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

*Jean-Pierre Flori and Hugues Randriam*

**Abstract: **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.

**Category / Keywords: **foundations / boolean functions

**Date: **received 16 May 2011

**Contact author: **flori at enst fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20110518:133427 (All versions of this report)

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