Techniques for fast exponentiation (multiplication) in various groups have been extensively studied for use in cryptographic primitives. Specifically the joint expression of two exponents (multipliers) plays an important role in the performances of the algorithms used. The crucial optimization relies in general on minimizing the joint Hamming weight of the exponents (multipliers).
J.A.Solinas suggested an optimal signed binary representation for pairs of integers, which is called a Joint Sparse Form (JSF) [25]. JSF is at most one bit longer than the binary expansion of the larger of the two integers, and the average joint Hamming density among Joint Sparse Form representations is 1/2.
This paper extends the Joint Sparse Form by using a window method, namely, presents a new representation for pairs of integers, which is called Width-3 Joint Sparse Form (JSF3), and proves that the representation is at most one bit longer than the binary expansion of the larger of the two integers and its average joint Hamming density is 37.1% via the method of stochastic process. So, Computing the form of uP+vQ by using JSF3 is almost 8.6% faster than that by using JSF.