{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T00:49:36Z","timestamp":1777337376957,"version":"3.51.4"},"publisher-location":"Cham","reference-count":85,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783031587504","type":"print"},{"value":"9783031587511","type":"electronic"}],"license":[{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024]]},"DOI":"10.1007\/978-3-031-58751-1_6","type":"book-chapter","created":{"date-parts":[[2024,4,28]],"date-time":"2024-04-28T03:01:57Z","timestamp":1714273317000},"page":"149-179","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["The Hardness of\u00a0LPN over\u00a0Any Integer Ring and\u00a0Field for\u00a0PCG Applications"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3843-1394","authenticated-orcid":false,"given":"Hanlin","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5991-7417","authenticated-orcid":false,"given":"Xiao","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7453-4043","authenticated-orcid":false,"given":"Kang","family":"Yang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9278-4521","authenticated-orcid":false,"given":"Yu","family":"Yu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,4,29]]},"reference":[{"key":"6_CR1","doi-asserted-by":"publisher","unstructured":"Abram, D., Scholl, P.: Low-communication multiparty triple generation for SPDZ from ring-LPN. In: Hanaoka, G., Shikata, J., Watanabe, Y. (eds.) PKC\u00a02022, Part\u00a0I. LNCS, vol. 13177, pp. 221\u2013251. Springer, Heidelberg (2022). https:\/\/doi.org\/10.1007\/978-3-030-97121-2_9","DOI":"10.1007\/978-3-030-97121-2_9"},{"key":"6_CR2","unstructured":"Akavia, A.: Learning noisy characters, multiplication codes, and cryptographic hardcore predicates. Ph.D. thesis, Massachusetts Institute of Technology (2008). https:\/\/people.csail.mit.edu\/akavia\/AkaviaPhDThesis.pdf"},{"key":"6_CR3","doi-asserted-by":"publisher","unstructured":"Al Jabri, A.: A statistical decoding algorithm for general linear block codes. In: Honary, B. (ed.) 8th IMA International Conference on Cryptography and Coding. LNCS, vol.\u00a02260, pp. 1\u20138. Springer, Heidelberg (2001). https:\/\/doi.org\/10.1007\/3-540-45325-3_1","DOI":"10.1007\/3-540-45325-3_1"},{"key":"6_CR4","doi-asserted-by":"publisher","unstructured":"Alekhnovich, M.: More on average case vs approximation complexity. In: 44th FOCS, pp. 298\u2013307. IEEE Computer Society Press (2003). https:\/\/doi.org\/10.1109\/SFCS.2003.1238204","DOI":"10.1109\/SFCS.2003.1238204"},{"key":"6_CR5","doi-asserted-by":"publisher","unstructured":"Applebaum, B.: Garbling XOR gates \u201cfor free\u201d in the standard model. J. Cryptol. 29(3), 552\u2013576 (2016). https:\/\/doi.org\/10.1007\/s00145-015-9201-9","DOI":"10.1007\/s00145-015-9201-9"},{"key":"6_CR6","doi-asserted-by":"publisher","unstructured":"Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography with constant input locality. In: Menezes, A. (ed.) CRYPTO\u00a02007. LNCS, vol.\u00a04622, pp. 92\u2013110. Springer, Heidelberg (2007). https:\/\/doi.org\/10.1007\/978-3-540-74143-5_6","DOI":"10.1007\/978-3-540-74143-5_6"},{"key":"6_CR7","doi-asserted-by":"publisher","unstructured":"Augot, D., Finiasz, M., Sendrier, N.: A family of fast syndrome based cryptographic hash functions. In: Dawson, E., Vaudenay, S. (eds.) Mycrypt 2005. LNCS, vol.\u00a03715, pp. 64\u201383. Springer, Heidelberg (2005). https:\/\/doi.org\/10.1007\/11554868_6","DOI":"10.1007\/11554868_6"},{"key":"6_CR8","doi-asserted-by":"publisher","unstructured":"Baum, C., Braun, L., Munch-Hansen, A., Razet, B., Scholl, P.: Appenzeller to Brie: efficient zero-knowledge proofs for mixed-mode arithmetic and Z2k. In: Vigna, G., Shi, E. (eds.) ACM CCS 2021, pp. 192\u2013211. ACM Press, November 2021. https:\/\/doi.org\/10.1145\/3460120.3484812","DOI":"10.1145\/3460120.3484812"},{"key":"6_CR9","doi-asserted-by":"publisher","unstructured":"Baum, C., Braun, L., Munch-Hansen, A., Scholl, P.: Moz$$\\mathbb{Z}_{2^k}$$arella: efficient vector-OLE and zero-knowledge proofs over $$\\mathbb{Z}_{2^k}$$. In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO\u00a02022, Part\u00a0IV. LNCS, vol. 