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 $D_n \approx E_n\!\,$. 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. 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.

References

• 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

This article incorporates material from computationally indistinguishable on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.