From 27f6b3d7dd540826eb977494fd291fa790d62b46 Mon Sep 17 00:00:00 2001 From: Christoph Fuerst Date: Mon, 3 Apr 2017 20:46:28 +0200 Subject: [PATCH] Corrected wrong invariant of LeftToRight, Updated Report, Implemented Algorithm for Discrete Log --- lefttoright.txt | 7 ++--- report/formal.pdf | Bin 297700 -> 297323 bytes report/formal.tex | 4 +-- src/primefactors.cpp | 75 ++++++++++++++++++++++++++++++++++++++++++++------- 4 files changed, 70 insertions(+), 16 deletions(-) diff --git a/lefttoright.txt b/lefttoright.txt index 285ec7b..73fce20 100644 --- a/lefttoright.txt +++ b/lefttoright.txt @@ -20,11 +20,7 @@ proc LeftToRightExponentation(x:Base,n:Exp): Result var locx:ℕ[N^(2*M)] = x; while locn > 0 do invariant (n ≥ locn) ∧ (locn ≥ 0) ∧ - (x^n = res*(locx^locn)) ∧ - (old_locx ≤ locx) ∧ - (res = 0 ∨ old_res ≤ res) ∧ - (((locn%2 = 1) ⇒ (old_res < res)) ∨ - ((locn%2 = 0) ⇒ (old_res = res))); + (x^n = res*(locx^locn)); decreases locn; { if locn%2 = 1 then @@ -38,3 +34,4 @@ proc LeftToRightExponentation(x:Base,n:Exp): Result } + diff --git a/report/formal.pdf b/report/formal.pdf index 8974510c74c08f21e578031fb17aeded6f2d6a7a..7a15370efe4d9a71a9cc9e5bf5bbe068519fdeb1 100644 GIT binary patch delta 63770 zcmV(#K;*yVl@jZc5|ASUGBz`lkp(D!-B(L<+%^op=U3>+orx5_L`g??Yq##WosPZJ z9^xKkncmfmR@{-esoh^6fS~r(P2=>ErWc6<0TB4`qIj}Qc=E#={>t}vi#PACM3qRM z86)##@stS7GtP^|lp-thDp@=x@7dK%3Fc;svteC#vs9Ua{f0Cxu53 z@NL)y@Cmkb$TqZowH{}GEPhIKkySz@snl5o#{~5?@r*4>^h=U&qrUm@vhi*?)SJ}{ zYSX`l zx~F9w4hE*XW+n@^8f@F!A=sRM=0&EsffK-ZrAYOy9c?`jF9}}vqi_AzTl#!*11<`7 z_p(N=hgk|w*R>7p&)s^n^%HVHxaT>)y}p~nTae^U+y50T2vNxuz-BTj1zB*h8QP9S z)msvfTogwj?ZFZzsj|vQLs$us3BWDY5N54OAz+~xT3=|k8Nk1OCQIgjk&r`>0v)`f z6oB;ybZbh1nP8JIz#XyxSn6)+V)O&uuWBE6hIrC;w2!nILXKOTP1gc3U>}S|gR?5n zV^sGkfd;;y+1V`BTCjOnk0T+RGqh>XI#G{IlA3{|o^~h@8jQ5GSl6(GN85nQXbnF$ z1Rb{>Sg)hYx-cw&wq4_Y6GL7CLc{>N+yKUE2GPoeV1KSGf`EJV!{~>)@uXsZE~45I zaWlAw&Z5;pV3LGDrRQv$qKDvW0igC_p^p5=4ey9Py6yu?*hmHoqQDSIkL}oOM+g^y 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a/report/formal.tex +++ b/report/formal.tex @@ -335,7 +335,7 @@ $$ The termination of the loop is a consequence of $\lfloor n/2^k\rfloor \rightarrow 0$ after finitely many steps, i.e. there exists $k\in\mathbb{N}$ such that $\lfloor n/2^k\rfloor = 0$. \newpage -\subsection{The RISCAL Formalization} +\subsection*{The RISCAL Formalization} {\scriptsize \verbatimboxed{../lefttoright.txt}} \subsection{Computing Integer Roots} @@ -396,7 +396,7 @@ y > a \Leftrightarrow (x^2+z) > a \Leftrightarrow x^2+2x+1 > a \Leftrightarrow ( $$ hence we have shown $x^2\leq a < (x+1)^2$, and taking square roots shows the claim. \newpage -\subsection{The RISCAL Formalization} +\subsection*{The RISCAL Formalization} %{\scriptsize \verbatiminput{../integerroot.txt}} {\scriptsize \verbatimboxed{../integerroot.txt}} \appendix diff --git a/src/primefactors.cpp b/src/primefactors.cpp index d4ad778..b1a3306 100644 --- a/src/primefactors.cpp +++ b/src/primefactors.cpp @@ -127,31 +127,81 @@ void tonelli(long a, long g, long p) } t = locpm1; - cout << "Express p-1 = " << p-1 << " as 2^" << s << "*" << t << endl; +// cout << "Express p-1 = " << p-1 << " as 2^" << s << "*" << t << endl; unsigned long* e = new unsigned long[s]; e[0] = 0; xGCD(g,p,u,v); gi = fmod(u,p)+p; - cout << "So far we got: t=" << t << " and s=" << 2 << " and gi=" << gi << endl; +// cout << "So far we got: t=" << t << " and s=" << 2 << " and gi=" << gi << endl; for(long i=1;i> input; cout << endl; - }while((input != 'A') && (input != 'C') && (input != 'D') ); + }while((input != 'A') && (input != 'C') && (input != 'D') && (input != 'E') ); switch(input) { @@ -222,10 +273,16 @@ int main(int argc, char** argv) } case 'D': { - cout << "Algorithm of Tonelli" << endl; +// cout << "Algorithm of Tonelli" << endl; tonelli(10,2,13); break; } + case 'E': + { + // cout << "Algorithm of Tonelli" << endl; + discrete_log(113, 3, 57); + break; + } default: break; } -- 2.1.4