After decades of evolution, the cellular system has become an indispensable part of modern life. Together with the convenience brought by the cellular system, many security issues have arisen. Message integrity protection is one of the urgent problems. The integrity of a message is usually protected by message authentication code (MAC). Forgery attacks are the primary threat to message integrity. By Simon's definition, forgery is twofold. The first is impersonation forgery, in which the opponent can forge a MAC without knowing any message-MAC pairs. The second is substitution forgery, in which the opponent can forge a MAC by knowing certain message-MAC pairs. In the 4G LTE system, MAC is applied not only to RRC control messages and user data, but also to authentication of the identities in the radio network during the authentication and key agreement (AKA) procedure. There is a set of functions used in AKA, which is called A3/A8. Originally, only one cipher suite called MILENAGE followed the definition of A3/A8. Recently, Vodafone has proposed another candidate called TUAK. This thesis first analyzes a MAC algorithm of the 4G LTE system called EIA1. The analysis shows that because of its linear structure, given two valid message-MAC pairs generated by EIA1, attackers can forge up to $2^{32}$ valid MACs by the algorithm called linear forgery attack proposed in this thesis. This thesis also proposes a well-designed scenario, in which attackers can apply the linear forgery attack to the real system. The second work presented in this thesis fixes the gap between the almost XOR universal property and the substitution forgery probability, and assesses the security of EIA1 under different attack models. After the security analysis, an optimized EIA1 using an efficient polynomial evaluation method is proposed. This polynomial evaluation method is analog to the fast Fourier transform. Compared with Horner's rule, which is used in the official implementation of EIA1, this method reduces the number of multiplications over ...