Computational indistinguishability

In computational complexity, if $\scriptstyle \{D_{n}\}_{{n\in {\mathbb {N}}}}$ and $\scriptstyle \{E_{n}\}_{{n\in {\mathbb {N}}}}$ are two distribution ensembles indexed by a security parameter n (which usually refers to the length of the input), then we say they are computationally indistinguishable if for any non-uniform probabilistic polynomial time algorithm A, the following quantity is a negligible function in n:

\delta (n)=\left|\Pr _{{x\gets D_{n}}}[A(x)=1]-\Pr _{{x\gets E_{n}}}[A(x)=1]\right|.

denoted $D_{n}\approx E_{n}$ .^[1] In other words, every efficient algorithm A's behavior does not significantly change when given samples according to D_n or E_n in the limit as $n\to \infty$ . Another interpretation of computational indistinguishability, is that polynomial-time algorithms actively trying to distinguish between the two ensembles cannot do so: That any such algorithm will only perform negligibly better than if one were to just guess.

Implicit in the definition is the condition that the algorithm, $A$ , must decide based on a single sample from one of the distributions. One might conceive of a situation in which the algorithm trying to distinguish between two distributions, could access as many samples as it needed. Hence two ensembles that cannot be distinguished by polynomial-time algorithms looking at multiple samples are deemed indistinguishable by polynomial-time sampling.^[2]^:107 If the polynomial-time algorithm can generate samples in polynomial time, or has access to a random oracle that generates samples for it, then indistinguishability by polynomial-time sampling is equivalent to computational indistinguishability.^[2]^:108

References

↑ Lecture 4 - Computational Indistinguishability, Pseudorandom Generators
1 2 Goldreich, O. (2003). Foundations of cryptography. Cambridge, UK: Cambridge University Press.

Donald Beaver and Silvio Micali and Phillip Rogaway, The Round Complexity of Secure Protocols (Extended Abstract), 1990, pp. 503–513
Shafi Goldwasser and Silvio Micali. Probabilistic Encryption. JCSS, 28(2):270–299, 1984
Oded Goldreich. Foundations of Cryptography: Volume 2 – Basic Applications. Cambridge University Press, 2004.
Jonathan Katz, Yehuda Lindell, "Introduction to Modern Cryptography: Principles and Protocols," Chapman & Hall/CRC, 2007

External links

Yehuda Lindell. Introduction to Cryptography

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[1] Lecture 4 - Computational Indistinguishability, Pseudorandom Generators

[Goldreich-2] 1 2 Goldreich, O. (2003). Foundations of cryptography. Cambridge, UK: Cambridge University Press.