{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T14:52:33Z","timestamp":1773931953612,"version":"3.50.1"},"reference-count":51,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2024,5,20]],"date-time":"2024-05-20T00:00:00Z","timestamp":1716163200000},"content-version":"vor","delay-in-days":140,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2024,1]]},"abstract":"<jats:p>In this paper, we develop the theoretical provisions of symmetric cryptographic algorithms based on the polynomial residue number system for the first time. The main feature of the proposed approach is that when reconstructing the polynomial based on the method of undetermined coefficients, multiplication is performed not on the found base numbers but on arbitrarily selected polynomials. The latter, together with pairwise coprime residues of the residue class system, serve as the keys of the cryptographic algorithm. Schemes and examples of the implementation of the developed polynomial symmetric encryption algorithm are presented. The analytical expressions of the cryptographic strength estimation are constructed, and their graphical dependence on the number of modules and polynomial powers is presented. Our studies show that the cryptanalysis of the proposed algorithm requires combinatorial complexity, which leads to an NP\u2010complete problem.<\/jats:p>","DOI":"10.1155\/2024\/4894415","type":"journal-article","created":{"date-parts":[[2024,5,20]],"date-time":"2024-05-20T21:50:27Z","timestamp":1716241827000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Symmetric Encryption Algorithms in a Polynomial Residue Number System"],"prefix":"10.1155","volume":"2024","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3446-1596","authenticated-orcid":false,"given":"I.","family":"Yakymenko","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8846-332X","authenticated-orcid":false,"given":"M.","family":"Karpinski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5381-9528","authenticated-orcid":false,"given":"R.","family":"Shevchuk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4469-8055","authenticated-orcid":false,"given":"M.","family":"Kasianchuk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2024,5,20]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.6028\/NIST.SP.800\u201012r1"},{"key":"e_1_2_11_2_2","doi-asserted-by":"publisher","DOI":"10.18372\/2225\u20105036.25.13841"},{"key":"e_1_2_11_3_2","first-page":"100","article-title":"Information security in an organization","volume":"24","author":"Alhassan M. M.","year":"2017","journal-title":"International Journal of Computer (IJC)"},{"key":"e_1_2_11_4_2","first-page":"469","article-title":"A survey on cryptography techniques","volume":"6","author":"Adki V.","year":"2016","journal-title":"International Journal of Advanced Research in Computer Science and Software Engineering"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.3390\/axioms10020116"},{"key":"e_1_2_11_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10559\u2010021\u201000358\u20106"},{"key":"e_1_2_11_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10559\u2010022\u201000494\u20107"},{"key":"e_1_2_11_8_2","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2019.2963826"},{"key":"e_1_2_11_9_2","volume-title":"Secret Keys for Nonpositional Cryptosystems. Development, Investigation and Implementation","author":"Biyashev R.","year":"2014"},{"key":"e_1_2_11_10_2","doi-asserted-by":"publisher","DOI":"10.2991\/amsee-16.2016.64"},{"key":"e_1_2_11_11_2","unstructured":"CheonJ. H. HongS. LeeC. andSonY. Polynomial functional encryption scheme with linear ciphertext size 2018 Cryptology ePrint Archive."},{"key":"e_1_2_11_12_2","doi-asserted-by":"publisher","DOI":"10.1515\/eng\u20102016\u20100034"},{"key":"e_1_2_11_13_2","doi-asserted-by":"crossref","unstructured":"BajardJ.