All-or-nothing transforms (AONTs) were proposed by Rivest as a message preprocessing technique for encrypting data to protect against brute-force attacks, and have numerous applications in cryptography and information security. Later the unconditionally secure AONTs and their combinatorial characterization were introduced by Stinson. Informally, a combinatorial AONT is an array with the unbiased requirements and its security properties in general depend on the prior probability distribution on the inputs s-tuples. Recently, it was shown by Esfahani and Stinson that a combinatorial AONT has perfect security provided that all the inputs s-tuples are equiprobable, and has weak security provided that all the inputs s-tuples are with non-zero probability. This paper aims to explore on the gap between perfect security and weak security for combinatorial (t, s, v)-AONTs. Concretely, we consider the typical scenario that all the s inputs take values independently (but not necessarily identically) and quantify the amount of information H(X|Y) about any t inputs X that is not revealed by any s – t outputs Y. In particular, we establish the general lower and upper bounds on H(X|Y) for combinatorial AONTs using information-theoretic techniques, and also show that the derived bounds can be attained in certain cases. Furthermore, the discussions are extended for the security properties of combinatorial asymmetric AONTs.
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