13510, pp. 329\u2013358. Springer, Heidelberg (2022). https:\/\/doi.org\/10.1007\/978-3-031-15985-5_12","DOI":"10.1007\/978-3-031-15985-5_12"},{"key":"6_CR10","doi-asserted-by":"publisher","unstructured":"Baum, C., Malozemoff, A.J., Rosen, M.B., Scholl, P.: Mac\u2019n\u2019Cheese: zero-knowledge proofs for boolean and arithmetic circuits with nested disjunctions. In: Malkin, T., Peikert, C. (eds.) CRYPTO\u00a02021, Part\u00a0IV. LNCS, vol. 12828, pp. 92\u2013122. Springer, Heidelberg, Virtual Event (2021). https:\/\/doi.org\/10.1007\/978-3-030-84259-8_4","DOI":"10.1007\/978-3-030-84259-8_4"},{"key":"6_CR11","doi-asserted-by":"publisher","unstructured":"Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in $$2^{n\/20}$$: how 1 + 1 = 0 improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT\u00a02012. LNCS, vol.\u00a07237, pp. 520\u2013536. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-29011-4_31","DOI":"10.1007\/978-3-642-29011-4_31"},{"key":"6_CR12","doi-asserted-by":"publisher","unstructured":"Beullens, W.: Improved cryptanalysis of UOV and Rainbow. In: Canteaut, A., Standaert, F.X. (eds.) EUROCRYPT\u00a02021, Part\u00a0I. LNCS, vol. 12696, pp. 348\u2013373. Springer, Heidelberg (2021). https:\/\/doi.org\/10.1007\/978-3-030-77870-5_13","DOI":"10.1007\/978-3-030-77870-5_13"},{"key":"6_CR13","doi-asserted-by":"publisher","unstructured":"Blum, A., Furst, M.L., Kearns, M.J., Lipton, R.J.: Cryptographic primitives based on hard learning problems. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol.\u00a0773, pp. 278\u2013291. Springer, Heidelberg (1994). https:\/\/doi.org\/10.1007\/3-540-48329-2_24","DOI":"10.1007\/3-540-48329-2_24"},{"key":"6_CR14","doi-asserted-by":"publisher","unstructured":"Bombar, M., Couteau, G., Couvreur, A., Ducros, C.: Correlated pseudorandomness from the hardness of quasi-abelian decoding. In: Handschuh, H., Lysyanskaya, A. (eds.) CRYPTO\u00a02023, Part\u00a0IV, pp. 567\u2013601. LNCS, Springer, Heidelberg (2023). https:\/\/doi.org\/10.1007\/978-3-031-38551-3_18","DOI":"10.1007\/978-3-031-38551-3_18"},{"key":"6_CR15","doi-asserted-by":"publisher","unstructured":"Boyle, E., Couteau, G., Gilboa, N., Ishai, Y.: Compressing vector OLE. In: Lie, D., Mannan, M., Backes, M., Wang, X. (eds.) ACM CCS 2018, pp. 896\u2013912. ACM Press, October 2018. https:\/\/doi.org\/10.1145\/3243734.3243868","DOI":"10.1145\/3243734.3243868"},{"key":"6_CR16","doi-asserted-by":"publisher","unstructured":"Boyle, E., et al.: Correlated pseudorandomness from expand-accumulate codes. In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO\u00a02022, Part\u00a0II. LNCS, vol. 13508, pp. 603\u2013633. Springer, Heidelberg (2022). https:\/\/doi.org\/10.1007\/978-3-031-15979-4_21","DOI":"10.1007\/978-3-031-15979-4_21"},{"key":"6_CR17","doi-asserted-by":"publisher","unstructured":"Boyle, E., et al.: Efficient two-round OT extension and silent non-interactive secure computation. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J. (eds.) ACM CCS 2019, pp. 291\u2013308. ACM Press, November 2019. https:\/\/doi.org\/10.1145\/3319535.3354255","DOI":"10.1145\/3319535.3354255"},{"key":"6_CR18","doi-asserted-by":"publisher","unstructured":"Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Kohl, L., Scholl, P.: Efficient pseudorandom correlation generators: silent OT extension and more. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO\u00a02019, Part\u00a0III. LNCS, vol. 11694, pp. 489\u2013518. Springer, Heidelberg (2019). https:\/\/doi.org\/10.1007\/978-3-030-26954-8_16","DOI":"10.1007\/978-3-030-26954-8_16"},{"key":"6_CR19","doi-asserted-by":"publisher","unstructured":"Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Kohl, L., Scholl, P.: Efficient pseudorandom correlation generators from ring-LPN. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO\u00a02020, Part\u00a0II. LNCS, vol. 12171, pp. 387\u2013416. Springer, Heidelberg (2020). https:\/\/doi.org\/10.1007\/978-3-030-56880-1_14","DOI":"10.1007\/978-3-030-56880-1_14"},{"key":"6_CR20","doi-asserted-by":"publisher","unstructured":"Boyle, E., Gilboa, N., Ishai, Y.: Function secret sharing. