This article is divided into two parts. The rst part describes the known candidates of trapdoor one-way permutations. The second part presents a new candidate trapdoor one-way permutation. This candidate is based on properties of multivariate polynomials on nite elds, and has similar characteristics to T. Matsumoto, H. Imai, and J. Patarin's schemes. What makes trapdoor one-way permutations particularly interesting is the fact that they immediately provide ciphering, signature, and authentication asymmetric schemes. Our candidate performs excellently in secret key, and secret key computations can be implemented in low-cost smart-cards, i.e. without co-processors. Key words : Trapdoor one-way permutations, multivariate polynomials, research of new asymmetric bijective schemes. Notes: This paper is the extended version of the paper with the same title published at ICICS'97. In this extended version, we have taken into account the recent results of . Part I Known candidates ..