In 2006, Tanaka has proposed an efficient variant of Maurer-Yacobi's identity-based non-interactive key sharing scheme. In Tanaka's scheme, the computational complexity to generate each user's secret information is much smaller than that of Maurer-Yacobi's scheme. Tanaka's original key sharing scheme does not provide completeness, and so Tanaka has corrected the original scheme to provide completeness. In this paper, we show that Tanaka's corrected key sharing scheme is not secure against collusion attacks. That is, two users can collaborate to factorize a system modulus with their secret information and thus break the key sharing scheme.

Hwang-Lo-Lin proposed a user identification scheme [3] based on the Maurer-Yacobi scheme [6] that is suitable for application to the mobile environment. Hwang-Lo-Lin argued that their scheme is secure against any attack. Against the Hwang-Lo-Lin argument, Liu-Horng-Liu showed that the Hwang-Lo-Lin scheme is insecure against a Liu-Horng-Liu attack mounted by an eavesdrop attacker. However, Liu-Horng-Liu did not propose any improved version of the original identification scheme which is still secure against the Liu-Horng-Liu attack. In this paper, we propose an identification scheme that can solve this problem and a non-interactive public key distribution scheme also.

A new simply implemented collusion-attack free identity-based non-interactive key sharing scheme (ID-NIKS) has been proposed. A common-key can be shared by executing only once a modular exponentiation which is equivalent to RSA deciphering, and the security depends on the difficulty of factoring and the discrete logarithm problem. Each user's secret information can be generated by solving two simple discrete logarithm problems and synthsizing their solutions by linear combination. The detail comparison with the Maurer-Yacobi's scheme including its modified versions shows that the computational complexity to generate each user's secret information is much smaller and the freedom to select system parameters is much greater than that of the Maurer-Yacobi's scheme. Then our proposed scheme can be implemented very easily and hence it is suitable for practical use.

Lee et al. recently proposed the first identity-based key agreement protocols for a multiple PKG environment where each PKG has different domain parameters in ICCSA 2005. However, this letter demonstrates that Lee et al.'s scheme does not include the property of implicit key authentication which is the fundamental security requirement, making it vulnerable to an impersonation attack.

In this paper an identity-based non-interactive key sharing scheme (IDNIKS) is proposed in order to realize the original concept of identity-based cryptosystem, of which secure realization scheme has not been proposed. First the necessary conditions for secure realization of IDNIKS are considered from two different poinrts of view: (i) the possibility to share a common-key non-interactively and (ii) the security for entity's conspiracy. Then a new non-interactive key sharing scheme is proposed, of which security depends on the difficulty of factoring. The most important contribution is to have succeeded in obtaining any entity's secret information as an exponent of the obtainer's identity information. The security of IDNIKS for entity's conspiracy is also considered in details.