1% ============================================================================2% UQO Working Paper 34 — Beyond the Square-Root Barrier3% ============================================================================4\documentclass[12pt,letterpaper]{article}5\usepackage[utf8]{inputenc}6\usepackage[T1]{fontenc}7\usepackage[english]{babel}8\usepackage[letterpaper,top=1in,bottom=1in,left=1in,right=1in]{geometry}9\usepackage{mathptmx}10\usepackage{microtype}11\usepackage{setspace}12\onehalfspacing13\usepackage{amsmath,amssymb,amsthm}14\usepackage{bm}15\usepackage{booktabs}16\usepackage{tabularx}17\usepackage{multirow}18\usepackage{threeparttable}19\usepackage{longtable,array,calc}20\usepackage{siunitx}21\sisetup{input-symbols = {()}, parse-numbers = false}22\usepackage{graphicx}23\usepackage{subcaption}24\usepackage{float}25\usepackage{tikz}26\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds,shapes.misc,shapes.geometric,matrix,chains}27\usepackage[dvipsnames]{xcolor}28\usepackage[colorlinks=true,linkcolor=NavyBlue,citecolor=NavyBlue,urlcolor=NavyBlue,breaklinks=true]{hyperref}29\usepackage[authoryear,round,semicolon]{natbib}30\bibliographystyle{aer}31\usepackage[font=small,labelfont=bf,labelsep=period,justification=justified,singlelinecheck=false]{caption}32\usepackage{fancyhdr}33\pagestyle{fancy}34\fancyhf{}35\renewcommand{\headrulewidth}{0pt}36\fancyfoot[C]{\thepage}37\usepackage{titlesec}38\titleformat{\section}{\large\bfseries}{\thesection.}{0.5em}{}39\titleformat{\subsection}{\normalsize\bfseries}{\thesubsection.}{0.5em}{}40\titleformat{\subsubsection}{\normalsize\itshape}{\thesubsubsection.}{0.5em}{}41\usepackage[toc,page]{appendix}42\usepackage{enumitem}43\setlist{nosep,leftmargin=*}44\usepackage{algorithm}45\usepackage{algpseudocode}46\usepackage{listings}47\lstset{basicstyle=\ttfamily\footnotesize,breaklines=true,columns=fullflexible,frame=single,framerule=0.2pt}4849\theoremstyle{plain}50\newtheorem{theorem}{Theorem}[section]51\newtheorem{proposition}[theorem]{Proposition}52\newtheorem{lemma}[theorem]{Lemma}53\newtheorem{corollary}[theorem]{Corollary}54\newtheorem{fact}[theorem]{Fact}55\theoremstyle{definition}56\newtheorem{definition}[theorem]{Definition}57\newtheorem{assumption}[theorem]{Assumption}58\newtheorem{hypothesis}[theorem]{Hypothesis}59\newtheorem{conjecture}[theorem]{Conjecture}60\newtheorem{observation}[theorem]{Observation}61\theoremstyle{remark}62\newtheorem{remark}[theorem]{Remark}63\newtheorem{example}[theorem]{Example}6465\newcommand{\F}{\mathbb{F}}66\newcommand{\Z}{\mathbb{Z}}67\newcommand{\Q}{\mathbb{Q}}68\newcommand{\N}{\mathbb{N}}69\newcommand{\R}{\mathbb{R}}70\newcommand{\Fp}{\mathbb{F}_p}71\newcommand{\E}{\mathcal{E}}72\newcommand{\Oinf}{\mathcal{O}}73\newcommand{\End}{\operatorname{End}}74\newcommand{\Aut}{\operatorname{Aut}}75\newcommand{\ord}{\operatorname{ord}}76\newcommand{\Norm}{\operatorname{N}}77\newcommand{\Tr}{\operatorname{Tr}}78\newcommand{\Ch}{\mathrm{Ch}}79\newcommand{\Maj}{\mathrm{Maj}}80\newcommand{\SigZ}{\Sigma_0}81\newcommand{\SigO}{\Sigma_1}82\newcommand{\sigz}{\sigma_0}83\newcommand{\sigo}{\sigma_1}84\newcommand{\rotr}{\mathrm{R}}85\newcommand{\shr}{\mathrm{S}}86\newcommand{\secp}{\textsf{secp256k1}}87\newcommand{\libsecp}{\texttt{libsecp256k1}}88\newcommand{\kk}{\texttt{k256}}89\newcommand{\code}[1]{\texttt{#1}}90\newcommand{\Hname}[1]{\textsf{#1}}91\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}92\graphicspath{{figures/}}9394% Numbers generated by scripts (never typed by hand): paper/tables/numbers.tex95\input{tables/numbers}96\input{tables/numbers_ecdlp}9798% ============================================================================99\newcommand{\WPnumber}{34}100\newcommand{\WPtitle}{Beyond the Square-Root Barrier: Automated Discovery of Algebraic and Algorithmic Shortcuts in Bitcoin Cryptography}101\newcommand{\WPsubtitle}{A hypothesis-driven, reproducible study of secp256k1, ECDSA, BIP340 Schnorr and SHA-256 with machine-checked identities and controlled negative results}102\newcommand{\WPdate}{November 2026}103\newcommand{\WPversion}{1.0}104\newcommand{\WPabstract}{\input{sections/abstract}}105\newcommand{\WPkeywords}{elliptic curve discrete logarithm; secp256k1; Pollard rho; endomorphisms; ECDSA; Schnorr signatures; BIP340; SHA-256; SAT-based cryptanalysis; automated mathematical discovery; Lean 4; reproducibility}106\newcommand{\WPjel}{C63, C88, D83 (MSC 2020: 94A60, 11T71, 11G20, 68V15, 68W30)}107\newcommand{\WPauthor}{Simon-Pierre Boucher}108\newcommand{\WPaffiliation}{%109 Département des sciences administratives\\110 Université du Québec en Outaouais%111}112\newcommand{\WPemail}{simon-pierre.boucher@uqo.ca}113\newcommand{\WPaddress}{%114 Gatineau -- Pavillon Alexandre-Taché\\115 283, boulevard Alexandre-Taché\\116 Gatineau, Québec, Canada J9A 1L8%117}118119\begin{document}120\input{sections/titlepage}121\tableofcontents122\newpage123\input{sections/01_introduction}124\input{sections/02_foundations}125\input{sections/03_literature}126\input{sections/04_secp256k1}127\input{sections/05_dlog_algorithms}128\input{sections/06_signatures}129\input{sections/07_sha256}130\input{sections/08_discovery_framework}131\input{sections/09_experimental_design}132\input{sections/10_results}133\input{sections/11_formal_negative}134\input{sections/12_implications}135\input{sections/13_limitations}136\input{sections/14_conclusion}137\newpage138\bibliography{references}139\input{appendix/appendix}140\end{document}141