-C. ImbertL. andPlantardT. Arithmetic operations in the polynomial modular number system 17th IEEE Symposium on Computer Arithmetic (ARITH\u203205) 2005 Cape Cod MA USA 206\u2013213 https:\/\/doi.org\/10.1109\/ARITH.2005.11.","DOI":"10.1109\/ARITH.2005.11"},{"key":"e_1_2_11_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/s13389\u2010019\u201000221\u20107"},{"key":"e_1_2_11_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2019.03.004"},{"key":"e_1_2_11_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-78967-3_9"},{"key":"e_1_2_11_17_2","first-page":"25","article-title":"An approach of cryptosystem using polynomials and Lucas numbers","volume":"44","author":"Ashok G.","year":"2023","journal-title":"Journal of Harbin Engineering University"},{"key":"e_1_2_11_18_2","doi-asserted-by":"publisher","DOI":"10.1080\/09720529.2022.2133237"},{"key":"e_1_2_11_19_2","doi-asserted-by":"crossref","unstructured":"MohanP. V. A. Specialized residue number systems 2016 Springer.","DOI":"10.1007\/978-3-319-41385-3"},{"key":"e_1_2_11_20_2","doi-asserted-by":"publisher","DOI":"10.1109\/TETC.2021.3073475"},{"key":"e_1_2_11_21_2","doi-asserted-by":"publisher","DOI":"10.1142\/p523"},{"key":"e_1_2_11_22_2","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2004.2"},{"key":"e_1_2_11_23_2","doi-asserted-by":"publisher","DOI":"10.7494\/csci.2019.20.3.3307"},{"key":"e_1_2_11_24_2","unstructured":"MilneJ. S. Algebraic number theory (v3.08) JS Milne 2020.http:\/\/www.jmilne.org\/math\/."},{"key":"e_1_2_11_25_2","unstructured":"FournarisA. P. PapachristodoulouL. BatinaL. andSklavosN. Secure and efficient RNS approach for elliptic curve cryptography In 6th Conference on Trustworthy Manufacturing and Utilization of Secure Devices (TRUDEVICE 2016) 2016 Barcelona 121\u2013126."},{"key":"e_1_2_11_26_2","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2007.70750"},{"key":"e_1_2_11_27_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVLSI.2014.2308322"},{"key":"e_1_2_11_28_2","doi-asserted-by":"publisher","DOI":"10.5755\/j01.eee.21.1.6657"},{"key":"e_1_2_11_29_2","doi-asserted-by":"crossref","unstructured":"GivakiK. HojabrR. NajafiH. KhonsariA. GholamrezayiM. H. GorginS. andRahmatiD. Using residue number systems to accelerate deterministic bit-stream multiplication 2019 IEEE 30th International Conference on Application-specific Systems Architectures and Processors (ASAP) 2019 New York NY USA https:\/\/doi.org\/10.1109\/ASAP.2019.00\u201033 2-s2.0-85072612141.","DOI":"10.1109\/ASAP.2019.00-33"},{"key":"e_1_2_11_30_2","unstructured":"TanW. ChiuS.-W. WangA. LaoY. andParhiK. K. PaReNTT: low-latency parallel residue number system and NTT-based long polynomial modular multiplication for homomorphic encryption 2023 https:\/\/arxiv.org\/abs\/2303.02237."},{"key":"e_1_2_11_31_2","unstructured":"LemaireE. Pretty modular symmetric encryption (PMSE) compact algorithm for embedded cryptography with quite low computational cost 2019 https:\/\/arxiv.org\/abs\/1905.08150."},{"key":"e_1_2_11_32_2","doi-asserted-by":"crossref","unstructured":"MiglioreV. RealM. M. LapotreV. TisserandA. FontaineC. andGogniatG. Fast polynomial arithmetic for somewhat homomorphic encryption operations in hardware with Karatsuba algorithm 2016 International Conference on Field-Programmable Technology (FPT) 2016 Xi\u2032an China 209\u2013212 https:\/\/doi.org\/10.1109\/FPT.2016.7929535 2-s2.0-85021456210.","DOI":"10.1109\/FPT.2016.7929535"},{"key":"e_1_2_11_33_2","doi-asserted-by":"crossref","unstructured":"Jayet-GriffonC. CornelieM.-A. MaistriP. Elbaz-VincentP. andLeveugleR. Polynomial multipliers for fully homomorphic encryption on FPGA 2015 International Conference on ReConFigurable Computing and FPGAs (ReConFig) 2015 Riviera Maya Mexico https:\/\/doi.org\/10.1109\/ReConFig.2015.7393335 2-s2.0-84964330931.","DOI":"10.1109\/ReConFig.2015.7393335"},{"key":"e_1_2_11_34_2","doi-asserted-by":"crossref","unstructured":"ShivdikarK. JonatanG. MoraE. LivesayN. AgrawalR. JoshiA. Abell\u00e1nJ. L. KimJ. andKaeliD. Accelerating polynomial multiplication for homomorphic encryption on GPUs 2022 IEEE International Symposium on Secure and Private Execution Environment Design (SEED) 2022 Storrs CT USA 61\u201372 