Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques
In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruence mod p to polynomial congruence mod pk. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the form axn≡b(mod pk), k≥1, where a,b, and n>0 are integ...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/575064 |
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| author | M. Khalid Mahmood Farooq Ahmad |
| author_facet | M. Khalid Mahmood Farooq Ahmad |
| author_sort | M. Khalid Mahmood |
| collection | DOAJ |
| description | In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruence mod p to polynomial congruence mod pk. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the form axn≡b(mod pk), k≥1, where a,b, and n>0 are integers which are not divisible by an odd prime p. It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the form ax-n≡b(mod pk) has also been addressed. |
| format | Article |
| id | doaj-art-4bb2f59b48d643bd8be50d10e1a49146 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-4bb2f59b48d643bd8be50d10e1a491462025-02-03T01:24:31ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/575064575064Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical TechniquesM. Khalid Mahmood0Farooq Ahmad1Department of Mathematics, University of the Punjab, Lahore 54590, PakistanFaculty of Information Technology, University of Central Punjab, Lahore 54500, PakistanIn this paper we put forward a family of algorithms for lifting solutions of a polynomial congruence mod p to polynomial congruence mod pk. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the form axn≡b(mod pk), k≥1, where a,b, and n>0 are integers which are not divisible by an odd prime p. It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the form ax-n≡b(mod pk) has also been addressed.http://dx.doi.org/10.1155/2014/575064 |
| spellingShingle | M. Khalid Mahmood Farooq Ahmad Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques Abstract and Applied Analysis |
| title | Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques |
| title_full | Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques |
| title_fullStr | Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques |
| title_full_unstemmed | Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques |
| title_short | Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques |
| title_sort | recursive elucidation of polynomial congruences using root finding numerical techniques |
| url | http://dx.doi.org/10.1155/2014/575064 |
| work_keys_str_mv | AT mkhalidmahmood recursiveelucidationofpolynomialcongruencesusingrootfindingnumericaltechniques AT farooqahmad recursiveelucidationofpolynomialcongruencesusingrootfindingnumericaltechniques |