{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T18:57:01Z","timestamp":1771613821604,"version":"3.50.1"},"reference-count":51,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AMC"],"published-print":{"date-parts":[[2025]]},"DOI":"10.3934\/amc.2024057","type":"journal-article","created":{"date-parts":[[2024,12,26]],"date-time":"2024-12-26T02:47:06Z","timestamp":1735181226000},"page":"1336-1359","source":"Crossref","is-referenced-by-count":1,"title":["Quantum secure digital signature scheme based on multivariate quadratic quasigroups (MQQ)"],"prefix":"10.3934","volume":"19","author":[{"given":"Satish","family":"Kumar","sequence":"first","affiliation":[]},{"given":"Indivar","family":"Gupta","sequence":"additional","affiliation":[{"name":"Scientific Analysis Group, Defence Research & Development Organisation, New Delhi-110054, India"}]},{"given":"Ashok","family":"Ji Gupta","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi-221005, India"}]}],"member":"2321","reference":[{"key":"key-10.3934\/amc.2024057-1","unstructured":"

C.-O. Chen, M.-S. Chen, J. Ding, F. Werner and B.-Y. Yang, Odd-char multivariate hidden field equations, Cryptology ePrint Archive<\/i>, 2008.<\/p>"},{"key":"key-10.3934\/amc.2024057-2","doi-asserted-by":"crossref","unstructured":"

M.-S. Chen, A. H\u00fclsing, J. Rijneveld, S. Samardjiska and P. Schwabe, From 5-pass-based identification to-based signatures, Application of Cryptology and Information Security, Springer<\/i>, 10032<\/b> (2016) 135-165.<\/p>","DOI":"10.1007\/978-3-662-53890-6_5"},{"key":"key-10.3934\/amc.2024057-3"},{"key":"key-10.3934\/amc.2024057-4","doi-asserted-by":"crossref","unstructured":"

J. Ding and D. Schmidt, Rainbow, a new multivariable polynomial signature scheme, Applied Cryptography and Network Security<\/i>, (2005), 164-175.<\/p>","DOI":"10.1007\/11496137_12"},{"key":"key-10.3934\/amc.2024057-5","doi-asserted-by":"crossref","unstructured":"

J. Ding, D. Schmidt and Z. Yin, Cryptanalysis of the new TTS scheme in CHES 2004, International Journal of Information Security, Springer<\/i>, 5<\/b> (2006), 231\u2013240.<\/p>","DOI":"10.1007\/s10207-006-0003-9"},{"key":"key-10.3934\/amc.2024057-6","doi-asserted-by":"crossref","unstructured":"

J. Ding and B.-Y. Yang, Multivariate public key cryptography, Post-Quantum Cryptography, Springer Berlin Heidelberg<\/i>, (2009) 193-241.<\/p>","DOI":"10.1007\/978-3-540-88702-7_6"},{"key":"key-10.3934\/amc.2024057-7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-74143-5_1"},{"key":"key-10.3934\/amc.2024057-8","unstructured":"

L. Euler, Commentationes arithmeticae collectae, Typis ac impensis Academiae Imperialis Scientiarum<\/i>, 2<\/b> (1849).<\/p>"},{"key":"key-10.3934\/amc.2024057-9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-46447-2_7"},{"key":"key-10.3934\/amc.2024057-10","doi-asserted-by":"crossref","unstructured":"

J.-C. Faug\u00e9re, R. S. \u00d8deg\u00e5rd, L. Perret and D. Gligoroski, Analysis of the MQQ public key cryptosystem, Cryptology and Network Security: 9th International Conference, CANS 2010<\/i>, Kuala Lumpur, Malaysia, Springer, (2010), 169-183.<\/p>","DOI":"10.1007\/978-3-642-17619-7_13"},{"key":"key-10.3934\/amc.2024057-11","doi-asserted-by":"crossref","unstructured":"

A. Ferozpuri and K. Gaj, High-speed FPGA implementation of the NIST round 1 rainbow signature scheme, 2018 International Conference on ReConFigurable Computing and FPGAs (ReConFig)<\/i>, IEEE, (2018), 1-8.<\/p>","DOI":"10.1109\/RECONFIG.2018.8641734"},{"key":"key-10.3934\/amc.2024057-12"},{"key":"key-10.3934\/amc.2024057-13","doi-asserted-by":"publisher","DOI":"10.1007\/11426639_20"},{"key":"key-10.3934\/amc.2024057-14","unstructured":"

D. Gligoroski, S. Markovski and S. J. Knapskog, Multivariate quadratic trapdoor functions based on multivariate quadratic quasigroups, Proceedings of the American Conference on Applied Mathematics<\/i>, (2008), 44-49.<\/p>"},{"key":"key-10.3934\/amc.2024057-15","unstructured":"

D. Gligoroski, S. Markovski and S. J. Knapskog, A public key block cipher based on multivariate quadratic quasigroups, preprint, arXiv: 0808.0247, 2008.<\/p>"},{"key":"key-10.3934\/amc.2024057-16","doi-asserted-by":"crossref","unstructured":"

