{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T13:14:58Z","timestamp":1777295698119,"version":"3.51.4"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T00:00:00Z","timestamp":1774915200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T00:00:00Z","timestamp":1774915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"PIA4 X7PQC"},{"DOI":"10.13039\/501100022107","name":"Universit\u00e9 de Limoges","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100022107","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Nowadays, equivalence problems are widely used in cryptography, most notably to establish cryptosystems such as digital signatures, with MEDS, LESS, PERK as the most recent ones. However, in the context of matrix codes, only the code equivalence problem has been studied, while the subcode equivalence is well-defined in the Hamming metric. In this work, we introduce two new problems: the Matrix Subcode Equivalence Problem and the Inhomogeneous Matrix Subcode Problem, to which we apply the Multi-Party-Computation-in-the-Head (MPCitH) paradigm to build a signature scheme. These new problems, closely related to the Matrix Code Equivalence problem, ask to find an isometry given a code\n                    <jats:italic>C<\/jats:italic>\n                    and a subcode\n                    <jats:italic>D<\/jats:italic>\n                    . Furthermore, we prove that the Matrix Subcode Equivalence Problem reduces to the Hamming Subcode Equivalence problem, which is known to be NP-Complete, thus introducing the matrix code version of the Permuted Kernel Problem. We also adapt the combinatorial and algebraic algorithms for the Matrix Code Equivalence problem to the subcode case, and we analyze their complexities. We find with this analysis that the algorithms perform much worse than in the code equivalence case, which is the same as what happens in the Hamming metric. Finally, our analysis of the attacks allows us to take parameters much smaller than in the Matrix Code Equivalence case. Coupled with the effectiveness of\n                    <jats:italic>Threshold-Computation-in-the-Head<\/jats:italic>\n                    or\n                    <jats:italic>VOLE-in-the-Head<\/jats:italic>\n                    , we obtain a signature size of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\approx $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mo>\u2248<\/mml:mo>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    4800 Bytes, with a public key of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\approx $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mo>\u2248<\/mml:mo>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    275 Bytes. We thus obtain a reasonable signature size, which brings diversity in the landscape of post-quantum signature schemes, by relying on a new hard problem. In particular, this new signature scheme performs better than SPHINCS+, with a smaller size of public key + signature. Our signature compares also well with other signature schemes: compared to MEDS, the signature is smaller, and we reduced the size of the sum of signature and public key by a factor close to 5. We also obtain a signature size that is almost half the size of the CROSS signature scheme.\n                  <\/jats:p>","DOI":"10.1007\/s10623-026-01836-8","type":"journal-article","created":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T15:25:09Z","timestamp":1774970709000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The matrix subcode equivalence problem and its application to signature with MPC-in-the-head"],"prefix":"10.1007","volume":"94","author":[{"given":"Magali","family":"Bardet","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Charles","family":"Brion","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Philippe","family":"Gaborit","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Mercedes","family":"Haiech","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Romaric","family":"Neveu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,3,31]]},"reference":[{"key":"1836_CR1","unstructured":"Aaraj N., Bettaieb S., Bidoux L., Budroni A., Dyseryn V., Esser A., Gaborit P., Kulkarni M., Mateu V., Palumbi M., Perin L., Tillich J.