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT\u00a02015, Part\u00a0II. LNCS, vol.\u00a09057, pp. 337\u2013367. Springer, Heidelberg (2015). https:\/\/doi.org\/10.1007\/978-3-662-46803-6_12","DOI":"10.1007\/978-3-662-46803-6_12"},{"key":"6_CR21","doi-asserted-by":"publisher","unstructured":"Brakerski, Z., Lyubashevsky, V., Vaikuntanathan, V., Wichs, D.: Worst-case hardness for LPN and cryptographic hashing via code smoothing. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT\u00a02019, Part\u00a0III. LNCS, vol. 11478, pp. 619\u2013635. Springer, Heidelberg (2019). https:\/\/doi.org\/10.1007\/978-3-030-17659-4_21","DOI":"10.1007\/978-3-030-17659-4_21"},{"key":"6_CR22","doi-asserted-by":"publisher","unstructured":"Briaud, P., \u00d8ygarden, M.: A new algebraic approach to the regular syndrome decoding problem and implications for PCG constructions. In: Hazay, C., Stam, M. (eds.) EUROCRYPT\u00a02023, Part\u00a0V. LNCS, vol. 14008, pp. 391\u2013422. Springer, Heidelberg (2023). https:\/\/doi.org\/10.1007\/978-3-031-30589-4_14","DOI":"10.1007\/978-3-031-30589-4_14"},{"key":"6_CR23","doi-asserted-by":"publisher","unstructured":"Bui, D., Couteau, G.: Improved private set intersection for sets with small entries. In: Boldyreva, A., Kolesnikov, V. (eds.) PKC\u00a02023, Part\u00a0II. LNCS, vol. 13941, pp. 190\u2013220. Springer, Heidelberg (2023). https:\/\/doi.org\/10.1007\/978-3-031-31371-4_7","DOI":"10.1007\/978-3-031-31371-4_7"},{"key":"6_CR24","doi-asserted-by":"publisher","unstructured":"Carozza, E., Couteau, G., Joux, A.: Short signatures from regular syndrome decoding in the head. In: Hazay, C., Stam, M. (eds.) EUROCRYPT\u00a02023, Part\u00a0V. LNCS, vol. 14008, pp. 532\u2013563. Springer, Heidelberg (2023). https:\/\/doi.org\/10.1007\/978-3-031-30589-4_19","DOI":"10.1007\/978-3-031-30589-4_19"},{"key":"6_CR25","doi-asserted-by":"publisher","unstructured":"Carrier, K., Debris-Alazard, T., Meyer-Hilfiger, C., Tillich, J.P.: Statistical decoding 2.0: reducing decoding to LPN. In: Agrawal, S., Lin, D. (eds.) ASIACRYPT\u00a02022, Part\u00a0IV. LNCS, vol. 13794, pp. 477\u2013507. Springer, Heidelberg (2022). https:\/\/doi.org\/10.1007\/978-3-031-22972-5_17","DOI":"10.1007\/978-3-031-22972-5_17"},{"issue":"205","key":"6_CR26","first-page":"333","volume":"62","author":"D Coppersmith","year":"1994","unstructured":"Coppersmith, D.: Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm. Math. Comput. 62(205), 333\u2013350 (1994)","journal-title":"Math. Comput."},{"key":"6_CR27","doi-asserted-by":"publisher","unstructured":"Couteau, G., Rindal, P., Raghuraman, S.: Silver: silent VOLE and oblivious transfer from hardness of decoding structured LDPC codes. In: Malkin, T., Peikert, C. (eds.) CRYPTO\u00a02021, Part\u00a0III. LNCS, vol. 12827, pp. 502\u2013534. Springer, Heidelberg, Virtual Event (2021). https:\/\/doi.org\/10.1007\/978-3-030-84252-9_17","DOI":"10.1007\/978-3-030-84252-9_17"},{"key":"6_CR28","doi-asserted-by":"publisher","unstructured":"Cramer, R., Damg\u00e5rd, I., Escudero, D., Scholl, P., Xing, C.: SPD $$\\mathbb{Z}_{2^k}$$: efficient MPC mod $$2^k$$ for dishonest majority. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO\u00a02018, Part\u00a0II. LNCS, vol. 10992, pp. 769\u2013798. Springer, Heidelberg (2018). https:\/\/doi.org\/10.1007\/978-3-319-96881-0_26","DOI":"10.1007\/978-3-319-96881-0_26"},{"key":"6_CR29","doi-asserted-by":"publisher","unstructured":"Cui, H., Wang, X., Yang, K., Yu, Y.: Actively secure half-gates with minimum overhead under duplex networks. In: Hazay, C., Stam, M. (eds.) EUROCRYPT\u00a02023, Part\u00a0II. LNCS, vol. 14005, pp. 35\u201367. Springer, Heidelberg (2023). https:\/\/doi.org\/10.1007\/978-3-031-30617-4_2","DOI":"10.1007\/978-3-031-30617-4_2"},{"key":"6_CR30","doi-asserted-by":"publisher","unstructured":"Damg\u00e5rd, I., Escudero, D., Frederiksen, T.K., Keller, M., Scholl, P., Volgushev, N.: New primitives for actively-secure MPC over rings with applications to private machine learning. In: 2019 IEEE Symposium on Security and Privacy, pp. 1102\u20131120. IEEE Computer Society Press, May 2019. https:\/\/doi.org\/10.1109\/SP.2019.00078","DOI":"10.1109\/SP.2019.00078"},{"key":"6_CR31","doi-asserted-by":"publisher","unstructured":"Damg\u00e5rd, I., Pastro, V., Smart, N.P., Zakarias, S.: Multiparty computation from somewhat homomorphic encryption. