[Resource Topic] 2011/245: On the Number of Carries Occuring in an Addition $\mod 2^k-1$

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On the Number of Carries Occuring in an Addition \mod 2^k-1

Authors: Jean-Pierre Flori, Hugues Randriam


In this paper we study the number of carries occurring while performing an addition modulo 2^k-1. For a fixed modular integer t, it is natural to expect the number of carries occurring when adding a random modular integer a to be roughly the Hamming weight of t. Here we are interested in the number of modular integers in \Zk producing strictly more than this number of carries when added to a fixed modular integer t \in \Zk. In particular it is conjectured that less than half of them do so. An equivalent conjecture was proposed by Tu and Deng in a different context~\cite{DCC:TD}. Although quite innocent, this conjecture has resisted different attempts of proof~\cite{DBLP:conf/seta/FloriRCM10, cryptoeprint:2010:170, cusick_combinatorial_2011, Carlet:Private} and only a few cases have been proved so far. The most manageable cases involve modular integers t whose bits equal to \textttup{0} are sparse. In this paper we continue to investigate the properties of \Ptk, the fraction of modular integers a to enumerate, for t in this class of integers. Doing so we prove that \Ptk has a polynomial expression and describe a closed form of this expression. This is of particular interest for computing the function giving \Ptk and studying it analytically. Finally we bring to light additional properties of \Ptk in an asymptotic setting and give closed forms for its asymptotic values.

ePrint: https://eprint.iacr.org/2011/245

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