A new fast, memory-efficient, and SPA-protected elliptic curve cryptography algorithm

Elliptic curve scalar multiplication kP, where k is a non-negative constant and P is a point on the elliptic curve, requires two different operations: addition (ADD) and doubling (DBL). To reduce the number of ADDs without increasing the number of DBLs, a recoding of k with fewer non-zero digits is needed. Based on Radix-2w arithmetic, we introduce a principled w-bit windowing method where the properties of speed, memory and security are described by exact analytic formulas as proof of superiority. In contrast to existing window algorithms, to minimize the number of ADDs, the window size (w) is controlled by an optimum depending on the bit length (l) of the scalar k. The number of pre-calculations required is minimal with respect to the value of w. The proposed method recodes the binary string k and evaluates the multiplication from right to left and left to right in the same way. The Radix-2w method is very easy to use and highly reconfigurable, allowing speed memory and speed security to compromise to meet various crypto system constraints. In addition, the method shows high resilience to side-channel attacks based on strength, timing and statistical analysis. All Radix-2w properties are confronted with standard windowing methods through an in-depth analysis of the complexity. A general comparison is made via NIST recommended GF (2l) finite fields.


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