Let K be a finite field of characteristic ≠ 2 and G the additive group of K × K. Let k_1, k_2 be integers not divisible by the characteristic p of K with（k_1, k_2） = 1. In 2004, Haddad and Helou constructed an...Let K be a finite field of characteristic ≠ 2 and G the additive group of K × K. Let k_1, k_2 be integers not divisible by the characteristic p of K with（k_1, k_2） = 1. In 2004, Haddad and Helou constructed an additive basis B of G for which the number of representations of g ∈ G as a sum b_1＋ b_2（b_1, b_2 ∈ B） is bounded by 18. For g ∈ G and B■G, let σk_1,k_2（B, g）be the number of solutions of g = k_1b_1 ＋ k_2b_2, where b_1, b_2 ∈ B. In this paper, we show that there exists a set B ？ G such that k_1 B ＋ k2 B = G and σk_1,k_2（B, g）≤16.展开更多
It is well known that almost all subset sum problems with density less than 0.9408… can be solved in polynomial time with an SVP oracle that can find a shortest vector in a special lattice.In this paper,the authors s...It is well known that almost all subset sum problems with density less than 0.9408… can be solved in polynomial time with an SVP oracle that can find a shortest vector in a special lattice.In this paper,the authors show that a similar result holds for the k-multiple subset sum problem which has k subset sum problems with exactly the same solution.Specially,for the single subset sum problem(k=1),a modified lattice is introduced to make the proposed analysis much simpler and the bound for the success probability tighter than before.Moreover,some extended versions of the multiple subset sum problem are also considered.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11471017)
文摘Let K be a finite field of characteristic ≠ 2 and G the additive group of K × K. Let k_1, k_2 be integers not divisible by the characteristic p of K with（k_1, k_2） = 1. In 2004, Haddad and Helou constructed an additive basis B of G for which the number of representations of g ∈ G as a sum b_1＋ b_2（b_1, b_2 ∈ B） is bounded by 18. For g ∈ G and B■G, let σk_1,k_2（B, g）be the number of solutions of g = k_1b_1 ＋ k_2b_2, where b_1, b_2 ∈ B. In this paper, we show that there exists a set B ？ G such that k_1 B ＋ k2 B = G and σk_1,k_2（B, g）≤16.
基金supported by the National Natural Science Foundation of China under Grant Nos.11201458,11471314in part by 973 Project under Grant No.2011CB302401in part by the National Center for Mathematics and Interdisciplinary Sciences,Chinese Academy of Sciences
文摘It is well known that almost all subset sum problems with density less than 0.9408… can be solved in polynomial time with an SVP oracle that can find a shortest vector in a special lattice.In this paper,the authors show that a similar result holds for the k-multiple subset sum problem which has k subset sum problems with exactly the same solution.Specially,for the single subset sum problem(k=1),a modified lattice is introduced to make the proposed analysis much simpler and the bound for the success probability tighter than before.Moreover,some extended versions of the multiple subset sum problem are also considered.
