algorithms.tex

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\usepackage{algorithm}
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\title{امضای دیجیتال مقاوم کوانتومی بر اساس همسانی های بین خم های سوپرسینگولار}
\author{مصطفی قربانی
	\\[1cm]{ استاد راهنما: دکتر حسن دقیق}}
%\author{مصطفی قربانی}
\date{}

\begin{document}
\maketitle


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% Algorithm 1 
% ===========================================================================================
\begin{algorithm}\label{algorithm_1}
	\caption{Prover : $P_{OE}$ on input $(x,w)$}
	%\begin{flushleft}
	%\resetlatinfont
	\begin{algorithmic}[1]
		\State // Create t.c proofs and hash each response
		\For{ $ i=1 \ \textbf{to} \ t $ }
			\State $com_i \leftarrow P_{\Sigma}^{1}(x,w)$
			\For{ $j=1 \ \textbf{to} \ c $ }
				\State $ch_{i,j} \leftarrow_{R} N_{ch} \setminus \{ch_{i,1} , \cdots , ch_{i,j-1} \}$
				\State $resp_{i,j} \leftarrow P_{\Sigma}^2 (x,w,com,ch_{i,j})$
				\State $h_{i,j} \leftarrow G(resp_{i,j})$ 
			\EndFor 
		\EndFor
		
		\State // Get challenge by hashing
		\State $ J_1 \parallel \cdots \parallel j_t \leftarrow H(x(com_i)_i , (ch_{i,j})_{i,j} , (h_{i,j})_{i,j}   ) $ \Comment{Get challenge by hashing}
		
		\State // return proof
		\State \textbf{return} $\pi \leftarrow ( (com_i)_i , (ch_{i,j})_{i,j} , (h_{i,j})_{i,j} , (resp_{i,J_i})_i  )$ \Comment{return proof}
	\end{algorithmic}
	%\end{flushleft}
\end{algorithm}

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% ===========================================================================================
% Algorithm 2 
% ===========================================================================================
\begin{algorithm}\label{algorithm_2}
	\caption{Verifier : $V_{OE}$ on input $(x,\pi) $
		where \newline
		\qquad $\pi =  ( (com_i)_i , (ch_{i,j})_{i,j} , (h_{i,j})_{i,j} , (resp_{i,J_i})_i  )$
		}
	%\begin{flushleft}
	%\resetlatinfont
	\begin{algorithmic}[1]
		\State // Compute the challenge hash
		\For{ $ i=1 \ \textbf{to} \ t $ }
		\State $ \textbf{check} ch_{i,1} , \cdots ch_{i,m} pairwise distinct $
		\State $ \textbf{check} h_{i,J_i} = G(resp_i) $
		\State $ \textbf{check} V_{\Sigma}(x,com_i,ch_{i,J_i} , resp_i) = 1 $
		\EndFor
		\If{ all checks succeed} 
		\textbf{return} 1
		\EndIf
		
	\end{algorithmic}
	%\end{flushleft}
\end{algorithm}

% ===========================================================================================
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% ===========================================================================================
% Algorithm 3 
% ===========================================================================================
\begin{algorithm}\label{algorithm_3}
	\caption{ $ KeyGen(\lambda) $}
	%\begin{flushleft}
	%\resetlatinfont
	\begin{algorithmic}[1]
		\State Pick a random point S of order $\ell_A^{e_A}$
		\State Compute the isogeny $\phi : E \rightarrow E / \langle S \rangle $
		\State $ pk \leftarrow (E / \langle S \rangle , \phi(P_B) , \phi(Q_B)) $
		\State $sk \leftarrow S$
		\State \textbf{return} $(pk,sk)$
		
	\end{algorithmic}
	%\end{flushleft}
\end{algorithm}

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% ===========================================================================================
% Algorithm 4 
% ===========================================================================================
\begin{algorithm}\label{algorithm_4}
	\caption{ $ Sign(sk,m)$ }
	%\begin{flushleft}
	%\resetlatinfont
	\begin{algorithmic}[1]
		\For {i=1 \textbf{to} $2\lambda$}
		\State Pick a random point R of order $\ell_B^{e_B}$
		\State Compute the isogeny $\psi : E \rightarrow E / \langle R \rangle $
		\State Compute either $ {\phi}' : E / \langle R \rangle \rightarrow E / \langle R ,S \rangle \ or \  {\psi}' : E / \langle S \rangle \rightarrow E / \langle R ,S \rangle $
		\State $(E_1,E_2) \leftarrow (E / \langle R \rangle , E / \langle R,S \rangle ) $
		\State $ com_i \leftarrow (E_1 , E_2 ) $
		\State $ch_{i,0} \leftarrow_{R} \{ 0,1\}$
		\State $ (resp_{i,0} , resp_{i,1}) \leftarrow ((R,\phi(R)) , \psi(S) ) $
		\If {$ch_{i,0} = 1$}
		\State swap $ (resp_{i,0} , resp_{i,1})$
		\EndIf 
		\State $h_{i,j} \leftarrow G(resp_{i,j})$
		\EndFor 
		\State $J_1 \parallel \cdots \parallel J_{2\lambda}  \leftarrow H( pk,m,(com_i)_i , (ch_{i,j})_{i,j} , (h_{i,j})_{i,j} )$
		\State \textbf{return} $ \sigma \leftarrow ( (com_i)_i , (ch_{i,j})_{i,j} , (h_{i,j})_{i,j} , (resp_{i,J_i})_i ) $
		
	\end{algorithmic}
	%\end{flushleft}
\end{algorithm}

% ===========================================================================================
% ===========================================================================================

% ===========================================================================================
% Algorithm 5 
% ===========================================================================================
\begin{algorithm}\label{algorithm_5}
	\caption{ $ Sign(sk,m)$ }
	%\begin{flushleft}
	%\resetlatinfont
	\begin{algorithmic}[1]
		\State $J_1 \parallel \cdots \parallel J_{2\lambda}  \leftarrow H( pk,m,(com_i)_i , (ch_{i,j})_{i,j} , (h_{i,j})_{i,j} )$
		\For {i=1 \textbf{to} $2\lambda$}
		\State \textbf{check} $h_{i,J_i} = G(resp_{i,J_i})$
		\If {$ ch_{i,J_i} = 0 $}
		\State Parse $(R,\phi(R)) \leftarrow resp_{i,J_i}$
		\State \textbf{check} $R , \phi(R)$ have order $\ell_B^{e_B}$
		\State \textbf{check} $R$ generates the kernel of the isogeny $E \leftarrow E_1$
		\State \textbf{check} $\phi(R)$ generates the kernel of the isogeny $E / \langle S \rangle \rightarrow E_2 $
		\ElsIf{}
		\State Parse $\psi(S) \rightarrow resp_{i,J_i}$
		\State \textbf{check} $\psi(S)$ has order $\ell_A^{e_A}$
		\State \textbf{check} $\psi(S)$ generates the kernel of the isogeny $E_1 \rightarrow E_2$
		\EndIf 
		\EndFor 
		\If{ all checks successd}
		\State \textbf{return} $1$
		\EndIf
		
	\end{algorithmic}
	%\end{flushleft}
\end{algorithm}

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\end{document}