We present the Double-Base Number system (DBNS) and its applications to cryptography, mainly to speed-up scalar multiplications on elliptic curves.

After a brief introduction, we focus on 2 practical contributions:

The first scalar multiplication algorithm having sublinear complexity. This method relies on a generalisation of the DBNS in the context of Koblitz curves

The fastest scalar multiplication algorithm for generic curves when some precomputations are available. This work relies on the so-called extended DBNS which is also a natural generalisation of the DBNS.