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I assume you are familiar with $P$ and $NP$. Also, my knowledge of SNARKs is based mostly on the work of Parno et al., other work may differ in some fine details. So, a SNARK is a succinct non-interactive argument of knowledge. Leaving the "knowledge" part aside for the moment, let's look at "plain" succinct non-interactive arguments (called SNARGs in the ...

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The initial idea of Fiat and Shamir was to eliminate the interaction in public coin protocols (note that public coin means that the random choices of the verifier are made public) and was used to convert three move public coin identification scheme into conceptually simple signature schemes (it has later been proven by Pointcheval and Stern that under the ...

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U-Prove TokenID is a hash output, so it may be not the best way to prove "not the same" statement. One would also consider inequality proof for a subset of user attributes instead. For each such attribute pair, "not the same" would mean an inverse exists for attribute difference, modulo group order. One would prove knowledge of such inverses while keeping ...

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To prove that product holds over integers, one would start from commitments with groups of a hidden order. That is, proving party should not know order of the group, which is the case with RSA-like multiplicative group. Consider Prover responses $\rho_x = tx + \alpha_x$, $\rho_y = ty + \alpha_y$, $\rho_z = tz + \alpha_z$ to Verifier challenge $t$ with ...

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