Combinatorial Constructions For Authentication Codes

Simmons frst introduced the concept of authentication codes in early1980s asthe cryptographic primitive for providing authentication in unconditionally secure sys-tems. Splitting authentication codes were also frst introduced by Simmons in1982.These codes are useful, inter alia, for the analysis of authentication codes with ar-bitration. This dissertation investigates t-fold perfect splitting authentication codes,t-fold optimal splitting authentication codes and t-fold perfect authentication code witharbitration.In Chapter2, we discovered that the information-theoretic bounds of the suc-cessful probability of optimal spoofng attack of order r and the number of encodingrules which were given by Pei et al. in an authentication codes without splitting arealso true in a splitting authentication codes. And we proved that splitting authenti-cation codes that are multi-fold perfect against spoofng can be characterized in termsof restricted strong partially balanced t-designs. We also investigate the existence ofrestricted strong partially balanced3-designs RSPBD3-(v, b,3×2; λ1, λ2,1,0)s, andshow that there exists an RSPBD3-(v, b,3×2; λ1, λ2,1,0) for any v≡1(mod8),v≥9and v≡33(mod48). As its application, we obtain a new infnite class of3-foldperfect splitting authentication codes.Splitting t-designs were frst formulated by Huber in recent investigation of t-foldoptimal splitting authentication codes. In Chapter3, we investigate the constructionand existence of splitting t-(v, u×k,1) designs and, show that there exists a splitting3-(v,3×2,1) design if and only if v≡2(mod8) and v≥10. As its application, weobtain a new infnite class of3-fold optimal splitting authentication codes.In Chapter4, we defne a new design, perfect strict restricted strong partially bal-anced t-design, and prove that the existence of perfect strict restricted strong partiallybalanced t-designs implies the existence of t-fold perfect authentication codes with ar-bitration. Further, we obtain some new infnity classes of t-fold perfect authentication codes with arbitration. Then we solve the open problem presented by Pei [D. Pei,Message Authentication Codes, Press: USCT, Hefei,2009] who pointed out that therehas not yet been able to construct t-fold perfect Cartesian authentication codes witharbitration for t>2.