCs and the total number of integers that we wish to represent a priori provides some guidance regarding the selection of suitable parameters. Experiments indicate that our new Near Optimal Controlled DBC approach provides a speedup of at least 10\% with respect to the NAF for sizes from 192 to 512 bits. Computations involve elliptic curves defined over \F_p, using the Inverted Edwards coordinate system and state of the art scalar multiplication techniques.

**ePrint:**
https://eprint.iacr.org/2014/371

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