-P.: PERK. NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 2) (2024).https:\/\/pqc-perk.org\/."},{"key":"1836_CR2","doi-asserted-by":"crossref","unstructured":"Adj G., Aragon N., Barbero S., Bardet M., Bellini E., Bidoux L., Chi-Dom\u00ednguez J.-J., Dyseryn V., Esser A., Feneuil T., Gaborit P., Neveu R., Rivain M., Rivera-Zamarripa L., Sanna C., Tillich J.-P., Verbel J., Zweydinger F.: Mirath. NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 2) (2024). https:\/\/pqc-mirath.org\/.","DOI":"10.46586\/tches.v2024.i2.304-328"},{"key":"1836_CR3","doi-asserted-by":"publisher","first-page":"564","DOI":"10.1007\/978-3-031-30589-4_20","volume-title":"Advances in Cryptology\u2014EUROCRYPT 2023","author":"C Aguilar-Melchor","year":"2023","unstructured":"Aguilar-Melchor C., Gama N., Howe J., H\u00fclsing A., Joseph D., Yue D.: The return of the SDitH. In: Hazay C., Stam M. (eds.) Advances in Cryptology\u2014EUROCRYPT 2023, pp. 564\u2013596. Springer, Cham (2023)."},{"key":"1836_CR4","unstructured":"Aguilar-Melchor C., Bettaieb S., Bidoux L., Feneuil T., Gaborit P., Gama N., Gueron S., Howe J., H\u00fclsing A., Joseph D., Joux A., Kulkarni M., Persichetti E., Randrianarisoa T.H., Rivain M., Yue D.: SD-in-the-Head Signature Scheme, NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 2) (2024). https:\/\/sdith.org\/."},{"key":"1836_CR5","unstructured":"Aragon N., Bardet M., Bidoux L., Chi-Dom\u00ednguez J.-J., Dyseryn V., Feneuil T., Gaborit P., Joux A., Neveu R., Rivain M., Tillich J.-P., Vincotte A.: RYDE. NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 2) (2024). https:\/\/pqc-ryde.org\/."},{"key":"1836_CR6","unstructured":"Aumasson J.-P., Bernstein D.J., Beullens W., Dobraunig C., Eichlseder M., Fluhrer S., Gazdag S.-L., H\u00fclsing A., Kampanakis P., K\u00f6lbl S., Lange T., Lauridsen M.M., Mendel F., Niederhagen R., Rechberger C., Rijneveld J., Schwabe P., Westerbaan B.: SPHINCS+, Submission to the 3rd round of the NIST post-quantum project (2022). https:\/\/sphincs.org\/."},{"key":"1836_CR7","unstructured":"Baldi M., Barenghi A., Battagliola M., Bitzer S., Karl P., Manganiello F., Pavoni A., Pelosi G., Santini P., Schupp J., Signorini E., Slaughter F., Wachter-Zeh A., Weger V.: CROSS: Codes and Restricted Objects Signature Scheme. NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 2) (2024). https:\/\/www.cross-crypto.com\/."},{"key":"1836_CR8","unstructured":"Baldi M., Barenghi A., Beckwith L., Biasse J.-F., Chou T., Esser A., Gaj K., Karl P., Mohajerani K., Pelosi G., Persichetti E., Saarinen M.-J., Santini P., Wallace R., Zweydinger F.: LESS: Linear Equivalence Signature Scheme. NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 2) (2024).https:\/\/www.less-project.com\/."},{"key":"1836_CR9","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/978-3-030-45374-9_17","volume-title":"Public-Key Cryptography\u2014PKC 2020","author":"C Baum","year":"2020","unstructured":"Baum C., Nof A.: Concretely-efficient zero-knowledge arguments for arithmetic circuits and their application to lattice-based cryptography. In: Kiayias A., Kohlweiss M., Wallden P., Zikas V. (eds.) Public-Key Cryptography\u2014PKC 2020, pp. 495\u2013526. Springer, Cham (2020)."