组合全有或全无变换的信息论安全性
On the Information-theoretic Security of Combinatorial All-or-nothing Transforms
摘要：
AONT（All-or-Nothing transforms）是Rivest提出的一种消息预处理技术，用于加密数据以防止蛮干攻击，在密码学和信息安全领域有着广泛的应用。后来Stinson引入了无条件安全AONT及其组合特征。非正式地，组合AONT是具有无偏要求的数组，并且其安全属性通常取决于输入$s$元组上的先验概率分布。最近，Esfahani和Stinson证明了组合AONT在所有输入$s$ -元组都是等概率的情况下具有完美安全性，并且在所有输入$s$ -元组都具有非零概率的情况下具有弱安全性。本文旨在探讨组合$(t,s,v)$ -AONT的完美安全性和弱安全性之间的差距。具体地，我们考虑所有$s$输入独立地取值（但不一定相同）的典型场景，并量化关于任何$t$输入$\mathcal{X}$的信息量$H(\mathcal{X}|\mathcal{Y})$，该信息量未被任何$s-t$输出$\mathcal{Y}$揭示。特别是，我们建立了一般的下限和上限$H(\mathcal{X}|\mathcal{Y})$组合AONT使用信息理论的技术，也表明，派生的界限可以达到在某些情况下。此外，讨论了组合非对称AONT的安全性质。
英文摘要：
All-or-nothing transforms (AONT) 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 AONT 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 $H(\mathcal{X}|\mathcal{Y})$ about any $t$ inputs $\mathcal{X}$ that is not revealed by any $s-t$ outputs $\mathcal{Y}$. In particular, we establish the general lower and upper bounds on $H(\mathcal{X}|\mathcal{Y})$ 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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