Abstract-Scalar recoding is popular to speed up ECC scalar multiplication: non-adjacent form, double-base number system, multi-base number system. But fast recoding methods require pre-computations: multiples of base point or off-line conversion. In this paper, we present a multi-base recoding method for ECC scalar multiplication based on i) a greedy algorithm starting least significant terms first, ii) cheap divisibility tests by multi-base elements and iii) fast exact divisions by multibase elements. Multi-base terms are obtained on-the-fly using a special recoding unit which operates in parallel to curve-level operations and at very high speed. This ensures that all recoding steps are performed fast enough to schedule the next curve-level operations without interruptions. The proposed method can be fully implemented in hardware without pre-computations. We report FPGA implementation details and very good performances compared to state-of-art results.
Abstract-The paper presents arithmetic level protections for ECC processor against some side channel attacks. The proposed protection is based on random recodings of the secret key in the double base number system (DBNS). DBNS is a highly redundant and sparse number system. Here, the high redundancy level of DBNS is used to randomly modify on-the-fly the ki digits during the scalar multiplication [k]P . The proposed solution leads to random numbers and orders of curve level operations (point addition, doubling and tripling) during the computation of [k]P operations. Our random recoding method provides [k]P computation time comparable to the best w-NAF recoding methods. But standard w-NAF recodings are deterministic ones while our solution is a random one.
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