An analysis of double base number systems and a sublinear scalar multiplication algorithm

In this paper we produce a practical and efficient algorithm to find a decomposition of type n = ∑_i=1^k2^si3 ^ti, sⁱ tⁱ ∈ ℕ ∪ {0} with k ≤ (c +o(1)) log n/log log n . It is conjectured that one can take c = 2 above. Then this decomposition is refined into an effective scalar multiplication algorithm to compute nP on some supersingular elliptic curves of characteristic 3 with running time bounded by O (log n/log log n) and essentially no storage. To our knowledge, this is the first instance of a scalar multiplication algorithm that requires o(log n) curve operations on an elliptic curve over double struck F sign_q, with log q ≈ log n and uses comparable storage as in the standard double-and-add algorithm. This leads to an efficient algorithm very useful for cryptographic protocols based on supersingular curves. This is for example the case of the well-studied (in the past four years) identity based schemes. The method carries over to any supersingular curve of fixed characteristic.