Cryptosystem

<h2 id="formal-definition">Formal definition</h2>
<p>Mathematically, a cryptosystem or encryption scheme can be defined as a <a href="/facts/Tuple/BI1HIxL8">tuple</a> 
  
    
      
        (
        
          
            P
          
        
        ,
        
          
            C
          
        
        ,
        
          
            K
          
        
        ,
        
          
            E
          
        
        ,
        
          
            D
          
        
        )
      
    
    {\displaystyle ({\mathcal {P}},{\mathcal {C}},{\mathcal {K}},{\mathcal {E}},{\mathcal {D}})}
  
 with the following properties.
</p>
<ol><li>
  
    
      
        
          
            P
          
        
      
    
    {\displaystyle {\mathcal {P}}}
  
 is a set called the "plaintext space". Its elements are called plaintexts.</li>
<li>
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 is a set called the "ciphertext space". Its elements are called ciphertexts.</li>
<li>
  
    
      
        
          
            K
          
        
      
    
    {\displaystyle {\mathcal {K}}}
  
 is a set called the "key space". Its elements are called keys.</li>
<li>
  
    
      
        
          
            E
          
        
        =
        {
        
          E
          
            k
          
        
        :
        k
        ∈
        
          
            K
          
        
        }
      
    
    {\displaystyle {\mathcal {E}}=\{E_{k}:k\in {\mathcal {K}}\}}
  
 is a set of functions 
  
    
      
        
          E
          
            k
          
        
        :
        
          
            P
          
        
        →
        
          
            C
          
        
      
    
    {\displaystyle E_{k}:{\mathcal {P}}\rightarrow {\mathcal {C}}}
  
. Its elements are called "encryption functions".</li>
<li>
  
    
      
        
          
            D
          
        
        =
        {
        
          D
          
            k
          
        
        :
        k
        ∈
        
          
            K
          
        
        }
      
    
    {\displaystyle {\mathcal {D}}=\{D_{k}:k\in {\mathcal {K}}\}}
  
 is a set of functions 
  
    
      
        
          D
          
            k
          
        
        :
        
          
            C
          
        
        →
        
          
            P
          
        
      
    
    {\displaystyle D_{k}:{\mathcal {C}}\rightarrow {\mathcal {P}}}
  
. Its elements are called "decryption functions".</li></ol>
<p>For each 
  
    
      
        e
        ∈
        
          
            K
          
        
      
    
    {\displaystyle e\in {\mathcal {K}}}
  
, there is 
  
    
      
        d
        ∈
        
          
            K
          
        
      
    
    {\displaystyle d\in {\mathcal {K}}}
  
 such that 
  
    
      
        
          D
          
            d
          
        
        (
        
          E
          
            e
          
        
        (
        p
        )
        )
        =
        p
      
    
    {\displaystyle D_{d}(E_{e}(p))=p}
  
 for all 
  
    
      
        p
        ∈
        
          
            P
          
        
      
    
    {\displaystyle p\in {\mathcal {P}}}
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>Note; typically this definition is modified in order to distinguish an encryption scheme as being either a <a href="/facts/Symmetric-key_algorithm/5zfe0b76">symmetric-key</a> or <a href="/facts/Public-key_cryptography/gowoFTxS">public-key</a> type of cryptosystem.
</p>
<h2 id="examples">Examples</h2>
<p>A classical example of a cryptosystem is the <a href="/facts/Caesar_cipher/Pjy93nA5">Caesar cipher</a>. A more contemporary example is the <a href="/facts/RSA_(cryptosystem)/XTzyfHuq">RSA</a> cryptosystem.
</p><p>Another example of a cryptosystem is the <a href="/facts/Advanced_Encryption_Standard/HdXGBY2Q">Advanced Encryption Standard</a> (AES). AES is a widely used symmetric encryption algorithm that has become the standard for securing data in various applications.
</p><p><a href="/facts/Paillier_cryptosystem/oLl6TR8H">Paillier cryptosystem</a> is another example used to preserve and maintain privacy and sensitive information. It is featured in electronic voting, electronic lotteries and electronic auctions.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_cryptosystems/l1yI2M2R">List of cryptosystems</a></li>
<li><a href="/facts/Semantic_security/PlW18aEh">Semantic security</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Menezes, A.; Oorschot, P. van; Vanstone, S. (1997). Handbook of Applied Cryptography (5th ed.). CRC Press. ISBN 0-8493-8523-7. <a href="0-8493-8523-7" target="_blank">0-8493-8523-7</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Buchmann, Johannes A. (13 July 2004). Introduction to Cryptography (2nd ed.). Springer. ISBN 0-387-20756-2. <a href="0-387-20756-2" target="_blank">0-387-20756-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Xia, Zhe; Yang, Xiaoyun; Xiao, Min; He, Debiao (2016). "Provably Secure Threshold Paillier Encryption Based on Hyperplane Geometry". In Liu, Joseph K.; Steinfeld, Ron (eds.). Information Security and Privacy. Lecture Notes in Computer Science. Vol. 9723. Cham: Springer International Publishing. pp. 73–86. doi:10.1007/978-3-319-40367-0_5. ISBN 978-3-319-40367-0. <a href="978-3-319-40367-0" target="_blank">978-3-319-40367-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Cryptosystem open-in-new

Cryptosystem