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO\u00a02012. LNCS, vol.\u00a07417, pp. 643\u2013662. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-32009-5_38","DOI":"10.1007\/978-3-642-32009-5_38"},{"key":"6_CR32","doi-asserted-by":"publisher","unstructured":"David, B., Dowsley, R., Nascimento, A.C.A.: Universally composable oblivious transfer based on a variant of LPN. In: Gritzalis, D., Kiayias, A., Askoxylakis, I.G. (eds.) CANS 2014. LNCS, vol.\u00a08813, pp. 143\u2013158. Springer, Heidelberg (2014). https:\/\/doi.org\/10.1007\/978-3-319-12280-9_10","DOI":"10.1007\/978-3-319-12280-9_10"},{"key":"6_CR33","doi-asserted-by":"publisher","unstructured":"Debris-Alazard, T., Tillich, J.: Statistical decoding. In: ISIT 2017 (2017). https:\/\/doi.org\/10.1109\/ISIT.2017.8006839","DOI":"10.1109\/ISIT.2017.8006839"},{"key":"6_CR34","doi-asserted-by":"publisher","unstructured":"Dittmer, S., Ishai, Y., Lu, S., Ostrovsky, R.: Authenticated garbling from simple correlations. In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO\u00a02022, Part\u00a0IV. LNCS, vol. 13510, pp. 57\u201387. Springer, Heidelberg (2022). https:\/\/doi.org\/10.1007\/978-3-031-15985-5_3","DOI":"10.1007\/978-3-031-15985-5_3"},{"key":"6_CR35","doi-asserted-by":"publisher","unstructured":"Dittmer, S., Ishai, Y., Lu, S., Ostrovsky, R.: Improving line-point zero knowledge: two multiplications for the price of one. In: Yin, H., Stavrou, A., Cremers, C., Shi, E. (eds.) ACM CCS 2022, pp. 829\u2013841. ACM Press, November 2022. https:\/\/doi.org\/10.1145\/3548606.3559385","DOI":"10.1145\/3548606.3559385"},{"key":"6_CR36","doi-asserted-by":"publisher","unstructured":"Dittmer, S., Ishai, Y., Ostrovsky, R.: Line-point zero knowledge and its applications. In: 2nd Conference on Information-Theoretic Cryptography (2021). https:\/\/doi.org\/10.4230\/LIPICS.ITC.2021.5","DOI":"10.4230\/LIPICS.ITC.2021.5"},{"key":"6_CR37","doi-asserted-by":"publisher","unstructured":"Dodis, Y., Kalai, Y.T., Lovett, S.: On cryptography with auxiliary input. In: Mitzenmacher, M. (ed.) 41st ACM STOC, pp. 621\u2013630. ACM Press, May\/June 2009. https:\/\/doi.org\/10.1145\/1536414.1536498","DOI":"10.1145\/1536414.1536498"},{"key":"6_CR38","unstructured":"Dumer, I.: On minimum distance decoding of linear codes. In: Proceedings of 5th Joint Soviet-Swedish International Workshop Information Theory (1991)"},{"key":"6_CR39","doi-asserted-by":"publisher","unstructured":"Esser, A., Bellini, E.: Syndrome decoding estimator. In: Hanaoka, G., Shikata, J., Watanabe, Y. (eds.) PKC\u00a02022, Part\u00a0I. LNCS, vol. 13177, pp. 112\u2013141. Springer, Heidelberg (2022). https:\/\/doi.org\/10.1007\/978-3-030-97121-2_5","DOI":"10.1007\/978-3-030-97121-2_5"},{"key":"6_CR40","doi-asserted-by":"publisher","unstructured":"Esser, A., K\u00fcbler, R., May, A.: LPN decoded. In: Katz, J., Shacham, H. (eds.) CRYPTO\u00a02017, Part\u00a0II. LNCS, vol. 10402, pp. 486\u2013514. Springer, Heidelberg (2017). https:\/\/doi.org\/10.1007\/978-3-319-63715-0_17","DOI":"10.1007\/978-3-319-63715-0_17"},{"key":"6_CR41","doi-asserted-by":"publisher","unstructured":"Esser, A., May, A., Zweydinger, F.: McEliece needs a break - solving McEliece-1284 and quasi-cyclic-2918 with modern ISD. In: Dunkelman, O., Dziembowski, S. (eds.) EUROCRYPT\u00a02022, Part\u00a0III. LNCS, vol. 13277, pp. 433\u2013457. Springer, Heidelberg, May\/June 2022. https:\/\/doi.org\/10.1007\/978-3-031-07082-2_16","DOI":"10.1007\/978-3-031-07082-2_16"},{"key":"6_CR42","doi-asserted-by":"publisher","unstructured":"Feneuil, T., Joux, A., Rivain, M.: Syndrome decoding in the head: shorter signatures from zero-knowledge proofs. In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO\u00a02022, Part\u00a0II. LNCS, vol. 13508, pp. 541\u2013572. Springer, Heidelberg (2022). https:\/\/doi.org\/10.1007\/978-3-031-15979-4_19","DOI":"10.1007\/978-3-031-15979-4_19"},{"key":"6_CR43","doi-asserted-by":"publisher","unstructured":"Finiasz, M., Sendrier, N.: Security bounds for the design of code-based cryptosystems. In: Matsui, M. (ed.) ASIACRYPT\u00a02009. LNCS, vol.