},{"key":"1836_CR10","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1007\/978-3-031-38554-4_19","volume-title":"Advances in Cryptology\u2014CRYPTO 2023","author":"C Baum","year":"2023","unstructured":"Baum C., Braun L., Saint Guilhem C.D., Kloo\u00df M., Orsini E., Roy L., Scholl P.: Publicly verifiable zero-knowledge and post-quantum signatures from VOLE-in-the-head. In: Handschuh H., Lysyanskaya A. (eds.) Advances in Cryptology\u2014CRYPTO 2023, pp. 581\u2013615. Springer, Cham (2023)."},{"key":"1836_CR11","unstructured":"Baum C., Beullens W., Braun L., Saint\u00a0Guilhem C.D., Kloo\u00df M., Majenz C., Mukherjee S., Orsini E., Ramacher S., Rechberger C., Roy L., Scholl P.: FAEST. NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 2) (2024). https:\/\/faest.info\/."},{"key":"1836_CR12","first-page":"463","volume-title":"Advances in Cryptology\u2014ASIACRYPT 2024","author":"C Baum","year":"2024","unstructured":"Baum C., Beullens W., Mukherjee S., Orsini E., Ramacher S., Rechberger C., Roy L., Scholl P.: One tree to rule them all: optimizing GGM trees and OWFs for post-quantum signatures. In: Chung K.-M., Sasaki Y. (eds.) Advances in Cryptology\u2014ASIACRYPT 2024, pp. 463\u2013493. Springer, Singapore (2024)."},{"key":"1836_CR13","doi-asserted-by":"crossref","unstructured":"Benadjila R., Bouillaguet C., Feneuil T., Rivain M.: MQ on my mind: post-quantum signatures from the non-structured multivariate quadratic problem. In: 2024 IEEE 9th European Symposium on Security and Privacy (EuroS&P), pp. 468\u2013485. IEEE (2024).","DOI":"10.1109\/EuroSP60621.2024.00032"},{"key":"1836_CR14","doi-asserted-by":"publisher","first-page":"3016","DOI":"10.1109\/TIT.2003.819322","volume":"49","author":"T Berger","year":"2003","unstructured":"Berger T.: Isometries for rank distance and permutation group of Gabidulin codes. IEEE Trans. Inf. Theory 49, 3016\u20133019 (2003). https:\/\/doi.org\/10.1109\/TIT.2003.819322.","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1836_CR15","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1007\/978-3-319-55589-8_15","volume-title":"Codes, Cryptology and Information Security","author":"TP Berger","year":"2017","unstructured":"Berger T.P., Gueye C.T., Klamti J.B.: A np-complete problem in coding theory with application to code based cryptography. In: El Hajji S., Nitaj A., Souidi E.M. (eds.) Codes, Cryptology and Information Security, pp. 230\u2013237. Springer, Cham (2017)."},{"key":"1836_CR16","unstructured":"Bettaieb S., Bidoux L., Gaborit P., Kulkarni M.: Modelings for generic PoK and Applications: Shorter SD and PKP based Signatures. Cryptology ePrint Archive, Paper 2024\/1668 (2024). https:\/\/eprint.iacr.org\/2024\/1668."},{"key":"1836_CR17","first-page":"387","volume-title":"Selected Areas in Cryptography","author":"W Beullens","year":"2020","unstructured":"Beullens W.: Not Enough LESS: An Improved Algorithm for Solving Code Equivalence Problems over $$\\mathbb{F} _q$$. In: Dunkelman O., Jacobson M.J. Jr., O\u2019Flynn C. (eds.) Selected Areas in Cryptography, pp. 387\u2013403. Springer, Cham (2020)."},{"key":"1836_CR18","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/978-3-030-35423-7_1","volume-title":"Progress in Cryptology\u2014INDOCRYPT 2019","author":"W Beullens","year":"2019","unstructured":"Beullens W., Faug\u00e8re J.-C., Koussa E., Macario-Rat G., Patarin J., Perret L.: PKP-based signature scheme. In: Hao F., Ruj S., Sen Gupta S. (eds.) Progress in Cryptology\u2014INDOCRYPT 2019, pp. 3\u201322. Springer, Cham (2019)."},{"key":"1836_CR19","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1007\/978-3-031-33017-9_2","volume-title":"Codes, Cryptology and Information Security","author":"L Bidoux","year":"2023","unstructured":"Bidoux L., Gaborit P.: Compact post-quantum signatures from proofs of knowledge leveraging structure for the PKP, SD and RSD problems. In: El Hajji S., Mesnager S., Souidi E.M. (eds.) Codes, Cryptology and Information Security, pp. 10\u201342. Springer, Cham (2023)."