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Since the complete solution for the existence of magic 2-dimensional rectangles in 1881, much attention has been paid on the existence of magic l-dimensional rectangles for l≥3. The existence problem for magic l-dimensional rectangles with even sizes has been solved completely for all integers l≥3. However, very little is known for the existence of magic l-dimensional rectangles (l≥3) with odd sizes except for some families and a few sporadic examples. In this paper, we focus our attention on the existence of magic 3-dimensional rectangles and prove that the necessary conditions for the existence of magic 3-dimensional rectangles are also sufficient. Our construction method is mainly based on a new concept, symmetric zero-sum subset partition, which plays a crucial role in the recursive constructions of magic 3-rectangles similar to that of PBD in the PBD-closure construction in combinatorial design theory.

Let (v,u×c,λ)-splitting BIBD denote a (v,u×c,λ)-splitting balanced incomplete block design of order v with block size u×c and index λ. Necessary conditions for the existence of a (v,u×c,λ)-splitting BIBD are v≥uc, λ(v−1)≡0 (mod c(u−1)) and λv(v−1)≡0 (mod (c2u(u−1))). We show in this paper that the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with possible exceptions when (1) (v,λ)∈{(55,1),(39,9k):k=1,2,…}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a (v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes.

Recently, the self-certified public key system, due to its advantages, has been paid a great deal of attention. Over the past years, many self-certified digital signature schemes have been proposed. However, to the best of our current knowledge, self-certified one-time short signature has not been investigated till now. Therefore, in this paper, we would like to propose an efficient self-certified one-time short signature scheme from bilinear pairing techniques. Because of its one-timeness and unforgeability, the proposed scheme can be suitable for the electronic cheque scenario. Moreover, the size of the proposed signature scheme is as short as 160 bits since it is based on the BLS short scheme, which shows that the proposed scheme has a great advantage while the bandwidth of a communication channel is limited.

The concept of designated verifier signatures was first introduced by Chaum and by Jakobsson, Sako and Im-pagliazzo independently in the middle of the 1990's, and then generalized by Desmedt at Crypto'03 rump session. Since a designated verifier signature can be only verified by a unique and specific user, the designated verifier signature scheme has been deeply studied and widely used in many applications for electronic voting or electronic auction. Most recently, Kang, Boyd, and Dawson proposed a novel identity-based strong designated verifier signature scheme from bilinear pairing and used the provable security technique to prove the security of the scheme. However, in this paper, we show that Kang et al.'s scheme can't resist some existential forgery attack and therefore propose an improved version with a slight modification to avoid the attack.