# 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 small probability.

## Formal definition

Let ${\displaystyle \scriptstyle \{D_{n}\}_{n\in \mathbb {N} }}$ and ${\displaystyle \scriptstyle \{E_{n}\}_{n\in \mathbb {N} }}$ 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:

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

denoted ${\displaystyle D_{n}\approx E_{n}}$.[1] In other words, every efficient algorithm A's behavior does not significantly change when given samples according to Dn or En in the limit as ${\displaystyle n\to \infty }$. 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.

## Related notions

Implicit in the definition is the condition that the algorithm, ${\displaystyle A}$, 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

## References

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