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For SAT problems in cryptanalysis, circuit design, and telecommunication, solutions can often be found more efficiently by representing them with a combination of exclusive OR (XOR) and conjunctive normal form (CNF) clauses. We propose a hardware accelerator architecture that natively embeds and solves such hybrid XOR\u2013CNF problems using in-memory computing hardware. To achieve this, we introduce an algorithm and demonstrate, both experimentally and through simulations, how it can be efficiently implemented with memristor crossbar arrays. Compared to the conventional approaches that translate XOR\u2013CNF problems to pure CNF problems, our simulations show that the accelerator improves computation speed, energy efficiency, and chip area utilization of in-memory accelerators by \u00a0~\u00a010\u00a0\u00d7 for a set of hard cryptographic benchmarking problems. Moreover, the accelerator achieves a \u00a0~\u00a010\u00a0\u00d7 speedup and a \u00a0~\u00a01000\u00a0\u00d7 gain in energy efficiency over state-of-the-art SAT solvers running on CPUs.<\/jats:p>","DOI":"10.1038\/s41467-026-69465-2","type":"journal-article","created":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T01:44:36Z","timestamp":1771465476000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Accelerating hybrid XOR\u2013CNF Boolean satisfiability problems natively with in-memory computing"],"prefix":"10.1038","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2625-9200","authenticated-orcid":false,"given":"Haesol","family":"Im","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8516-9785","authenticated-orcid":false,"given":"Fabian","family":"B\u00f6hm","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4501-8672","authenticated-orcid":false,"given":"Giacomo","family":"Pedretti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Noriyuki","family":"Kushida","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7524-4240","authenticated-orcid":false,"given":"Moslem","family":"Noori","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3820-7851","authenticated-orcid":false,"given":"Elisabetta","family":"Valiante","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiangyi","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-0556-8981","authenticated-orcid":false,"given":"Chan-Woo","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4591-6277","authenticated-orcid":false,"given":"Tinish","family":"Bhattacharya","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xia","family":"Sheng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5091-3674","authenticated-orcid":false,"given":"Jim","family":"Ignowski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arne","family":"Heittmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1382-3677","authenticated-orcid":false,"given":"John Paul","family":"Strachan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Masoud","family":"Mohseni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Raymond","family":"Beausoleil","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7301-8610","authenticated-orcid":false,"given":"Thomas Van","family":"Vaerenbergh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4878-1868","authenticated-orcid":false,"given":"Ignacio","family":"Rozada","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,19]]},"reference":[{"key":"69465_CR1","doi-asserted-by":"crossref","unstructured":"Cook, S. 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ADC Performance Survey 1997-2024. Available: https:\/\/github.com\/bmurmann\/ADC-survey (2025)."},{"key":"69465_CR66","unstructured":"Andrulis, T., Chen, R. Lee, H.-S. Emer, J. S. & Sze, V. Modeling Analog-Digital-Converter Energy and Area for Compute-In-Memory Accelerator Design. arXiv https:\/\/arxiv.org\/abs\/2404.06553 (2024)."},{"key":"69465_CR67","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1145\/3007787.3001139","volume":"44","author":"A Shafiee","year":"2016","unstructured":"Shafiee, A. et al. ISAAC: a convolutional neural network accelerator with in-situ analog arithmetic in crossbars. ACM SIGARCH Comput. Architecture N. 44, 14\u201326 (2016).","journal-title":"ACM SIGARCH Comput. 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