D. Gligoroski, R. S. \u00d8deg\u00e5rd, R. E. Jensen, L. Perret, J. C. Faug\u00e9re, S. J. Knapskog and S. Markovski, MQQ-SIG: An ultra-fast and provably CMA resistant digital signature scheme, International Conference on Trusted Systems, Springer<\/i>, (2011), 184-203.<\/p>","DOI":"10.1007\/978-3-642-32298-3_13"},{"key":"key-10.3934\/amc.2024057-17","unstructured":"

D. Gligoroski and S. Samardjiska, The multivariate probabilistic encryption scheme MQQ-ENC, IACR Cryptology ePrint Archive<\/i>, 2012.<\/p>"},{"key":"key-10.3934\/amc.2024057-18"},{"key":"key-10.3934\/amc.2024057-19"},{"key":"key-10.3934\/amc.2024057-20","unstructured":"

Y. Hashimoto, On the security of HMFEv, Cryptology ePrint Archive<\/i>, 2017.<\/p>"},{"key":"key-10.3934\/amc.2024057-21","unstructured":"

A. D. Keedwell and J. D\u00e9nes, Latin Squares and Their Applications<\/i>, 2$^{nd}$<\/tex-math><\/inline-formula> edition, Elsevier, 2015.<\/p>"},{"key":"key-10.3934\/amc.2024057-22","doi-asserted-by":"crossref","unstructured":"

A. Kipnis, J. Patarin and L. Goubin, Unbalanced oil and vinegar signature schemes, Advances in Cryptology\u2014EUROCRYPT '99<\/i>, Springer, 1592<\/b> (1999), 206-222.<\/p>","DOI":"10.1007\/3-540-48910-X_15"},{"key":"key-10.3934\/amc.2024057-23","doi-asserted-by":"crossref","unstructured":"

A. Kipnis and A. Shamir, Cryptanalysis of the oil and vinegar signature scheme, Advances in Cryptology\u2014CRYPTO '98<\/i>, Springer, 1462<\/b> (1998), 257-266.<\/p>","DOI":"10.1007\/BFb0055733"},{"key":"key-10.3934\/amc.2024057-24","doi-asserted-by":"crossref","unstructured":"

A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem by relinearization, Annual International Cryptology Conference<\/i>, Springer, 1666<\/b> (1999), 19-30.<\/p>","DOI":"10.1007\/3-540-48405-1_2"},{"key":"key-10.3934\/amc.2024057-25","doi-asserted-by":"publisher","DOI":"10.1080\/01611194.2022.2081824"},{"key":"key-10.3934\/amc.2024057-26","doi-asserted-by":"crossref","unstructured":"

S. Kumar, H. Singh, I. Gupta and A. J. Gupta, MDS codes based on orthogonality of quasigroups, Applicable Algebra in Engineering, Communication and Computing<\/i>, (2023), 1-22.<\/p>","DOI":"10.1007\/s00200-023-00631-5"},{"key":"key-10.3934\/amc.2024057-27"},{"key":"key-10.3934\/amc.2024057-28","doi-asserted-by":"crossref","unstructured":"

S. E. Mohamed, J. Ding and J. Buchmann, Algebraic Cryptanalysis of MQQ Public Key Cryptosystem by MutantXL<\/i>, IACR Cryptology ePrint Archive, 2008.<\/p>","DOI":"10.1007\/978-3-642-10433-6_26"},{"key":"key-10.3934\/amc.2024057-29","doi-asserted-by":"crossref","unstructured":"

R. A. Mollin and S. Charles, On permutation polynomials over finite fields, International Journal of Mathematics and Mathematical Sciences<\/i>, Hindawi 10<\/b> (1987), 535-543.<\/p>","DOI":"10.1155\/S0161171287000644"},{"key":"key-10.3934\/amc.2024057-30","doi-asserted-by":"crossref","unstructured":"

R. Moufang, Zur struktur von alternativk\u00f6rpern, Mathematische Annalen<\/i>, Springer, 110<\/b> (1935), 416-430.<\/p>","DOI":"10.1007\/BF01448037"},{"key":"key-10.3934\/amc.2024057-31","unstructured":"

R. \u00d8deg\u00e5rd, L. Perret, J. C. Faug\u00e9re and D. Gligoroski, Analysis of the MQQ public key cryptosystem, Conference on Symbolic Computation and Cryptography<\/i>, 2010.<\/p>"},{"key":"key-10.3934\/amc.2024057-32","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008341625464"},{"key":"key-10.3934\/amc.2024057-33"},{"key":"key-10.3934\/amc.2024057-34","unstructured":"

J. Patarin, The Oil and Vinegar Signature Scheme<\/i>, Presented at the Dagstuhl Workshop on Cryptography September, 1997.<\/p>"},{"key":"key-10.3934\/amc.2024057-35","doi-asserted-by":"crossref","unstructured":"

