  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.
