{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T12:54:11Z","timestamp":1765976051640,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2014,9,25]],"date-time":"2014-09-25T00:00:00Z","timestamp":1411603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area means finding the majority of points on the elliptic curve and minimizing the time to compute the solution in contrast with the necessary time to compute the inverse solution. We can compute the positive solution of PDE (partial differential equation) like oscillations of f(s)\/s around the principal eigenvalue \u03bb1 of -\u0394 in H 0 1 (\u03a9).Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption\/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves.<\/jats:p>","DOI":"10.3390\/e16095144","type":"journal-article","created":{"date-parts":[[2014,9,25]],"date-time":"2014-09-25T12:17:36Z","timestamp":1411647456000},"page":"5144-5158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Nonlinearities in Elliptic Curve Authentication"],"prefix":"10.3390","volume":"16","author":[{"given":"Ramzi","family":"Alsaedi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah Campus, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"given":"Nicolae","family":"Constantinescu","sequence":"additional","affiliation":[{"name":"Department of Informatics, University of Craiova, Street A.I Cuza 13, 200585 Craiova, Romania"}]},{"given":"Vicen\u0163iu","family":"R\u0101dulescu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah Campus, P.O. Box 80203, Jeddah 21589, Saudi Arabia"},{"name":"Institute of Mathematics \"Simion Stoilow\" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2014,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1145\/322510.322514","article-title":"Public-key cryptography and password protocols","volume":"2","author":"Halevi","year":"1999","journal-title":"ACM Trans. Inf. Syst. Secur"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bellovin, S.M., and Merritt, M. (1992, January 4\u20136). Encrypted key exchange: Password-based protocols secure against dictionary attacks. Oakland, CA, USA.","DOI":"10.1145\/168588.168618"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bellovin, S.M., and Merritt, M. (1993, January 3\u20135). Augmented encrypted key exchange: A password-based protocol secure against dictionary attacks and password file compromise. Fairfax, VA, USA.","DOI":"10.1145\/168588.168618"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1145\/242896.242897","article-title":"Strong password-only authenticated key exchange","volume":"26","author":"Jablon","year":"1996","journal-title":"ACM SIGCOMM Comput. Commun. Rev"},{"key":"ref_5","unstructured":"Wu, T. (1998, January 11\u201313). The secure remote password protocol. San Diego, CA, USA."},{"key":"ref_6","unstructured":"Mohr, A. (2007). A Survey of Zero-Knowledge Proofs with Applications to Cryptography, Southern Illinois University."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1090\/S0025-5718-1987-0866109-5","article-title":"Elliptic curve cryptosystems","volume":"48","author":"Koblitz","year":"1987","journal-title":"Math. Comput"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Williams, H.C. (1986). Advances in Cryptology\u2014CRYPTO \u201885, Springer.","DOI":"10.1007\/3-540-39799-X"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Roy, B., and Okamoto, E. (2000). Progress in Cryptology\u2014INDOCRYPT 2000, Springer.","DOI":"10.1007\/3-540-44495-5"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/PL00003816","article-title":"Parallel Collision Search with Cryptanalytic Applications","volume":"12","author":"Wiener","year":"1999","journal-title":"J. Cryptol"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s001459900052","article-title":"The Discrete Logarithm Problem on Elliptic Curves of Trace One","volume":"12","author":"Smart","year":"1999","journal-title":"J. Cryptol"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Menezes, A., Okamoto, T., and Vanstone, S. (1991, January 5\u20138). Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field. New Orleans, LA, USA.","DOI":"10.1145\/103418.103434"},{"key":"ref_13","unstructured":"Constantinescu, N. (2009). Criptografie, Romanian Academy."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1016\/j.jmaa.2005.11.006","article-title":"Positive solutions of elliptic problems with locally oscillating nonlinearities","volume":"323","author":"Obersnel","year":"2006","journal-title":"J. Math. Anal. Appl"},{"key":"ref_15","first-page":"83","article-title":"Some remarks on the solvability of the nonlinear two-point boundary value problems","volume":"10","author":"Njoku","year":"1991","journal-title":"J. Niger. Math. Soc"},{"key":"ref_16","first-page":"63","article-title":"On the solvability of a semilinear two-point BVP around the first eigenvalue","volume":"2","author":"Fernandes","year":"1989","journal-title":"Differ. Integr. Equ"},{"key":"ref_17","first-page":"483","article-title":"Elliptic curves over finite fields and the computation of square roots mod p","volume":"44","author":"Schoof","year":"1985","journal-title":"Math. Comp"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Cohen, H., and Frey, G. (2006). Handbook of Elliptic and Hyperelliptic Curve Cryptography, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420034981"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Cohen, H. (1993). A Course in Computational Algebraic Number Theory, Springer-Verlag.","DOI":"10.1007\/978-3-662-02945-9"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Rao, J.R., and Sunar, B. (2005). Cryptographic Hardware and Embedded Systems\u2014CHES 2005, Springer.","DOI":"10.1007\/11545262"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/BF02351717","article-title":"Zero knowledge proofs of identity","volume":"1","author":"Feige","year":"1987","journal-title":"J. Cryptol"},{"key":"ref_22","unstructured":"Menezes, A.J., van Oorschot, P.C., and Vanstone, S.A. (2001). Handbook of Applied Cryptography, Chapman and Hall\/CRC. [5th ed]."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1515\/gcc-2012-0009","article-title":"No-leak Authentication by the Sherlockk Holmes Method","volume":"4","author":"Grigoriev","year":"2012","journal-title":"Groups Complex. Cryptol"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1090\/S0025-5718-1985-0777282-X","article-title":"Multiplication Modular without Trial Division","volume":"44","author":"Montgomery","year":"1985","journal-title":"Math. Comput"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s001459900045","article-title":"Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two","volume":"11","author":"Muller","year":"1998","journal-title":"J. Cryptol"},{"key":"ref_26","unstructured":"Naccache, D., and Paar, C. (2001). Cryptographic Hardware and Embedded Systems\u2014CHES \u201801, Springer."},{"key":"ref_27","unstructured":"Ciet, M. (2003). Aspects of Fast and Secure Arithmetics for Elliptic Curve Cryptography. [Ph.D. Thesis, Universite Catholique de Louvain]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/9\/5144\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:16:19Z","timestamp":1760217379000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/9\/5144"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,25]]},"references-count":27,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2014,9]]}},"alternative-id":["e16095144"],"URL":"https:\/\/doi.org\/10.3390\/e16095144","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,9,25]]}}}