Efficient Algorithm for Multiplication of Numbers in Zeckendorf Representation

In the Zeckendorf representation an integer is expressed as a sum of Fibonacci numbers in which no two are consecutive. We show O(n log n) algorithm for multiplication of two n-digit numbers in Zeckendorf representation. For this purpose we investigate a relationship between the numeral system using Zeckendorf representations and the golden ratio numeral system. We also show O(n) algorithms for converting numbers between these systems

Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

Abstract. In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a mod-ified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as Pn i=1 2 bi3ti where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we pro-pose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups

Heap games, numeration systems and sequences

We propose and analyse a 2-parameter family of 2-player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an efficient strategy or not. We introduce yet another class of sequences, and demonstrate its equivalence with the class of sequences defined for the strategy of our games.Comment: To appear in Annals of Combinatoric

Implementing Write Compression in Flash Memory Using Zeckendorf Two-Round Rewriting Codes

Flash memory has become increasingly popular as the underlying storage technology for high-performance nonvolatile storage devices. However, while flash offers several benefits over alternative storage media, a number of limitations still exist within the current technology. One such limitation is that programming (altering a bit from its default value) and erasing (returning a bit to its default value) are asymmetric operations in flash memory devices: a flash memory can be programmed arbitrarily, but can only be erased in relatively large batches of storage bits called blocks, with block sizes ranging from 512K up to several megabytes. This creates a situation where relatively small write operations to the drive can potentially require reading out, erasing, and rewriting many times more data than the initial operation would normally require if that write would result in a bit erase operation. Prior work suggests that the performance impact of these costly block erase cycles can be mitigated by using a rewriting code, increasing the number of writes that can be performed on the same location in memory before an erase operation is required. This paper provides an implementation of this rewriting code, both as a software program written in C and as a SystemVerilog FPGA circuit specification, and discusses many of the additional design considerations that would be necessary to integrate such a rewriting code with current file storage techniques

On Aperiodic Subtraction Games with Bounded Nim Sequence

Subtraction games are a class of impartial combinatorial games whose positions correspond to nonnegative integers and whose moves correspond to subtracting one of a fixed set of numbers from the current position. Though they are easy to define, sub- traction games have proven difficult to analyze. In particular, few general results about their Sprague-Grundy values are known. In this paper, we construct an example of a subtraction game whose sequence of Sprague-Grundy values is ternary and aperiodic, and we develop a theory that might lead to a generalization of our construction.Comment: 45 page