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Quantum network coding with the assistance of auxiliary resources can achieve perfect transmission of the quantum state. This paper suggests a novel perfect network coding scheme to efficiently solve the quantum $k$-pair problem, in which only a few assisting resources are introduced. Specifically, only one pair of maximally entangled state needs to be pre-shared between two intermediate nodes, and only $O(k)$ of classical information is transmitted though the network. Moreover, the classical communication used in our protocol does not cause transmission congestion, providing better adaptability to large-scale quantum $k$-pair networks.Through relevant analyses and comparisons, we demonstrate that our proposed scheme saves resources and has good application value, thereby showing its high efficiency. Furthermore, the proposed scheme achieves 1-max flow quantum communication, and the achievable rate region result is extended from its counterpart over the butterfly network.

Our concern, in the paper, is to construct information-theoretic secure network coding in the presence of eavesdroppers. Based on the generalized attack model and all-or-nothing transformation, the security of network coding in the generalized combination network can be characterized by the network capacity and the min-cut bound of wiretapping set from the source. Furthermore, it can be extended to any directed acyclic networks with single source. Compared with the traditional results, we construct information-theoretic secure network coding without additional encryptions and giving up any capacity.

The development of quantum computation casts serious threats to the securities of most existing public-key cryptosystems. Braid-based cryptography is one of the alternatives that have potential advantages in resisting quantum attacks. In this paper, the state of the art of braid cryptography is surveyed, and then a new cryptographic problem—conjugate adjoining problem related to braid groups is proposed. Based on this problem, we design a new braid-based signature scheme. This scheme is efficient and provably secure in the random oracle model. Further, we present the comparison between braid-based signatures and RSA-based ones. The signing process of the braid-based schemes is more efficient than that of RSA-based ones, while the verifying process of the braid-based ones is observably slow. Hence, braid-based signatures are suitable for scenarios where the signing process has to be as quick as possible but delays are permitted in the verifying process, for example, in off-line e-cash systems. The key sizes in braid-based schemes are considerably large—about 2K bits in the case of secret keys and 12K bits in the case of public keys. However, braid operations are much simpler and more efficient than modular exponential operations. Therefore, braid-based schemes can be embedded into devices with low computational ability and large memory space. The capability of braid cryptosystems to resist currently known quantum attacks is also discussed from the perspective of hidden subgroup problems.