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Full-Text Articles in Algebra
The Digraph Of The Square Mapping On Elliptic Curves, Katrina Glaeser
The Digraph Of The Square Mapping On Elliptic Curves, Katrina Glaeser
Mathematical Sciences Technical Reports (MSTR)
Consider a subgroup of an elliptic curve generated by a point P of order n. It is possible to match any point Q to an integer k (mod n) such that Q = kP using a brute force method. By observing patterns in the digraph of the squaring map on the integers modulo n it is possible to perform this matching. These techniques can be applied to solving the Elliptic Curve Discrete Log Problem given a complete graph of the square mapping k P -> k^2 P for the elliptic curve points.
Discrete Logarithm Over Composite Moduli, Marcus L. Mace
Discrete Logarithm Over Composite Moduli, Marcus L. Mace
Mathematical Sciences Technical Reports (MSTR)
In an age of digital information, security is of utmost importance. Many encryption schemes, such as the Diffie-Hellman Key Agreement and RSA Cryptosystem, use a function which maps x to y by a modular power map with generator g. The inverse of this function - trying to find x from y - is called the discrete logarithm problem. In most cases, n is a prime number. In some cases, however, n may be a composite number. In particular, we will look at when n = p^b for a prime p. We will show different techniques of obtaining graphs of this …
Structural Properties Of Power Digraphs Modulo N, Joseph Kramer-Miller
Structural Properties Of Power Digraphs Modulo N, Joseph Kramer-Miller
Mathematical Sciences Technical Reports (MSTR)
We define G(n, k) to be a directed graph whose set of vertices is {0, 1, ..., n−1} and whose set of edges is defined by a modular relation. We say that G(n, k) is symmetric of order m if we can partition G(n, k) into subgraphs, each containing m components, such that all the components in a subgraph are isomorphic. We develop necessary and sufficient conditions for G(n, k) to contain symmetry when n is odd and square-free. Additionally, we use group theory to describe the structural properties of the subgraph of G(n, k) containing only those vertices relatively …