On Repeated Squarings in Binary Fields

Abstract

Abstract. In this paper, we discuss the problem of computing repeated squarings (exponentiations to a power of 2) in finite fields with polynomial basis. Repeated squarings have importance, especially, in elliptic curve cryptography where they are used in computing inversions in the field and scalar multiplications on Koblitz curves. We explore the problem specifically from the perspective of efficient implementation using field-programmable gate arrays (FPGAs) where the look-up table (LUT) structure helps to reduce both area and delay overheads. In fact, we show that the optimum construction depends on the size of the LUTs. We propose several repeated squarer architectures and demonstrate their practicability for FPGA-based implementations. Finally, we show that the proposed repeated squarers can offer significant speedups and even improve resistivity against side-channel attacks.

Accelerating Scalar Conversion for Koblitz Curve Cryptoprocessors on Hardware Platforms

“…Since squaring in F 2 m is cheap [1], [19], computing the Frobenius map of a point is also easy. The map operator τ satisfies the relation τ 2 + 2 = µτ , where µ = (−1)…”

Pushing the Limits of High-Speed GF(2 m ) Elliptic Curve Scalar Multiplication on FPGAs

“…For example, in GF (2 163 ) and FPGAs having 4 input LUTs (such as Xilinx Virtex 4), the optimal choice for n is 2. More details on choosing n can be found in [8]. The number of cascades, u s , depends on the critical delay of the ECM and will be discussed in Section 4.…”