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The double-base number system and its application to elliptic curve cryptography. (English) Zbl 1133.14034
This paper, an extension of a previous paper of the authors [Advances in Cryptology, ASIACRYPT’96, LNCS vol. 3788, Springer, 59–78 (2005; Zbl 1154.94388)], presents a representation of a natural number as a sum of powers of 2 and 3 and its application to scalar multiplication on elliptic curves.
The scalar multiplication of a point on an elliptic curve by a natural number \(n\) is the basic operation in the discrete logarithm based Elliptic Curve Cryptography. To minimize its computational cost, alternatives to the usual binary representation of \(n\), such as Non-Adjacent Form (NAF) or windows methods (w-NAF), as well as different coordinate representations of the points of the curve have been proposed. For a good overview of the problem to see the book of D. Hankerson, A. Menezes and S. Vanstone [Guide to elliptic curve cryptography. Springer Professional Computing. New York, NY: Springer. (2004; Zbl 1059.94016)].
In the present paper the authors propose to use the representation of the scalar \(n\) as a sum of mixed powers of 2 and 3. This double-base number system (DBNS), in the authors’ words, “leads to fewer point additions that others classical methods”, and it can also provides protection against various kinds of side channels attacks.
Section 2 studies the DBNS representations and gives a greedy algorithm (Algorithm 1) to find a sparse signed DBNS representation (although maybe not a minimal one). Section 2 also recalls the basic of elliptic curves and the cost of point addition and point doubling in both affine and Jacobian projective coordinates.
Section 3 studies efficient ways to perform elliptic curve operations (point tripling, point quadrupling, etc) to be applied in the DBNS scalar multiplication algorithm presented in Section 4. In order to reduce the number of operations the algorithm uses a modified DBNS representation of the scalar \(n\) (Algorithm 2).
The authors illustrate the efficiency of the proposed algorithm in Section 5, providing results of numerical experiments over 10.000 randomly chosen 160 bits integers and giving the comparison with others classical (double-and-add, NAF, w-NAF) and recent representation methods.

MSC:
14G50 Applications to coding theory and cryptography of arithmetic geometry
94A60 Cryptography
11A63 Radix representation; digital problems
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Software:
gmp
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