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Using Hybrid Argument, I want to prove following equivalence: $$\{X,Y,Z\}\equiv \{X',Y',Z'\}$$

by proving following equivalences:
$\{X,Y,Z\}\equiv \{X,Y,Z'\}$(1)
$\{X,Y,Z'\}\equiv \{X,Y',Z'\}$(2)
$\{X,Y',Z'\}\equiv \{X',Y',Z'\}$ (3)
I know how to prove $\{Z\}\equiv\{Z'\}$, under what condition $\{Z\}\equiv\{Z'\}$ implies (1) ?

(I've changed my post since my problem has nothing to do with SMC (maybe it has nothing to do with cryptography either) and the way I presented the problem was confusing. )

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  • $\begingroup$ In the third line what is $c(k.y)$? $\endgroup$
    – pg1989
    Jul 16, 2016 at 1:11
  • $\begingroup$ If X and Y are independent from Z you can make a simple reduction showing that distinguishing between {X,Y,Z} and {X,Y,Z′} implies distinguishing between Z and Z' (i.e. a proof by contradiction). If not, it is still possible that your claim holds, but you cannot use such a generic argument. $\endgroup$
    – antosecret
    Jul 17, 2016 at 5:40
  • $\begingroup$ For example, here is a case where your claim does not hold: if you choose z' at random (independently of x and y) and z as x + y where x comes from X and y comes from Y, then the two distributions in (1) are distinguishable while Z and Z' aren't. $\endgroup$
    – antosecret
    Jul 17, 2016 at 5:42
  • $\begingroup$ @antosecret Thanks, more specifically my question is, should Z be independent from {X} and {Y}, or be independent from {X,Y}. I guess (considering your example) the second one is correct $\endgroup$
    – Mhy
    Jul 17, 2016 at 13:13
  • $\begingroup$ @Mhy You are right. $\endgroup$
    – antosecret
    Jul 17, 2016 at 19:36

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