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You are probably looking for what is called "anonymous credentials". An anonymous credentials system relates three types of parties: authorities, users, and verifiers. An authority (Alice) can issue a credential to a user (Bob), that certifies that the user satisfies some property (in your case, that would be "is trusted"). Credentials are unforgeable. ...

It may be an interesting avenue to explore. I've read the highlight reel of the digital cash literature but I do not know it well enough to know how well these issues have been addressed. A few things to consider for #2: In a quick read of Lucre, it seems that the payee does no verification before passing the coin onto the mint. It seems, if the mint has ...

What you are describing is an anonymous credential system. There are two different ways to go about making these and two actually systems that use those techniuqes. Microsoft's U-prove and IBM's Idemix. If you're interested in smart card usage, you probably prefer U-prove as it tends to work better with smart cards. Its described in a by its original ...

If Bob and Charlie will share a secret, then it information-theoretic privacy might be possible, otherwise the best that can be hoped for is computational anonymity. I haven't come up up with any way to achieve information-theoretic privacy when Bob and Charlie share a secret, so I will only be addressing computational privacy. I assume that Bob can ...

If you can find such an efficiently computable function (other than the trivial solution $\hat{e}( x, y ) = 1\ \ $ for all $x$, $y$), then you have shown that the decisional Diffie-Hellman problem is easy. That is, given $g$, $g^a$, $g^b$, $g^c$, you can check whether $$ab = c$$ simply by testing: $$\hat{e}( g^a, g^b ) = \hat{e}( g, g^c)$$ The ...