We revisit the problem of proving that a user algorithm selected and correctly used a truly random seed in the generation of her cryptographic key. A first approach was proposed in 2002 by Juels and Guajardo for the validation of RSA secret keys. We present a new security model and general tools to efficiently prove that a private key was generated at random according to a prescribed process, without revealing any further information about the private key. We give a generic protocol for all key-generation algorithms based on probabilistic circuits and prove its security. We also propose a new protocol for factoring-based cryptography that we prove secure in the aforementioned model. This latter relies on a new efficient zero-knowledge argument for the double discrete logarithm problem that achieves an exponential improvement in communication complexity compared to the state of the art, and is of independent interest.
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