Browsing by Author "Pal, Ankan"

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High order elements in finite fields arising from recursive towers
(Springer, 2022-04) Pal, Ankan
We illustrate a general technique to construct towers of fields producing high order elements in , for odd q, and in , for . These towers are obtained recursively by , for odd q, or , for , where v(x) is a polynomial of small degree over the prime field and belongs to the finite field extension , for an odd q, or to . Several examples are provided to show the numerical efficacy of our method. Using the techniques of Burkhart et al. (Des Codes Cryptogr 51(3):301–314, 2009) we prove similar lower bounds on the orders of the groups generated by , or by the discriminant of the polynomial. We also provide a general framework which can be used to produce many different examples, with the numerical performance of our best examples being slightly better than in the cases analyzed in Burkhart et al. (2009).
Moments of Gaussian hypergeometric functions over finite fields
(Project Euclid, 2023-09) Pal, Ankan
We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions n+1Fn, n≥1, over finite fields with q elements where q is an odd prime. This enables us to find an estimate for the value 6F5(1). In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value 3F2(−1). These formulas also allow us to express the product of certain 2F1 and n+1Fn functions in terms of finite field Appell series which generalizes current formulas for products of 2F1 functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.
A new method for geometric interpretation of elliptic curve discrete logarithm problem
(2019-09) Pal, Ankan
In this paper, we intend to study the geometric meaning of the discrete logarithm problem defined over an Elliptic Curve. The key idea is to reduce the Elliptic Curve Discrete Logarithm Problem (EC-DLP) into a system of equations. These equations arise from the interesection of quadric hypersurfaces in an affine space of lower dimension. In cryptography, this interpretation can be used to design attacks on EC-DLP. Presently, the best known attack algorithm having a sub-exponential time complexity is through the implementation of Summation Polynomials and Weil Descent. It is expected that the proposed geometric interpretation can result in faster reduction of the problem into a system of equations. These overdetermined system of equations are hard to solve. We have used F4 (Faugere) algorithms and got results for primes less than 500,000. Quantum Algorithms can expedite the process of solving these over-determined system of equations. In the absence of fast algorithms for computing summation polynomials, we expect that this could be an alternative. We do not claim that the proposed algorithm would be faster than Shor's algorithm for breaking EC-DLP but this interpretation could be a candidate as an alternative to the 'summation polynomial attack' in the post-quantum era.