Security and efficiency analysis of a certificateless encryption scheme based on Chebyshev polynomial

In 2014, Algehawi and Samsuddin proposed a certificateless encryption scheme based on Chebyshev polynomials. The authors based their construction on the Al-Riyami and Paterson original design for certificateless encryption scheme but replaced bilinear pairings with the Chebyshev polynomial and claimed that their scheme is secure because Al-Riyami and Paterson´s scheme is secure. In this paper, we provide a cryptanalysis for their scheme and reveal some negative results. We present attacks of both Type-1 (modelling malicious users) and Type-2 (modelling passive but curious servers) against the scheme, and also comment on the (in)efficiency of the scheme.