J. Patarin, Hidden fields equations (HFE) and isomorphisms of polynomials (IP): Two new families of asymmetric algorithms, International Conference on the Theory and Applications of Cryptographic Techniques<\/i>, Springer, (1996) 33-48.<\/p>","DOI":"10.1007\/3-540-68339-9_4"},{"key":"key-10.3934\/amc.2024057-36"},{"key":"key-10.3934\/amc.2024057-37","doi-asserted-by":"crossref","unstructured":"

J. Patarin, N. Courtois and L. Goubin, QUARTZ, 128-bit long digital signatures,, Topics in Cryptology-CT-RSA 2001<\/i>, Springer, (2001), 282-297.<\/p>","DOI":"10.1007\/3-540-45353-9_21"},{"key":"key-10.3934\/amc.2024057-38","unstructured":"

A. Petzoldt, Selecting and reducing key sizes for multivariate cryptography, PhD Thesis, Darmstadt Tuprints<\/i>, 2013.<\/p>"},{"key":"key-10.3934\/amc.2024057-39"},{"key":"key-10.3934\/amc.2024057-40"},{"key":"key-10.3934\/amc.2024057-41","doi-asserted-by":"crossref","unstructured":"

A. Petzoldt, M. S. Chen, B. Y. Yang, C. Tao and J. Ding, Design principles for HFEv-based multivariate signature schemes, Advances in Cryptology-ASIACRYPT 2015<\/i> Springer, 9452<\/b> (2015), 311-334.<\/p>","DOI":"10.1007\/978-3-662-48797-6_14"},{"key":"key-10.3934\/amc.2024057-42","doi-asserted-by":"crossref","unstructured":"

K. Sakumoto, T. Shirai and H. Hiwatari, Public-key identification schemes based on multivariate quadratic polynomials, Advances in Cryptology-CRYPTO 2011<\/i>, Springer, 6841<\/b> (2011), 706-723.<\/p>","DOI":"10.1007\/978-3-642-22792-9_40"},{"key":"key-10.3934\/amc.2024057-43","doi-asserted-by":"crossref","unstructured":"

S. Samardjiska, Y. Chen and D. Gligoroski, Algorithms for construction of Multivariate Quadratic Quasigroups (MQQs) and their parastrophe operations in arbitrary Galois fields, Journal of Information Assurance and Security<\/i>, 7<\/b> (2012).<\/p>","DOI":"10.1109\/ISIAS.2011.6122839"},{"key":"key-10.3934\/amc.2024057-44","unstructured":"

S. Samardjiska, S. Markovski and D. Gligoroski, Multivariate quasigroups defined by t-functions, Conference on Symbolic Computation and Cryptography<\/i>, (2010), 117.<\/p>"},{"key":"key-10.3934\/amc.2024057-45","doi-asserted-by":"crossref","unstructured":"

V. Shcherbacov, Elements of Quasigroup Theory and Applications<\/i>, CRC Press, Boca Raton, FL, 2017.<\/p>","DOI":"10.1201\/9781315120058"},{"key":"key-10.3934\/amc.2024057-46"},{"key":"key-10.3934\/amc.2024057-47","unstructured":"

E. Thomae, About the security of multivariate quadratic public key schemes, PhD Thesis, Ruhr-Universit$\\ddot{a}$<\/tex-math><\/inline-formula>t Bochum, Universit$\\ddot{a}$<\/tex-math><\/inline-formula>tsbibliothek, 2013.<\/p>"},{"key":"key-10.3934\/amc.2024057-48","doi-asserted-by":"crossref","unstructured":"

E. Thomae and C. Wolf, Cryptanalysis of enhanced TTS, STS and all its variants, or: Why cross-terms are important, Progress in Cryptology-AFRICACRYPT 2012<\/i>, Springer, 7374<\/b> (2012), 188-202.<\/p>","DOI":"10.1007\/978-3-642-31410-0_12"},{"key":"key-10.3934\/amc.2024057-49"},{"key":"key-10.3934\/amc.2024057-50","unstructured":"

C. Wolf and B. Preneel, Taxonomy of Public Key Schemes Based on the Problem of Multivariate Quadratic Equations<\/i>, IACR Cryptology ePrint Archive, 2005.<\/p>"},{"key":"key-10.3934\/amc.2024057-51","doi-asserted-by":"crossref","unstructured":"

B.-Y. Yang and J.-M. Chen, Building secure tame-like multivariate public-key cryptosystems: The new TTS, Australasian Conference on Information Security and Privacy<\/i>, Springer, (2005), 518-531.<\/p>","DOI":"10.1007\/11506157_43"}],"container-title":["Advances in Mathematics of Communications"],"original-title":[],"deposited":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T09:36:50Z","timestamp":1745228210000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.aimsciences.org\/\/article\/doi\/10.3934\/amc.2024057"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"references-count":51,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025]]}},"URL":"https:\/\/doi.org\/10.3934\/amc.2024057","relation":{},"ISSN":["1930-5346","1930-5338"],"issn-type":[{"value":"1930-5346","type":"print"},{"value":"1930-5338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025]]}}}