Computational indistinguishability

In computational complexity and cryptography, two families of distributions are computationally indistinguishable if no efficient algorithm can tell the difference between them except with negligible probability.

Let ${\displaystyle \scriptstyle \{D_{n}\}_{n\in \mathbb {N} }}${\displaystyle \scriptstyle \{D_{n}\}_{n\in \mathbb {N} }} and ${\displaystyle \scriptstyle \{E_{n}\}_{n\in \mathbb {N} }}${\displaystyle \scriptstyle \{E_{n}\}_{n\in \mathbb {N} }} be two distribution ensembles indexed by a security parameter n (which usually refers to the length of the input); 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:

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

denoted ${\displaystyle D_{n}\approx E_{n}}${\displaystyle 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 ${\displaystyle n\to \infty }${\displaystyle 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, ${\displaystyle A}${\displaystyle 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}

• Yehuda Lindell. Introduction to Cryptography
• 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

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