Let$N(d,d^perp)$denote the minimum length$n$of a linear code$C$with$d$and$d^bot$, where$d$is the minimum Hamming distance of$C$and$d^bot$is the minimum Hamming distance of$C^bot$. In this correspondence, we show lower bounds and an upper bound on$N(d,d^perp)$. Further, for small values of$d$and$d^perp$, we determine$N(d,d^perp)$and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa

An explicit study of min-wise independent permutation families, together with their variants --- k-restricted, approximate, etc. --- was initiated by Broder, et al[4]. In this paper, we give a lower bound for the size of k-restricted min-wise independent permutation family. A family F of permutations on [0,n-1]=(0,1,...,n-1) is said to be k-restricted min-wise independent if for any subset X ⊆ [0,n-1] with |X| ≤ k and any x ∈ X, Pr[min(π(X))=π(x)] = 1/|X|, when π is randomly chosen from F according to a probability distribution D on the family F. For the minimum size of a family of k-restricted min-wise independent permutations, upper bounds of O(nk) for any fixed k have been shown for uniform and biased probability distributions on F. We show that if a family F of permutations on [0,n-1] is k-restricted min-wise independent, then |F| ≥ m(n-1,k-1), where m(n,d) = ∑i=0d/2(ni) if d is even; m(n,d)= ∑i=0(d-1)/2(ni) + (n-1(d-1)/2) otherwise. The lower bound for the size of F still holds when we allow an arbitrary probability distribution on F. Our proof technique is based on linear algebra methods, and can be regarded as a generalization of the result by Alon, Babai, and Itai[1], i.e., if random variables X1,X2,...,Xn: Ω → (0,1) are k-wise independent and Pr[Xi=1] = pi is neither 0 nor 1, then |Ω| ≥ m(n,k). By applying our proof technique, we also derive lower bounds for the sample size of the related notions, e.g., k-wise symmetrically independent distributions, k-rankwise independent permutation families, etc.

In a modern network system, data security technologies such as cryptosystems, signature schemes, etc., are indispensable for reliable data transmission. In particular, for a large-scale network, ID-based systems such as the ID-based cryptosystem, the ID-based signature scheme, or the ID-based key distribution system are among the better countermeasures for establishing efficient and secure data transmission systems. The concept of an ID-based cryptosystem has been proposed by A. S?hamir (1985), and it is advantageous to public-key cryptosystems because a large public-key file is not required for such a system. An ID-based cryptosystem based on the discrete logarithm problem is proposed which is one of the earliest realizations in Shamir's sense. The security against a conspiracy of some entities in the proposed system is considered, along with the possibility of establishing a more secure system.>