Perfect security

Perfect security or perfect secrecy is a term coined by Claude Shannon from information theory and cryptology . A perfectly secure encryption method is characterized in that a generated with him ciphertext (also referred to as a secret text or cipher text) no conclusions about the corresponding plaintext permits. With such a method, it is mathematically proven that an attacker who knows the ciphertext cannot gain any further information about it apart from the length of the plaintext. He cannot decipher the key text or even break the entire procedure .

definition

We need the components of an encryption process: Let be the set of all possible plaintexts, the set of keys and the set of all possible ciphers. These sets each have a finite number of elements. In addition, assume an encryption function and the corresponding decryption function. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle E\colon {\mathcal {P}}\times {\mathcal {K}}\to {\mathcal {C}}}$ ${\displaystyle D\colon {\mathcal {C}}\times {\mathcal {K}}\to {\mathcal {P}}}$

As a rule, plain texts do not occur with the same probability . This depends, for example, on the language used or a protocol that the conversation follows. denote the probability with which a plaintext occurs. ${\displaystyle Pr(p)}$ ${\displaystyle p\in {\mathcal {P}}}$

An encryption method is called perfectly secure if the occurrence of a certain plain text is stochastically independent of the fact that a certain cipher is present. The equation therefore applies to all plaintexts and every cipher . ${\displaystyle p\in {\mathcal {P}}}$ ${\displaystyle c\in {\mathcal {C}}}$ ${\displaystyle Pr(p|c)=Pr(p)}$

If an attacker intercepts a ciphertext , it is not possible for him to evaluate statistical abnormalities of the plaintext space. If he tries to guess the corresponding plain text, he is only correct with the probability that he would also achieve if he did not know the intercepted key text and blindly guessed a plain text. ${\displaystyle c}$ ${\displaystyle c}$

Shannon's theorem

In 1949 Shannon proved the following theorem, which explains the conditions under which an encryption method is perfectly secure.

Be and for all was . The encryption method is perfectly safe , if and only if the probability distribution on the key space the uniform distribution , and if for each plaintext and ciphertext each key exactly there, so . ${\displaystyle |{\mathcal {P}}|=|{\mathcal {K}}|=|{\mathcal {C}}|<\infty }$ ${\displaystyle p\in {\mathcal {P}}}$ ${\displaystyle Pr(p)>0}$ ${\displaystyle p\in {\mathcal {P}}}$ ${\displaystyle c\in {\mathcal {C}}}$ ${\displaystyle k\in {\mathcal {K}}}$ ${\displaystyle E(p,k)=c}$

practice

From Shannon's theorem it follows that with perfectly secure methods the probability distribution on the key space must be the uniform distribution. Statistical deviations can lead to the system becoming insecure. Therefore, cryptographically secure random number generators should be used to generate keys .

Shannon was able to show that perfectly secure encryption methods actually exist. One example of this is the one-time pad . Using such perfectly safe procedures is usually cumbersome in practice. They are therefore rarely used in real-time applications such as the Internet.

literature

• Johannes Buchmann : Introduction to Cryptography . Springer, Berlin, 2010, ISBN 978-3-642-11185-3
• Claude E. Shannon: Communication Theory of Secrecy Systems . In: AT&T Corporation (Ed.): Bell System Technical Journal . 28, No. 4, USA, October 1949, pp. 656-715. Retrieved November 17, 2013.