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how to prove definition 3.8 and 3.9 are equivalence ?

image of definitions

picture is from book an introduction to modern cryptography (2nd edition) by j. katz and y. lindell pdf

and https://repo.zenk-security.com/Cryptographie%20.%20Algorithmes%20.%20Steganographie/Introduction%20to%20Modern%20Cryptography.pdf is link to first edition which definitions can be find on pages 63 and 64 of book as definition 3.9 and 3.10

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  • $\begingroup$ Hint: Derive the left side of the second inequality from the first left side by being explicit about the hidden random bit the challenger chooses. $\endgroup$ – SEJPM Apr 17 at 17:40
  • $\begingroup$ @mike, first of all: are you sure that the link to this .pdf copy of the cited book is really legal? You have to consider that one of the book's authors is a frequent contributor here in Crypto S.E. $\endgroup$ – McFly Apr 20 at 2:36
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Intuitive idea: the first definition is saying that an adversary can not do significantly better than random guessing. The second definition says that for two different inputs, the adversary would have the same probability of guessing 1. So to prove the equivalence, start with the first definition. Run this privacy experiment with two different inputs, subtract them, this is your second definition.

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  • $\begingroup$ what do you mean by two different inputs ? @Hasan Iqbal $\endgroup$ – mike Apr 17 at 18:26

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