},{"key":"1836_CR20","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1007\/978-981-96-0888-1_2","volume-title":"Advances in Cryptology\u2014ASIACRYPT 2024","author":"L Bidoux","year":"2025","unstructured":"Bidoux L., Feneuil T., Gaborit P., Neveu R., Rivain M.: Dual support decomposition in the head: shorter signatures from rank SD and MinRank. In: Chung K.-M., Sasaki Y. (eds.) Advances in Cryptology\u2014ASIACRYPT 2024, pp. 38\u201369. Springer, Singapore (2025)."},{"key":"1836_CR21","unstructured":"Bl\u00e4ser M., Duong D.H., Narayanan A.K., Plantard T., Qiao Y., Sipasseuth A., Tang G.: ALTEQ (2023). https:\/\/pqcalteq.github.io\/."},{"key":"1836_CR22","unstructured":"Bouillaguet C.: Algorithms for some hard problems and cryptographic attacks against specific cryptographic primitives. Theses, Universit\u00e9 Paris Diderot (Paris 7) (November 2011). https:\/\/theses.hal.science\/tel-03630843."},{"key":"1836_CR23","unstructured":"Chou T., Niederhagen R., Persichetti E., Ran L., Randrianarisoa T.H., Reijnders K., Samardjiska S., Trimoska M.: MEDS: Matrix Equivalence Digital Signature Scheme. NIST\u2019s Post-Quantum Cryptography Standardization of Additional Digital Signature Schemes Project (Round 1) (2023). https:\/\/www.meds-pqc.org\/."},{"key":"1836_CR24","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1007\/978-3-031-37679-5_2","volume-title":"Progress in Cryptology\u2014AFRICACRYPT 2023","author":"T Chou","year":"2023","unstructured":"Chou T., Niederhagen R., Persichetti E., Randrianarisoa T.H., Reijnders K., Samardjiska S., Trimoska M.: Take your meds: digital signatures from matrix code equivalence. In: El Mrabet N., De Feo L., Duquesne S. (eds.) Progress in Cryptology\u2014AFRICACRYPT 2023, pp. 28\u201352. Springer, Cham (2023)."},{"key":"1836_CR25","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/978-3-031-62743-9_4","volume-title":"Post-Quantum Cryptography","author":"T Chou","year":"2024","unstructured":"Chou T., Niederhagen R., Ran L., Samardjiska S.: Reducing signature size of matrix-code-based signature schemes. In: Saarinen M.-J., Smith-Tone D. (eds.) Post-Quantum Cryptography, pp. 107\u2013134. Springer, Cham (2024)."},{"key":"1836_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-025-01576-1","author":"T Chou","year":"2025","unstructured":"Chou T., Persichetti E., Santini P.: On linear equivalence, canonical forms, and digital signatures. Des. Codes Cryptogr. (2025). https:\/\/doi.org\/10.1007\/s10623-025-01576-1.","journal-title":"Des. Codes Cryptogr."},{"key":"1836_CR27","unstructured":"Chou T., Niederhagen R., Persichetti E., Ran L., Randrianarisoa T.H., Reijnders K., Samardjiska S., Trimoska M.: NIST first round for additional post-quantum signatures\u2014official comments. https:\/\/csrc.nist.gov\/csrc\/media\/Projects\/pqc-dig-sig\/documents\/round-1\/official-comments\/MEDS-round1-dig-sig-official-comment.pdf."},{"key":"1836_CR28","unstructured":"Couvreur A., Debris-Alazard T., Gaborit P.: On the hardness of code equivalence problems in rank metric (2021). arXiv:2011.04611."},{"key":"1836_CR29","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/978-981-99-8721-4_14","volume-title":"Advances in Cryptology\u2014ASIACRYPT 2023","author":"T Feneuil","year":"2023","unstructured":"Feneuil T., Rivain M.: Threshold linear secret sharing to the rescue of MPC-in-the-head. In: Guo J., Steinfeld R. (eds.) Advances in Cryptology\u2014ASIACRYPT 2023, pp. 441\u2013473. Springer, Singapore (2023)."},{"issue":"3","key":"1836_CR30","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1007\/s00145-025-09543-8","volume":"38","author":"T Feneuil","year":"2025","unstructured":"Feneuil T., Rivain M.: Threshold computation in the head: improved framework for post-quantum signatures and zero-knowledge arguments. J. Cryptol. 38(3), 28 (2025).","journal-title":"J. Cryptol."