S. M. Dehnavi, M. R. Mirzaee Shamsabad and A. Mahmoodi Rishakani
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 75—83
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Quadratic functions have applications in cryptography. In this paper, we investigate the modular quadratic equation ax2 + bx + c = 0 (mod 2n), and provide a complete analysis of it. More precisely, we determine when this equation has a solution and in the case that it has a solution, we give not only the number of solutions, but also the set of solutions, in O(n) time. One of the interesting results of our research is that, if this equation has a solution, then the number of solutions is a power of two. Most notably, as an application, we characterize the number of fixed-points of quadratic permutation polynomials over ℤ2n, which are used in symmetric cryptography.
- Quadratic equation mod 2n
- Number of solutions
- Set of solutions
- Number of fixedpoints
2010 Mathematics Subject Classification
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Cite this paperAPA
Dehnavi, S. M., Mirzaee Shamsabad, M. R., & Mahmoodi Rishakani, A. (2019). Complete solving the quadratic equation mod 2n. Notes on Number Theory and Discrete Mathematics, 25(1), 75-83, doi: 10.7546/nntdm.2019.25.1.75-83.Chicago
Dehnavi, S. M., M. R. Mirzaee Shamsabad and A. Mahmoodi Rishakani. “Complete Solving the Quadratic Equation mod 2n.” Notes on Number Theory and Discrete Mathematics 25, no. 1 (2019): 75-83, doi: 10.7546/nntdm.2019.25.1.75-83.MLA
Dehnavi, S. M., M. R. Mirzaee Shamsabad and A. Mahmoodi Rishakani. “Complete Solving the Quadratic Equation mod 2n.” Notes on Number Theory and Discrete Mathematics 25.1 (2019): 75-83. Print, doi: 10.7546/nntdm.2019.25.1.75-83.