{"date_updated":"2023-09-05T15:12:27Z","article_processing_charge":"No","month":"10","day":"13","type":"conference","abstract":[{"text":"A proof of exponentiation (PoE) in a group G of unknown order allows a prover to convince a verifier that a tuple (x,q,T,y)∈G×N×N×G satisfies xqT=y. This primitive has recently found exciting applications in the constructions of verifiable delay functions and succinct arguments of knowledge. The most practical PoEs only achieve soundness either under computational assumptions, i.e., they are arguments (Wesolowski, Journal of Cryptology 2020), or in groups that come with the promise of not having any small subgroups (Pietrzak, ITCS 2019). The only statistically-sound PoE in general groups of unknown order is due to Block et al. (CRYPTO 2021), and can be seen as an elaborate parallel repetition of Pietrzak’s PoE: to achieve λ bits of security, say λ=80, the number of repetitions required (and thus the blow-up in communication) is as large as λ.\r\n\r\nIn this work, we propose a statistically-sound PoE for the case where the exponent q is the product of all primes up to some bound B. We show that, in this case, it suffices to run only λ/log(B) parallel instances of Pietrzak’s PoE, which reduces the concrete proof-size compared to Block et al. by an order of magnitude. Furthermore, we show that in the known applications where PoEs are used as a building block such structured exponents are viable. Finally, we also discuss batching of our PoE, showing that many proofs (for the same G and q but different x and T) can be batched by adding only a single element to the proof per additional statement.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"https://eprint.iacr.org/2022/1021"}],"department":[{"_id":"KrPi"}],"date_published":"2022-10-13T00:00:00Z","publication_status":"published","scopus_import":"1","publisher":"Springer Nature","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_created":"2023-01-12T12:12:07Z","oa":1,"oa_version":"Preprint","alternative_title":["LNCS"],"status":"public","quality_controlled":"1","title":"Practical statistically-sound proofs of exponentiation in any group","publication_identifier":{"eisbn":["9783031159794"],"isbn":["9783031159787"],"eissn":["1611-3349"],"issn":["0302-9743"]},"external_id":{"isi":["000886792700013"]},"page":"370-399","doi":"10.1007/978-3-031-15979-4_13","author":[{"orcid":"0000-0003-2027-5549","id":"0f78d746-dc7d-11ea-9b2f-83f92091afe7","full_name":"Hoffmann, Charlotte","first_name":"Charlotte","last_name":"Hoffmann"},{"full_name":"Hubáček, Pavel","first_name":"Pavel","last_name":"Hubáček"},{"full_name":"Kamath, Chethan","first_name":"Chethan","last_name":"Kamath"},{"last_name":"Klein","first_name":"Karen","full_name":"Klein, Karen"},{"last_name":"Pietrzak","first_name":"Krzysztof Z","full_name":"Pietrzak, Krzysztof Z","id":"3E04A7AA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9139-1654"}],"publication":"Advances in Cryptology – CRYPTO 2022","language":[{"iso":"eng"}],"_id":"12176","conference":{"start_date":"2022-08-15","name":"CRYYPTO: International Cryptology Conference","location":"Santa Barbara, CA, United States","end_date":"2022-08-18"},"citation":{"mla":"Hoffmann, Charlotte, et al. “Practical Statistically-Sound Proofs of Exponentiation in Any Group.” <i>Advances in Cryptology – CRYPTO 2022</i>, vol. 13508, Springer Nature, 2022, pp. 370–99, doi:<a href=\"https://doi.org/10.1007/978-3-031-15979-4_13\">10.1007/978-3-031-15979-4_13</a>.","ieee":"C. Hoffmann, P. Hubáček, C. Kamath, K. Klein, and K. Z. Pietrzak, “Practical statistically-sound proofs of exponentiation in any group,” in <i>Advances in Cryptology – CRYPTO 2022</i>, Santa Barbara, CA, United States, 2022, vol. 13508, pp. 370–399.","chicago":"Hoffmann, Charlotte, Pavel Hubáček, Chethan Kamath, Karen Klein, and Krzysztof Z Pietrzak. “Practical Statistically-Sound Proofs of Exponentiation in Any Group.” In <i>Advances in Cryptology – CRYPTO 2022</i>, 13508:370–99. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/978-3-031-15979-4_13\">https://doi.org/10.1007/978-3-031-15979-4_13</a>.","ista":"Hoffmann C, Hubáček P, Kamath C, Klein K, Pietrzak KZ. 2022. Practical statistically-sound proofs of exponentiation in any group. Advances in Cryptology – CRYPTO 2022. CRYYPTO: International Cryptology Conference, LNCS, vol. 13508, 370–399.","apa":"Hoffmann, C., Hubáček, P., Kamath, C., Klein, K., &#38; Pietrzak, K. Z. (2022). Practical statistically-sound proofs of exponentiation in any group. In <i>Advances in Cryptology – CRYPTO 2022</i> (Vol. 13508, pp. 370–399). Santa Barbara, CA, United States: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-031-15979-4_13\">https://doi.org/10.1007/978-3-031-15979-4_13</a>","ama":"Hoffmann C, Hubáček P, Kamath C, Klein K, Pietrzak KZ. Practical statistically-sound proofs of exponentiation in any group. In: <i>Advances in Cryptology – CRYPTO 2022</i>. Vol 13508. Springer Nature; 2022:370-399. doi:<a href=\"https://doi.org/10.1007/978-3-031-15979-4_13\">10.1007/978-3-031-15979-4_13</a>","short":"C. Hoffmann, P. Hubáček, C. Kamath, K. Klein, K.Z. Pietrzak, in:, Advances in Cryptology – CRYPTO 2022, Springer Nature, 2022, pp. 370–399."},"year":"2022","volume":13508,"intvolume":"     13508","acknowledgement":"We would like to thank the authors of [BHR+21] for clarifying several questions we had\r\nregarding their results. Pavel Hubá£ek was supported by the Grant Agency of the Czech\r\nRepublic under the grant agreement no. 19-27871X and by the Charles University project\r\nUNCE/SCI/004. Chethan Kamath is supported by Azrieli International Postdoctoral Fellowship\r\nand ISF grants 484/18 and 1789/19. Karen Klein was supported in part by ERC CoG grant\r\n724307 and conducted part of this work at Institute of Science and Technology Austria.","isi":1}