},{"key":"1836_CR31","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1007\/978-3-031-15979-4_19","volume-title":"Advances in Cryptology\u2014CRYPTO 2022","author":"T Feneuil","year":"2022","unstructured":"Feneuil T., Joux A., Rivain M.: Syndrome decoding in the head: shorter signatures from zero-knowledge proofs. In: Dodis Y., Shrimpton T. (eds.) Advances in Cryptology\u2014CRYPTO 2022, pp. 541\u2013572. Springer, Cham (2022)."},{"key":"1836_CR32","doi-asserted-by":"crossref","unstructured":"Fiat A., Shamir A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko A.M. (ed.) CRYPTO\u201986. LNCS, vol. 263, pp. 186\u2013194. Springer, Heidelberg (1987).","DOI":"10.1007\/3-540-47721-7_12"},{"key":"1836_CR33","unstructured":"Huth J., Joux A.: VOLE-in-the-head signatures from Subfield Bilinear Collisions . Cryptology ePrint Archive, Paper 2024\/1537 (2024). https:\/\/eprint.iacr.org\/2024\/1537."},{"key":"1836_CR34","first-page":"21","volume-title":"39th ACM STOC","author":"Y Ishai","year":"2007","unstructured":"Ishai Y., Kushilevitz E., Ostrovsky R., Sahai A.: Zero-knowledge from secure multiparty computation. In: Johnson D.S., Feige U. (eds.) 39th ACM STOC, pp. 21\u201330. ACM Press, New York (2007)."},{"key":"1836_CR35","doi-asserted-by":"publisher","unstructured":"Katz J., Kolesnikov V., Wang X.: Improved non-interactive zero knowledge with applications to post-quantum signatures. In: Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security. CCS \u201918, pp. 525\u2013537. Association for Computing Machinery, New York (2018). https:\/\/doi.org\/10.1145\/3243734.3243805 .","DOI":"10.1145\/3243734.3243805"},{"issue":"3","key":"1836_CR36","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1109\/TIT.1982.1056498","volume":"28","author":"J Leon","year":"1982","unstructured":"Leon J.: Computing automorphism groups of error-correcting codes. IEEE Trans. Inf. Theory 28(3), 496\u2013511 (1982). https:\/\/doi.org\/10.1109\/TIT.1982.1056498.","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1836_CR37","unstructured":"MacWilliams F.J.: Combinatorial properties of elementary abelian groups. PhD thesis, Radcliff college (1962)."},{"key":"1836_CR38","doi-asserted-by":"publisher","unstructured":"Narayanan A.K., Qiao Y., Tang G.: Algorithms for matrix code and alternating trilinear form equivalences via new isomorphism invariants. In: Advances in Cryptology\u2014EUROCRYPT 2024: 43rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Zurich, Switzerland, May 26\u201330, 2024, Proceedings, Part III, pp. 160\u2013187. Springer, Berlin (2024). https:\/\/doi.org\/10.1007\/978-3-031-58734-4_6.","DOI":"10.1007\/978-3-031-58734-4_6"},{"key":"1836_CR39","unstructured":"NIST: Call for Additional Digital Signature Schemes for the Post-Quantum Cryptography Standardization Process. https:\/\/csrc.nist.gov\/Projects\/pqc-dig-sig\/standardization\/call-for-proposals."},{"key":"1836_CR40","unstructured":"NIST: Call for Post-Quantum Cryptography Standardization. https:\/\/csrc.nist.gov\/Projects\/post-quantum-cryptography\/post-quantum-cryptography-standardization\/Call-for-Proposals."},{"issue":"5","key":"1836_CR41","doi-asserted-by":"publisher","first-page":"1602","DOI":"10.1109\/18.623157","volume":"43","author":"E Petrank","year":"1997","unstructured":"Petrank E., Roth R.M.: Is code equivalence easy to decide? IEEE Trans. Inf. Theory 43(5), 1602\u20131604 (1997). https:\/\/doi.org\/10.1109\/18.623157.","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1836_CR42","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1007\/978-981-96-0944-4_3","volume-title":"Advances in Cryptology\u2014ASIACRYPT 2024","author":"L Ran","year":"2025","unstructured":"Ran L., Samardjiska S.: Rare structures in tensor graphs. In: Chung K.-M., Sasaki Y. (eds.) Advances in Cryptology\u2014ASIACRYPT 2024, pp. 66\u201396. Springer, Singapore (2025)."