# what's the reason of the notational difference between statistical and computational indistinguishabilities?

Statistical: $$|\Pr[E_K(m_0)\in T]-\Pr[E_K(m_1)\in T]|\leq\epsilon$$

Computational: $$|\Pr[A(E_K(m_0))=1]-\Pr[A(E_K(m_1))=1]|\leq\epsilon(n)$$

What is the $$1$$ doing there? Why isn't it $$Pr[A(E_K(m_0))\in T]$$? Is there something deep behind these different notations? or is it simply a convention of the PPT $$A$$ having a boolean as a return value?

• Where did you read? 1 means the adversary is successful. – kelalaka Apr 11 at 19:15
• @kelalaka - where did i read what? – ihadanny Apr 11 at 19:17

To formalize this idea, you need to talk about distinguishers, which are algorithms that receive an input from one of the two distributions and try to guess which distribution it comes from. This would the algorithm $$A$$ in your definition. The intuition of the definition is that such algorithm will output some fixed value ($$1$$, in this case) with almost the same probability, independently from which of the two distributions the input is sampled from.