**The Distinction Between Fixed and Random Generators in Group-Based Assumptions**

*James Bartusek and Fermi Ma and Mark Zhandry*

**Abstract: **There is surprisingly little consensus on the precise role of the generator g in group-based assumptions such as DDH. Some works consider g to be a fixed part of the group description, while others take it to be random. We study this subtle distinction from a number of angles.

- In the generic group model, we demonstrate the plausibility of groups in which random-generator DDH (resp. CDH) is hard but fixed-generator DDH (resp. CDH) is easy. We observe that such groups have interesting cryptographic applications.

- We find that seemingly tight generic lower bounds for the Discrete-Log and CDH problems with preprocessing (Corrigan-Gibbs and Kogan, Eurocrypt 2018) are not tight in the sub-constant success probability regime if the generator is random. We resolve this by proving tight lower bounds for the random generator variants; our results formalize the intuition that using a random generator will reduce the effectiveness of preprocessing attacks.

- We observe that DDH-like assumptions in which exponents are drawn from low-entropy distributions are particularly sensitive to the fixed- vs. random-generator distinction. Most notably, we discover that the Strong Power DDH assumption of Komargodski and Yogev (Komargodski and Yogev, Eurocrypt 2018) used for non-malleable point obfuscation is in fact false precisely because it requires a fixed generator. In response, we formulate an alternative fixed-generator assumption that suffices for a new construction of non-malleable point obfuscation, and we prove the assumption holds in the generic group model. We also give a generic group proof for the security of fixed-generator, low-entropy DDH (Canetti, Crypto 1997).

**Category / Keywords: **foundations / Diffie-Hellman, preprocessing, point obfuscation, generic group model, non-malleability

**Date: **received 22 Feb 2019, last revised 5 Mar 2019

**Contact author: **fermima1 at gmail com, bartusek james at gmail com, mzhandry at princeton edu

**Available format(s): **PDF | BibTeX Citation

**Note: **Fixed an error in the proof of Theorem 3, added relevant missing citations.

**Version: **20190305:185353 (All versions of this report)

**Short URL: **ia.cr/2019/202

[ Cryptology ePrint archive ]