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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 ...


Key rings are mostly used for PGP (Pretty Good Privacy) or it is also used for signing/verifying and encrypting/decrypting text.

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