},{"key":"1836_CR43","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1007\/s10623-023-01338-x","volume":"92","author":"K Reijnders","year":"2024","unstructured":"Reijnders K., Samardjiska S., Trimoska M.: Hardness estimates of the code equivalence problem in the rank metric. Des. Codes Cryptogr. 92, 833\u2013862 (2024). https:\/\/doi.org\/10.1007\/s10623-023-01338-x.","journal-title":"Des. Codes Cryptogr."},{"key":"1836_CR44","doi-asserted-by":"publisher","unstructured":"Saint\u00a0Guilhem C., Orsini E., Tanguy T.: Limbo: efficient Zero-knowledge MPCitH-based arguments. In: Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security. CCS \u201921, pp. 3022\u20133036. Association for Computing Machinery, New York (2021). https:\/\/doi.org\/10.1145\/3460120.3484595.","DOI":"10.1145\/3460120.3484595"},{"issue":"3","key":"1836_CR45","doi-asserted-by":"publisher","first-page":"2254","DOI":"10.1109\/TIT.2023.3323068","volume":"70","author":"P Santini","year":"2024","unstructured":"Santini P., Baldi M., Chiaraluce F.: Computational hardness of the permuted kernel and subcode equivalence problems. IEEE Trans. Inf. Theory 70(3), 2254\u20132270 (2024). https:\/\/doi.org\/10.1109\/TIT.2023.3323068.","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"1836_CR46","doi-asserted-by":"publisher","first-page":"1193","DOI":"10.1109\/18.850662","volume":"46","author":"N Sendrier","year":"2000","unstructured":"Sendrier N.: Finding the permutation between equivalent linear codes: the support splitting algorithm. IEEE Trans. Inf. Theory 46(4), 1193\u20131203 (2000). https:\/\/doi.org\/10.1109\/18.850662.","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1836_CR47","doi-asserted-by":"publisher","first-page":"606","DOI":"10.1007\/0-387-34805-0_54","volume-title":"Advances in Cryptology\u2014CRYPTO\u2019 89 Proceedings","author":"A Shamir","year":"1990","unstructured":"Shamir A.: An efficient identification scheme based on permuted kernels. In: Brassard G. (ed.) Advances in Cryptology\u2014CRYPTO\u2019 89 Proceedings, pp. 606\u2013609. Springer, New York (1990)."},{"key":"1836_CR48","doi-asserted-by":"publisher","unstructured":"Shor P.W.: Algorithms for quantum computation: discrete logarithms and factoring . In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124\u2013134 (1994). https:\/\/doi.org\/10.1109\/SFCS.1994.365700.","DOI":"10.1109\/SFCS.1994.365700"},{"key":"1836_CR49","doi-asserted-by":"crossref","unstructured":"Stern J.: A new identification scheme based on syndrome decoding. In: International Cryptology Conference (CRYPTO) (1993).","DOI":"10.1007\/3-540-48329-2_2"},{"key":"1836_CR50","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: Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security. CCS \u201921, pp. 2986\u20133001. Association for Computing Machinery, New York (2021). https:\/\/doi.org\/10.1145\/3460120.3484556.","DOI":"10.1145\/3460120.3484556"}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-026-01836-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-026-01836-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-026-01836-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T12:15:38Z","timestamp":1777292138000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-026-01836-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,31]]},"references-count":50,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["1836"],"URL":"https:\/\/doi.org\/10.1007\/s10623-026-01836-8","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"value":"0925-1022","type":"print"},{"value":"1573-7586","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,31]]},"assertion":[{"value":"17 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 January 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 March 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 March 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"76"}}