From 2f878c8130646c1ad4cb0e8db76e68ab8fe95abf Mon Sep 17 00:00:00 2001 From: Christoph Fuerst Date: Sun, 9 Apr 2017 09:02:13 +0200 Subject: [PATCH] Finalized formalization of discrete_log --- discrete_log.txt | 2 ++ report/formal.pdf | Bin 351029 -> 358002 bytes report/formal.tex | 46 ++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 48 insertions(+) diff --git a/discrete_log.txt b/discrete_log.txt index 30b5143..c5c11a5 100644 --- a/discrete_log.txt +++ b/discrete_log.txt @@ -88,6 +88,7 @@ proc DiscreteLog(p:prime,g:nat,a:nat): res { // result found ret = i*m + j; + print ret; // abort loop // Q: Is there a more comfortable way (e.g. break??) i = m-1; @@ -100,3 +101,4 @@ proc DiscreteLog(p:prime,g:nat,a:nat): res return ret; } + diff --git a/report/formal.pdf b/report/formal.pdf index 55e74ad16fe14333d5ed897a7b00bcfd5bd1fdc3..600a46b5c7a5ceb2872314cd056de53dadefba95 100644 GIT binary patch delta 139551 zcmZshQ*fYNu&!g~Jtpl$Vb)x{OZT8?F!@#Y_r1% 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An algorithm to compute the discrete logarithm is the Baby-Step +Giant-Step algorithm as sketeched before. A possible formalization might be formulated +as follows: +{\scriptsize \verbatimboxed{../discrete_log.txt}} +Some comments on the formalization: +\begin{itemize} + \item The first argument is the characteristic of the finite field, which is a prime. + Below 10, the primes are 2,3,5,7. Obviously, for $p=2$ we have no elements of + order $p-1$; + \item The condition that $g$ has to be of order $p-1$ is a major restriction. + The only elements that satisfy this are given by +\end{itemize} +\begin{center} +\begin{tabular}{|c|c|} +\hline +$p$ & Elements of order $p-1$\\ +\hline +3 & 2 \\ +5 & 2,3\\ +7 & 3,5\\ +\hline +\end{tabular} +\end{center} +\begin{itemize} + \item The element $a$ has obviously to be non-zero in $\mathbb{Z}_p$. +\end{itemize} +Having this thoughts in mind, we expect that the set of possible inputs might +be shrinked dramatically. Indeed, the tuples with values less than 10, which +are in our scope might be explictly enumerated by +\begin{center} +\begin{tabular}{c} +$(p,g,a)$\\ +\hline +$(3,2,1)$, $(3,2,2)$\\ +$(5,2,1)$,$(5,2,2)$,$(5,2,3)$,$(5,2,4)$\\ +$(5,3,1)$, $(5,3,2)$, $(5,3,3)$, $(5,3,4)$\\ + $(7,3,1)$, $(7,3,2)$, $(7,3,3)$, $(7,3,4)$, $(7,3,5)$, $(7,3,6)$ \\ +$(7,5,1)$, $(7,5,2)$, $(7,5,3)$, $(7,5,4)$, $(7,5,5)$, $(7,5,6)$ +\end{tabular} +\end{center} +which gives a total of 22 elements to check. \appendix + + \begin{thebibliography}{9} \bibitem{hardylittlewood} G.H. Hardy and E.M. Wright. \textit{An Introduction to the Theory of Numbers}. -- 2.1.4