Wansu Bao

Xiangqun Fu, Wansu Bao, Jianhong Shi et al.

Considering the difficult problem under classical computing model can be solved by the quantum algorithm in polynomial time, t-multiple discrete logarithm problems presented. The problem is non-degeneracy and unique solution. We talk about what the parameter effects the problem solving difficulty. Then we pointed out that the index-calculus algorithm is not suitable for the problem, and two sufficient conditions of resistance to the quantum algorithm for the hidden subgroup problem are given.

Heliang Huang, Wansu Bao

Solving systems of m multivariate quadratic equations in n variables (MQ-problem) over finite fields is NP-hard. The security of many cryptographic systems is based on this problem. Up to now, the best algorithm for solving the underdefined MQ-problem is Hiroyuki Miura et al.'s algorithm, which is a polynomial-time algorithm when \[n \ge m(m + 3)/2\] and the characteristic of the field is even. In order to get a wider applicable range, we reduce the underdefined MQ-problem to the problem of finding square roots over finite field, and then combine with the guess and determine method. In this way, the applicable range is extended to \[n \ge m(m + 1)/2\], which is the widest range until now. Theory analysis indicates that the complexity of our algorithm is \[O(q{n^ω}m{(\log {\kern 1pt} {\kern 1pt} q)^2}){\kern 1pt} \] when characteristic of the field is even and \[O(q{2^m}{n^ω}m{(\log {\kern 1pt} {\kern 1pt} q)^2})\] when characteristic of the field is odd, where \[2 \le ω\le 3\] is the complexity of Gaussian elimination.

Wansu Bao, Heliang Huang

Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Grobner bases algorithm is a well-known method to solve this problem. However, a serious drawback exists in the Grobner bases based algebraic attacks, namely, any information won't be got if we couldn't work out the Grobner bases of the polynomial equations system. In this paper, firstly, a generalized model of Grobner basis algorithms is presented, which provides us a platform to analyze and solve common problems of the algorithms. Secondly, we give and prove the degree bound of the polynomials appeared during the computation of Grobner basis after field polynomials is added. Finally, by detecting the temporary basis during the computation of Grobner bases and then extracting the univariate polynomials contained unique solution in the temporary basis, a heuristic strategy named Middle-Solving is presented to solve these polynomials at each iteration of the algorithm. Farther, two specific application mode of Middle-Solving strategy for the incremental and non-incremental Grobner bases algorithms are presented respectively. By using the Middle-Solving strategy, even though we couldn't work out the final Grobner bases, some information of the variables still leak during the computational process.

Xiangqun Fu, Wansu Bao, Jianhong Shi et al.

In order to research the security of the knapsack problem under quantum algorithm attack, we study the quantum algorithm for knapsack problem over Z_r based on the relation between the dimension of the knapsack vector and r. First, the oracle function is designed based on the knapsack vector B and S, and the quantum algorithm for the knapsack problem over Z_r is presented. The observation probability of target state is not improved by designing unitary transform, but oracle function. Its complexity is polynomial. And its success probability depends on the relation between n and r. From the above discussion, we give the essential condition for the knapsack problem over Z_r against the existing quantum algorithm attacks, i.e. r<O(2^n). Then we analyze the security of the Chor-Rivest public-key crypto.

Heliang Huang, Wansu Bao

Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Faugere's F4 is a well-known Grobner bases algorithm to solve this problem. However, a serious drawback exists in the Grobner bases based algebraic attacks, namely, any information won't be got if we couldn't work out the Grobner bases of the polynomial equations system. In this paper, we in-depth research the F4 algorithm over GF(2). By using S-polynomials to replace critical pairs and computing the normal form of the productions with respect to the field equations in certain steps, many "redundant" reductors are avoided during the computation process of the F4 algorithm. By slightly modifying the logic of F4 algorithm, we solve the univariate polynomials appeared in the algorithm and then back-substitute the values of the solved variables at each iteration of the algorithm. We call our improvements Middle-Solving F4. The heuristic strategy of Middle-Solving overcomes the drawback of algebraic attacks and well suits algebraic attacks. It has never been applied to the Grobner bases algorithm before. Experiments to some Hidden Field Equation instances and some classical benchmarks (Cyclic 6, Gonnet83) show that Middle-Solving F4 is faster and uses less memory than Faugere's F4.