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In this paper, we extend a recent piece of work on low-weight polynomial form integers (LWPFIs). We present a new coefficient reduction algorithm based on the Montgomery reduction algorithm and provide its detailed analysis results. We give a condition for eliminating the final subtractions at the end of our Mont-gomery reduction algorithm adapted to… (More)

We present efficient squaring formulae based on the Toom-Cook multiplication algorithm. The latter always requires at least one non-trivial constant division in the interpolation step. We show such non-trivial divisions are not needed in the case two operands are equal for three, four and five-way squarings. Our analysis shows that our 3-way squaring… (More)

In 1999, Jerome Solinas introduced families of moduli called the generalized Mersenne numbers (GMNs), which are expressed in low-weight polynomial form, p = f (t), where t is limited to a power of 2. GMNs are very useful in elliptic curve cryptosystems over prime fields, since only integer additions and subtractions are required in modular reductions.… (More)

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