We consider the secure coded caching problem proposed by Ravindrakumaret. alwhere no user can obtain information about files other than the one requested. We first propose three new schemes for 1) the general case with arbitraryNfiles andKusers; 2) cache sizeM= 1,N= 2 files and arbitraryKusers; and 3) worst-case delivery rateR= 1, arbitraryNfiles andKusers, respectively. Then we derive some new converse results for 1) the general case with arbitraryNfiles andKusers; 2) cache sizeM= 1 with arbitraryNfiles andKusers; 3) worst-case delivery rateR= 1 with arbitraryNfiles andKusers; and 4) cache sizeM∈ [1,K/K-1] withN= 2 files and arbitraryKusers. As a result, we obtain 1) the two exact end-points for the optimal memory-rate tradeoff curve for arbitrary number of users and files; 2) a segment of the optimal memory-rate tradeoff curve, whereM∈ [1,K/K-1], for the case ofN= 2 files and arbitrary number of users; and 3) a multiplicative-gap-10 result, i.e., we show that the proposed achievable schemes achieve a ratio less than 10 with respect to the cut-set bound.