\u00a05912, pp. 88\u2013105. Springer, Heidelberg (2009). https:\/\/doi.org\/10.1007\/978-3-642-10366-7_6","DOI":"10.1007\/978-3-642-10366-7_6"},{"key":"6_CR44","doi-asserted-by":"publisher","unstructured":"Fischer, J.B., Stern, J.: An efficient pseudo-random generator provably as secure as syndrome decoding. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol.\u00a01070, pp. 245\u2013255. Springer, Heidelberg (1996). https:\/\/doi.org\/10.1007\/3-540-68339-9_22","DOI":"10.1007\/3-540-68339-9_22"},{"key":"6_CR45","doi-asserted-by":"publisher","unstructured":"Franzese, N., Katz, J., Lu, S., Ostrovsky, R., Wang, X., Weng, C.: Constant-overhead zero-knowledge for RAM programs. In: Vigna, G., Shi, E. (eds.) ACM CCS 2021, pp. 178\u2013191. ACM Press, November 2021. https:\/\/doi.org\/10.1145\/3460120.3484800","DOI":"10.1145\/3460120.3484800"},{"key":"6_CR46","unstructured":"Hamdaoui, Y., Sendrier, N.: A non asymptotic analysis of information set decoding. Cryptology ePrint Archive, Report 2013\/162 (2013). https:\/\/eprint.iacr.org\/2013\/162"},{"key":"6_CR47","doi-asserted-by":"publisher","unstructured":"Hazay, C., Orsini, E., Scholl, P., Soria-Vazquez, E.: TinyKeys: a new approach to efficient multi-party computation. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO\u00a02018, Part\u00a0III. LNCS, vol. 10993, pp. 3\u201333. Springer, Heidelberg (2018). https:\/\/doi.org\/10.1007\/978-3-319-96878-0_1","DOI":"10.1007\/978-3-319-96878-0_1"},{"issue":"4","key":"6_CR48","doi-asserted-by":"publisher","first-page":"1732","DOI":"10.1007\/s00145-020-09355-y","volume":"33","author":"C Hazay","year":"2020","unstructured":"Hazay, C., Scholl, P., Soria-Vazquez, E.: Low cost constant round MPC combining BMR and oblivious transfer. J. Cryptol. 33(4), 1732\u20131786 (2020). https:\/\/doi.org\/10.1007\/s00145-020-09355-y","journal-title":"J. Cryptol."},{"key":"6_CR49","doi-asserted-by":"publisher","unstructured":"Hopper, N.J., Blum, M.: Secure human identification protocols. In: Boyd, C. (ed.) ASIACRYPT\u00a02001. LNCS, vol.\u00a02248, pp. 52\u201366. Springer, Heidelberg (2001). https:\/\/doi.org\/10.1007\/3-540-45682-1_4","DOI":"10.1007\/3-540-45682-1_4"},{"key":"6_CR50","unstructured":"Hou, X., et al.: CipherGPT: secure two-party GPT inference. Cryptology ePrint Archive, Paper 2023\/1147 (2023). https:\/\/eprint.iacr.org\/2023\/1147"},{"key":"6_CR51","unstructured":"Huang, Z., Lu, W.J., Hong, C., Ding, J.: Cheetah: lean and fast secure two-party deep neural network inference. In: Butler, K.R.B., Thomas, K. (eds.) USENIX Security 2022, pp. 809\u2013826. USENIX Association, August 2022"},{"key":"6_CR52","doi-asserted-by":"publisher","unstructured":"Jain, A., Lin, H., Sahai, A.: Indistinguishability obfuscation from well-founded assumptions. In: Khuller, S., Williams, V.V. (eds.) 53rd ACM STOC, pp. 60\u201373. ACM Press, June 2021. https:\/\/doi.org\/10.1145\/3406325.3451093","DOI":"10.1145\/3406325.3451093"},{"key":"6_CR53","doi-asserted-by":"publisher","unstructured":"Jain, A., Krenn, S., Pietrzak, K., Tentes, A.: Commitments and efficient zero-knowledge proofs from learning parity with noise. In: Wang, X., Sako, K. (eds.) ASIACRYPT\u00a02012. LNCS, vol.\u00a07658, pp. 663\u2013680. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-34961-4_40","DOI":"10.1007\/978-3-642-34961-4_40"},{"issue":"3","key":"6_CR54","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1007\/s00145-010-9061-2","volume":"23","author":"J Katz","year":"2010","unstructured":"Katz, J., Shin, J.S., Smith, A.: Parallel and concurrent security of the HB and HB+ protocols. J. Cryptol. 23(3), 402\u2013421 (2010). https:\/\/doi.org\/10.1007\/s00145-010-9061-2","journal-title":"J. Cryptol."