Rongquan Feng

Lattice-based key exchange on small integer solution problem

In this paper, we propose a new hard problem, called bilateral inhomogeneous small integer solution (Bi-ISIS), which can be seen as an extension of the small integer solution problem on lattices. The main idea is that, instead of choosing a rectangle matrix, we choose a square matrix with small rank to generate Bi-ISIS problem without affecting the hardness of the underlying SIS problem. Based on this new problem, we present two new hardness problems: computational Bi-ISIS and decisional problems. As a direct application of these problems, we construct a new lattice-based key exchange (KE) protocol, which is analogous to the classic Diffie-Hellman KE protocol. We prove the security of this protocol and show that it provides better security in case of worst-case hardness of lattice problems, relatively efficient implementations, and great simplicity.

On the existence of some specific elements in finite fields of characteristic 2

Abstract Let q be a power of 2, n be a positive integer, and let F q n be the finite field with q n elements. In this paper, we consider the existence of some specific elements in F q n . The main results obtained in this paper are listed as follows: (1) There is an element ξ in F q n such that both ξ and ξ + ξ − 1 are primitive elements of F q n if q = 2 s , and n is an odd number no less than 13 and s > 4 . (2) For q = 2 s , and any odd n, there is an element ξ in F q n such that ξ is a primitive normal element and ξ + ξ − 1 is a primitive element of F q n if either n | ( q − 1 ) , and n ⩾ 33 , or n ∤ ( q − 1 ) , and n ⩾ 30 , s ⩾ 6 .

Twisted jacobi intersections curves

In this paper, the twisted Jacobi intersections which contains Jacobi intersections as a special case is introduced We show that every elliptic curve over the prime field with three points of order 2 is isomorphic to a twisted Jacobi intersections curve Some fast explicit formulae for twisted Jacobi intersections curves in projective coordinates are presented These explicit formulae for addition and doubling are almost as fast as the Jacobi intersections In addition, the scalar multiplication can be more effective in twisted Jacobi intersections than in Jacobi intersections Moreover, we propose new addition formulae which are independent of parameters of curves and more effective in reality than the previous formulae in the literature.

Graham’s pebbling conjecture on product of complete bipartite graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for any connected graphs G and H, f( G x H) ⩽ f( G) f( H). We show that Graham’s conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are complete bipartite graphs.

Error Correction for Cooperative Data Exchange

This paper considers the problem of error correction for a cooperative data exchange (CDE) system, where some clients are compromised or failed and send false messages. Assuming each client possesses a subset of the total messages, we analyze the error correction capability when every client is allowed to broadcast only one linearly-coded message. Our error correction capability bound determines the maximum number of clients that can be compromised or failed without jeopardizing the final decoding solution at each client. We show that deterministic, feasible linear codes exist that can achieve the derived bound. We also evaluate random linear codes, where the coding coefficients are drawn randomly, and then develop the probability for a client to withstand a certain number of compromised or failed peers and successfully deduce the complete message for any network size and any initial message distributions.

Isomorphism Classes of Concrete Graph Coverings

Hofmeister introduced the notion of a concrete (resp., concrete regular) covering of a graph G and gave formulas for enumerating the isomorphism classes of concrete (resp., concrete regular) coverings of G [Ars Combin., 32 (1991), pp. 121--127; SIAM J. Discrete Math., 8 (1995), pp. 51--61]. In this paper, we show that the number of the isomorphism classes of n-fold concrete (resp., concrete regular) coverings of G is equal to that of the isomorphism classes of n-fold (resp., regular) coverings of a new graph, the join

The Complexity of Network Coding With Two Unit-Rate Multicast Sessions

G+\infty

Solving the two simple multicast network coding problem in time O(|E|)

of G and an extra vertex

On the ranks of bent functions

\infty

Twisted Jacobi intersections curves

. As a consequence, we can enumerate the isomorphism classes of concrete (resp., concrete regular) coverings of a graph by using known formulas for enumerating the isomorphism classes of coverings (resp., regular coverings) of a graph.