A Preliminary Study on Methods to Eliminate Short Fruitless Cycles for Pollards Rho Method for ECDLP over BN Curves This paper discusses the conditions of fruitless cycles for Pollardu0027s rho method with skew Frobenius mapping for elliptic curve discrete logarithm problem (ECDLP) over a BarretoNaehrig (BN) curve. When a random walk pass achieves a fruitless cycle, the random walk pass must restart with a different starting point. There can be several methods to eliminate fruitless cycles of short lengths for BN curves based on the construction of a random walk table. Therefore, in this research, the authors give an analysis of the previous methods for a BN curve of order 349 with the mapping with experiments. The results show that several fruitless cycles of length two and three cannot be eliminated by the previous methods. The authors investigated the reason for degradation. As a result of the analysis, a new condition where a part of the fruitless cycles of any lengths occur is derived, and a new method to eliminate the fruitless cycles is proposed.

Fixed Polarity Pascal Transforms with Symbolic Computer Algebra Applications The fixed polarity forms of the ReedMuller (RM) transform exist in 2n different polarities. The integervalued Pascal transform is related to the binaryvalued RM transform through the Sierpinski fractal, calculated by performing the modulo2 operation on Pascalxe2x80x99s triangle, as it appears in the lower triangular portion of the positivepolarity RM transform. We generalize the relationship between the fixedpolarity forms of the RM transform and introduce associated forms of the Pascal transform that are characterized by a polarity value allowing for a family of fixedpolarity Pascal (FPP) transform matrices to be defined. We observe and prove several properties of the FPP transforms and their inverses. An application of FPP transforms in the area of symbolic computer algebra that enables very fast decomposition of realvalued polynomials as weighted sums of different binomials raised to a power as compared to manual symbolic manipulation is described. The decomposition weights can be considered to be the inverse FPP spectrum with respect to a realvalued polynomial since they are computed using one of the linear orthogonal FPP transformation matrices.