},{"key":"6_CR55","doi-asserted-by":"publisher","unstructured":"Keller, M., Orsini, E., Scholl, P.: Actively secure OT extension with optimal overhead. In: Gennaro, R., Robshaw, M.J.B. (eds.) CRYPTO\u00a02015, Part\u00a0I. LNCS, vol.\u00a09215, pp. 724\u2013741. Springer, Heidelberg (2015). https:\/\/doi.org\/10.1007\/978-3-662-47989-6_35","DOI":"10.1007\/978-3-662-47989-6_35"},{"key":"6_CR56","doi-asserted-by":"publisher","unstructured":"Keller, M., Orsini, E., Scholl, P.: MASCOT: faster malicious arithmetic secure computation with oblivious transfer. In: Weippl, E.R., Katzenbeisser, S., Kruegel, C., Myers, A.C., Halevi, S. (eds.) ACM CCS 2016, pp. 830\u2013842. ACM Press, October 2016. https:\/\/doi.org\/10.1145\/2976749.2978357","DOI":"10.1145\/2976749.2978357"},{"key":"6_CR57","unstructured":"Lin, F., Xing, C., Yao, Y.: More efficient zero-knowledge protocols over $$\\mathbb{Z}_{2^k}$$ via galois rings. Cryptology ePrint Archive, Report 2023\/150 (2023). https:\/\/eprint.iacr.org\/2023\/150"},{"key":"6_CR58","unstructured":"Liu, H., Wang, X., Yang, K., Yu, Y.: The hardness of LPN over any integer ring and field for PCG applications. Cryptology ePrint Archive, Report 2022\/712 (2022). https:\/\/eprint.iacr.org\/2022\/712"},{"key":"6_CR59","doi-asserted-by":"publisher","unstructured":"May, A., Meurer, A., Thomae, E.: Decoding random linear codes in $$\\tilde{\\cal{O}}(2^{0.054n})$$. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT\u00a02011. LNCS, vol.\u00a07073, pp. 107\u2013124. Springer, Heidelberg (2011). https:\/\/doi.org\/10.1007\/978-3-642-25385-0_6","DOI":"10.1007\/978-3-642-25385-0_6"},{"issue":"5","key":"6_CR60","doi-asserted-by":"publisher","first-page":"3927","DOI":"10.1109\/TIT.2018.2804444","volume":"64","author":"CA Melchor","year":"2018","unstructured":"Melchor, C.A., Blazy, O., Deneuville, J., Gaborit, P., Z\u00e9mor, G.: Efficient encryption from random quasi-cyclic codes. IEEE Trans. Inf. Theory 64(5), 3927\u20133943 (2018). https:\/\/doi.org\/10.1109\/TIT.2018.2804444","journal-title":"IEEE Trans. Inf. Theory"},{"key":"6_CR61","doi-asserted-by":"publisher","unstructured":"Meyer-Hilfiger, C., Tillich, J.: Rigorous foundations for dual attacks in coding theory. In: Rothblum, G.N., Wee, H. (eds.) TCC 2023. LNCS, vol. 14372, pp. 3\u201332. Springer, Heidelberg (2023). https:\/\/doi.org\/10.1007\/978-3-031-48624-1_1","DOI":"10.1007\/978-3-031-48624-1_1"},{"key":"6_CR62","doi-asserted-by":"publisher","unstructured":"Micciancio, D., Mol, P.: Pseudorandom knapsacks and the sample complexity of LWE search-to-decision reductions. In: Rogaway, P. (ed.) CRYPTO\u00a02011. LNCS, vol.\u00a06841, pp. 465\u2013484. Springer, Heidelberg (2011). https:\/\/doi.org\/10.1007\/978-3-642-22792-9_26","DOI":"10.1007\/978-3-642-22792-9_26"},{"key":"6_CR63","doi-asserted-by":"publisher","unstructured":"Misoczki, R., Tillich, J., Sendrier, N., Barreto, P.S.L.M.: MDPC-McEliece: new McEliece variants from moderate density parity-check codes. In: Proceedings of the 2013 IEEE International Symposium on Information Theory, 2013. pp. 2069\u20132073. IEEE (2013). https:\/\/doi.org\/10.1109\/ISIT.2013.6620590","DOI":"10.1109\/ISIT.2013.6620590"},{"key":"6_CR64","doi-asserted-by":"publisher","unstructured":"Nielsen, J.B., Nordholt, P.S., Orlandi, C., Burra, S.S.: A new approach to practical active-secure two-party computation. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO\u00a02012. LNCS, vol.\u00a07417, pp. 681\u2013700. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-32009-5_40","DOI":"10.1007\/978-3-642-32009-5_40"},{"key":"6_CR65","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1109\/TIT.1962.1057777","volume":"8","author":"E Prange","year":"1962","unstructured":"Prange, E.: The use of information sets in decoding cyclic codes. IRE Trans. Inf. Theory 8, 5\u20139 (1962). https:\/\/doi.org\/10.1109\/TIT.1962.1057777","journal-title":"IRE Trans. Inf. Theory"},{"key":"6_CR66","doi-asserted-by":"publisher","unstructured":"Raghuraman, S., Rindal, P.: Blazing fast PSI from improved OKVS and subfield VOLE. In: Yin, H., Stavrou, A., Cremers, C., Shi, E. (eds.) ACM