https:\/\/doi.org\/10.1109\/SEED55351.2022.00013.","DOI":"10.1109\/SEED55351.2022.00013"},{"key":"e_1_2_11_35_2","doi-asserted-by":"publisher","DOI":"10.37190\/oa200303"},{"key":"e_1_2_11_36_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2013.2283674"},{"key":"e_1_2_11_37_2","doi-asserted-by":"publisher","DOI":"10.15587\/1729-4061.2022.251913"},{"key":"e_1_2_11_38_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071\u2010018\u20104310\u20102"},{"key":"e_1_2_11_39_2","doi-asserted-by":"publisher","DOI":"10.15294\/sji.v7i1.24006"},{"key":"e_1_2_11_40_2","doi-asserted-by":"publisher","DOI":"10.1177\/1687814018781638"},{"key":"e_1_2_11_41_2","first-page":"2696","article-title":"Avalanche analysis of variant polynomials for AES","volume":"12","author":"Alshammari K. F.","year":"2021","journal-title":"Turkish Journal of Computer and Mathematics Education (TURCOMAT)"},{"key":"e_1_2_11_42_2","doi-asserted-by":"crossref","unstructured":"ChuJ.andBenaissaM. A novel architecture of implementing error detecting AES using PRNS 2011 14th Euromicro Conference on Digital System Design 2011 Oulu Finland https:\/\/doi.org\/10.1109\/DSD.2011.90 2-s2.0-80054976034.","DOI":"10.1109\/DSD.2011.90"},{"key":"e_1_2_11_43_2","unstructured":"El-KassarA. N. HaratyR. AwadY. andDebnathN. Modified RSA in the domains of Gaussian integers and polynomials over finite fields 2005 In CAINE."},{"key":"e_1_2_11_44_2","doi-asserted-by":"crossref","unstructured":"YakymenkoI. KasianchukM. ShylinskaI. ShevchukR. YatskivV. andKarpinskiM. Polynomial Rabin cryptosystem based on the operation of addition 2022 12th International Conference on Advanced Computer Information Technologies (ACIT) 2022 Ruzomberok Slovakia 345\u2013350 https:\/\/doi.org\/10.1109\/ACIT54803.2022.9913089.","DOI":"10.1109\/ACIT54803.2022.9913089"},{"key":"e_1_2_11_45_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304\u20103975(00)00425\u20104"},{"key":"e_1_2_11_46_2","unstructured":"DossoF. Y. BerzatiA. El MrabetN. andProyJ. PMNS revisited for consistent redundancy and equality test 2023 Cryptology ePrint Archive."},{"key":"e_1_2_11_47_2","doi-asserted-by":"crossref","unstructured":"DidierL.-S. DossoF.-Y. El MrabetN. MarrezJ. andV\u00e9ronP. Randomization of arithmetic over polynomial modular number system 2019 IEEE 26th Symposium on Computer Arithmetic (ARITH) 2019 Kyoto Japan 199\u2013206 https:\/\/doi.org\/10.1109\/ARITH.2019.00048.","DOI":"10.1109\/ARITH.2019.00048"},{"key":"e_1_2_11_48_2","doi-asserted-by":"crossref","unstructured":"KarpinskiM. RajbaS. ZawislakS. WarwasK. KasianchukM. IvasievS. andYakymenkoI. A method for decimal number recovery from its residues based on the addition of the product modules 2019 10th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS) 2019 Metz France 13\u201317 https:\/\/doi.org\/10.1109\/IDAACS.2019.8924395.","DOI":"10.1109\/IDAACS.2019.8924395"},{"key":"e_1_2_11_49_2","doi-asserted-by":"publisher","DOI":"10.11648\/j.se.20170505.12"},{"key":"e_1_2_11_50_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10559\u2010014\u20109654\u20100"},{"key":"e_1_2_11_51_2","first-page":"74","article-title":"Algorithm for the synthesis of irreducible polynomials of linear complexity","volume":"22","author":"Beletsky A.","year":"2020","journal-title":"Ukrainian Information Security Research Journal"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2024\/4894415.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2024\/4894415.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2024\/4894415","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T14:40:01Z","timestamp":1725892801000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2024\/4894415"}},"subtitle":[],"editor":[{"given":"Saeid","family":"Abbasbandy","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2024,1]]},"references-count":51,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["10.1155\/2024\/4894415"],"URL":"https:\/\/doi.org\/10.1155\/2024\/4894415","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1]]},"assertion":[{"value":"2023-10-21","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-04-24","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-05-20","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"4894415"}}