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+\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{bm} +\usepackage[T1]{fontenc} +\usepackage{geometry} +\geometry{verbose,tmargin=2cm,bmargin=2cm,lmargin=2cm,rmargin=2cm} +\usepackage{orcidlink} + +\usepackage{setspace} +\AtBeginDocument{\let~=\nobreakspace} +\spacing{1.5} + +\usepackage{lineno} +\linenumbers + +\hypersetup{ + unicode=false, % non-Latin characters in Acrobat's bookmarks + pdftoolbar=true, % show Acrobat's toolbar? + pdfmenubar=true, % show Acrobat's menu? + pdffitwindow=false, % window fit to page when opened +% pdfstartview={FitW}, % fits the width of the page to the window + pdftitle={Closed-form Expression of Monero's wallet2 Decoy Selection Algorithm}, % title + pdfauthor={Rucknium}, % author + pdfsubject={}, % subject of the document + pdfcreator={Rucknium}, % creator of the document + pdfproducer={}, % producer of the document + pdfkeywords={}, % list of keywords + pdfnewwindow=true, % links in new window + colorlinks=false, % false: boxed links; true: colored links + linkcolor=red, % color of internal links + citecolor=green, % color of links to bibliography + filecolor=magenta, % color of file links + urlcolor=cyan % color of external links +} + + + + +\begin{document} + +\title{Closed-form Expression of Monero's \texttt{wallet2}\\Decoy Selection Algorithm\\\vspace{.3cm} +\large Draft v0.1\vspace{-.715cm}} +\author{Rucknium\orcidlink{https://orcid.org/0000-0001-5999-8950} } +\date{October 2023} + + +\maketitle + +\section{Modified Log-gamma distribution} + +Let $G(x;\alpha,\beta)$ be the Log-gamma cumulative distribution +function (CDF) + +\begin{equation} +G(x;\alpha,\beta)=\dfrac{\gamma\left(\alpha,\beta\ln\left(x\right)\right)}{\Gamma\left(\alpha\right)} +\end{equation} + +where $\alpha$ is the shape parameter, $\beta$ is the rate parameter, +$\gamma$ is the lower incomplete gamma function, and $\Gamma$ is +the gamma function. + +In the \texttt{wallet2} code, $\alpha=19.28$ and $\beta=1.61$. The +$G(x;\alpha,\beta)$ is adjusted in the code to: +\begin{enumerate} +\item Eliminate the portion of the distribution that would be younger than +the youngest spendable output and reallocate it to a uniform distribution +in the \texttt{RECENT\_SPEND\_WINDOW}. The\texttt{ $\frac{x/v_{t}}{1800}\cdot G(1200)$} +term of the expression below performs the reallocation. +\item Re-scale the output index distance unit into seconds. The $x\cdot v_{t}$ +term does the re-scaling. +\item Eliminate the portion of the distribution that would be older than +the oldest spendable output and reallocate it to the rest of the distribution. +Terms with $z_{t}$ perform this reallocation. +\end{enumerate} +$G^{*}(x;\alpha,\beta,v_{t},z_{t})$ is the modified CDF that \texttt{wallet2} +uses to randomly generate an output index: + +\begin{equation} +G^{*}(x;\alpha,\beta,v_{t},z_{t})=\begin{cases} +0 & \textrm{if }x\cdot v_{t}<0\\ +\left(G(x\cdot v_{t}+1200)-G(1200)+\frac{x\cdot v_{t}}{1800}\cdot G(1200)\right)/G\left(z_{t}\cdot v_{t}\right) & \textrm{if }0\leq x\cdot v_{t}\leq1800\\ +G\left(x\cdot v_{t}+1200\right)/G\left(z_{t}\cdot v_{t}\right) & \textrm{if }1800