CCS 2022, pp. 2505\u20132517. ACM Press, November 2022. https:\/\/doi.org\/10.1145\/3548606.3560658","DOI":"10.1145\/3548606.3560658"},{"key":"6_CR67","doi-asserted-by":"publisher","unstructured":"Raghuraman, S., Rindal, P., Tanguy, T.: Expand-convolute codes for pseudorandom correlation generators from LPN. In: CRYPTO\u00a02023, Part\u00a0IV, pp. 602\u2013632. LNCS, Springer, Heidelberg (2023). https:\/\/doi.org\/10.1007\/978-3-031-38551-3_19","DOI":"10.1007\/978-3-031-38551-3_19"},{"key":"6_CR68","doi-asserted-by":"publisher","unstructured":"Rindal, P., Schoppmann, P.: VOLE-PSI: fast OPRF and circuit-PSI from vector-OLE. In: Canteaut, A., Standaert, F.X. (eds.) EUROCRYPT\u00a02021, Part\u00a0II. LNCS, vol. 12697, pp. 901\u2013930. Springer, Heidelberg (2021). https:\/\/doi.org\/10.1007\/978-3-030-77886-6_31","DOI":"10.1007\/978-3-030-77886-6_31"},{"key":"6_CR69","doi-asserted-by":"publisher","unstructured":"Schoppmann, P., Gasc\u00f3n, A., Reichert, L., Raykova, M.: Distributed vector-OLE: improved constructions and implementation. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J. (eds.) ACM CCS 2019, pp. 1055\u20131072. ACM Press, November 2019. https:\/\/doi.org\/10.1145\/3319535.3363228","DOI":"10.1145\/3319535.3363228"},{"key":"6_CR70","doi-asserted-by":"publisher","unstructured":"Sendrier, N.: Decoding one out of many. In: Yang, B.Y. (ed.) Post-Quantum Cryptography - 4th International Workshop, PQCrypto 2011, pp. 51\u201367. Springer, Heidelberg, November\/December 2011. https:\/\/doi.org\/10.1007\/978-3-642-25405-5_4","DOI":"10.1007\/978-3-642-25405-5_4"},{"key":"6_CR71","doi-asserted-by":"publisher","unstructured":"Stern, J.: A method for finding codewords of small weight. In: Coding Theory and Applications, vol.\u00a0388 (1988). https:\/\/doi.org\/10.1007\/BFB0019850","DOI":"10.1007\/BFB0019850"},{"issue":"5","key":"6_CR72","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1006\/JSCO.2002.0533","volume":"33","author":"E Thom\u00e9","year":"2002","unstructured":"Thom\u00e9, E.: Subquadratic computation of vector generating polynomials and improvement of the block Wiedemann algorithm. J. Symb. Comput. 33(5), 757\u2013775 (2002). https:\/\/doi.org\/10.1006\/JSCO.2002.0533","journal-title":"J. Symb. Comput."},{"key":"6_CR73","doi-asserted-by":"publisher","unstructured":"Torres, R.C., Sendrier, N.: Analysis of information set decoding for a sub-linear error weight. In: Takagi, T. (ed.) Post-Quantum Cryptography - 7th International Workshop, PQCrypto 2016, pp. 144\u2013161. Springer, Heidelberg (2016). https:\/\/doi.org\/10.1007\/978-3-319-29360-8_10","DOI":"10.1007\/978-3-319-29360-8_10"},{"key":"6_CR74","doi-asserted-by":"publisher","unstructured":"Wang, X., Ranellucci, S., Katz, J.: Authenticated garbling and efficient maliciously secure two-party computation. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 21\u201337. ACM Press, October\/November 2017. https:\/\/doi.org\/10.1145\/3133956.3134053","DOI":"10.1145\/3133956.3134053"},{"key":"6_CR75","doi-asserted-by":"publisher","unstructured":"Wang, X., Ranellucci, S., Katz, J.: Global-scale secure multiparty computation. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 39\u201356. ACM Press, October\/November 2017. https:\/\/doi.org\/10.1145\/3133956.3133979","DOI":"10.1145\/3133956.3133979"},{"key":"6_CR76","doi-asserted-by":"publisher","unstructured":"Weng, C., Yang, K., Katz, J., Wang, X.: Wolverine: fast, scalable, and communication-efficient zero-knowledge proofs for boolean and arithmetic circuits. In: 2021 IEEE Symposium on Security and Privacy, pp. 1074\u20131091. IEEE Computer Society Press, May 2021. https:\/\/doi.org\/10.1109\/SP40001.2021.00056","DOI":"10.1109\/SP40001.2021.00056"},{"key":"6_CR77","unstructured":"Weng, C., Yang, K., Xie, X., Katz, J., Wang, X.: Mystique: efficient conversions for zero-knowledge proofs with applications to machine learning. In: Bailey, M., Greenstadt, R. (eds.) USENIX Security 2021, pp. 501\u2013518. USENIX Association, August 2021"},{"key":"6_CR78","doi-asserted-by":"publisher","unstructured":"Weng, C., Yang, K., Yang, Z., Xie, X., Wang, X.: AntMan: interactive zero-knowledge proofs with sublinear communication. In: Yin, H., Stavrou, A., Cremers, C., Shi, E. (eds.) ACM CCS 2022, pp. 2901\u20132914. ACM Press, November 2022. https:\/\/doi.org\/10.1145\/3548606.3560667","DOI":"10.1145\/3548606.3560667"},{"issue":"1","key":"6_CR79","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1109\/TIT.1986.1057137","volume":"32","author":"DH Wiedemann","year":"1986","unstructured":"Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory 32(1), 54\u201362 (1986)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"6_CR80","doi-asserted-by":"publisher","unstructured":"Yang, K., Sarkar, P., Weng, C., Wang, X.: QuickSilver: efficient and affordable zero-knowledge proofs for circuits and polynomials over any field. In: Vigna, G., Shi, E. (eds.) ACM CCS 2021, pp. 2986\u20133001. ACM Press, November 2021. https:\/\/doi.org\/10.1145\/3460120.3484556","DOI":"10.1145\/3460120.3484556"},{"key":"6_CR81","doi-asserted-by":"publisher","unstructured":"Yang, K., Wang, X., Zhang, J.: More efficient MPC from improved triple generation and authenticated garbling. In: Ligatti, J., Ou, X., Katz, J., Vigna, G. (eds.) ACM CCS 2020, pp. 1627\u20131646. ACM Press, November 2020. https:\/\/doi.org\/10.1145\/3372297.3417285","DOI":"10.1145\/3372297.3417285"},{"key":"6_CR82","doi-asserted-by":"publisher","unstructured":"Yang, K., Weng, C., Lan, X., Zhang, J., Wang, X.: Ferret: fast extension for correlated OT with small communication. In: Ligatti, J., Ou, X., Katz, J., Vigna, G. (eds.) ACM CCS 2020, pp. 1607\u20131626. ACM Press, November 2020. https:\/\/doi.org\/10.1145\/3372297.3417276","DOI":"10.1145\/3372297.3417276"},{"key":"6_CR83","doi-asserted-by":"publisher","unstructured":"Yu, Y., Steinberger, J.P.: Pseudorandom functions in almost constant depth from low-noise LPN. In: Fischlin, M., Coron, J.S. (eds.) EUROCRYPT\u00a02016, Part\u00a0II. LNCS, vol.\u00a09666, pp. 154\u2013183. Springer, Heidelberg (2016). https:\/\/doi.org\/10.1007\/978-3-662-49896-5_6","DOI":"10.1007\/978-3-662-49896-5_6"},{"key":"6_CR84","doi-asserted-by":"publisher","unstructured":"Yu, Y., Zhang, J., Weng, J., Guo, C., Li, X.: Collision resistant hashing from sub-exponential learning parity with noise. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT\u00a02019, Part\u00a0II. LNCS, vol. 11922, pp. 3\u201324. Springer, Heidelberg (2019). https:\/\/doi.org\/10.1007\/978-3-030-34621-8_1","DOI":"10.1007\/978-3-030-34621-8_1"},{"key":"6_CR85","unstructured":"Zichron, L.: Locally computable arithmetic pseudorandom generators. Master\u2019s thesis, School of Electrical Engineering, Tel Aviv University (2017)"}],"container-title":["Lecture Notes in Computer Science","Advances in Cryptology \u2013 EUROCRYPT 2024"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-58751-1_6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T00:05:18Z","timestamp":1777334718000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-58751-1_6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024]]},"ISBN":["9783031587504","9783031587511"],"references-count":85,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-58751-1_6","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024]]},"assertion":[{"value":"29 April 2024","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"EUROCRYPT","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Annual International Conference on the Theory and Applications of Cryptographic Techniques","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Zurich","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Switzerland","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2024","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"26 May 2024","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"30 May 2024","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"43","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"eurocrypt2024","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/eurocrypt.iacr.org\/2024\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}