Novel improvements and extensions of the extractable results about (leakage-resilient) privacy schemes with imperfect randomness

Traditional cryptographic primitives usually take for granted the availability of perfect randomness. Unfortunately, in reality one must deal with various imperfect randomness (e.g., physical sources, secrets with partial leakage, biometric data). Bosley and Dodis in TCC’07 [BD07] proposed that private-key encryption requires extractable randomness and hoped their result would arouse more interest in exploring the extent to which cryptographic primitives can be grounded on imperfect randomness. Aggarwal et al. in TCC’22 [ACOR22] observed leakage-resilient secret sharing requires extractable randomness. Partially motivated by these, we study improvements and extensions of the extractable results proposed before. We consider the generalized (leakage-resilient) privacy schemes (including encryption, perfectly binding commitment, threshold secret sharing). We get the new results below. Firstly, we explore extractable results about the generalized privacy schemes using two methods: one is an improved and generalized method based on [BD07] by combining different Chernoff Bounds; the other creatively employs Lemma 3 of [ACOR22]. Afterwards, we improve and extend the above results grounded on the Rényi entropy. In particular, (a) substituting the collision entropy for the min-entropy, we obtain tighter bounds than the counterpart of Lemma 3 in [ACOR22]; (b) replacing the min-entropy with the Rényi entropy, we give a tricky and detailed proof for generalized version of Lemma 4 of [ACOR22], while the coupling argument in that proof of [ACOR22] is used directly without explanation, which is unclear and hard to understand. Finally, we propose the extractable results about the generalized leakage-resilient privacy schemes using two methods: one extends Theorem 1(a) of [BD07]; the other uses more generalized, more intuitive, and simpler proof ideas than the counterpart of [ACOR22]. Furthermore, we present concrete and essential restrictions on the parameters by proving the main theorem other than [ACOR22] that proposed unspecific parameters.