计量逻辑学中的雪崩逻辑公式

Avalanche Logic Formulae in Quantitative Logic

将密码学中满足严格雪崩准则的布尔函数的概念引入到计量逻辑学之中,提出了雪崩逻辑公式的概念,并研究了雪崩逻辑公式的真度及其性质。证明了至少含有三个原子公式的雪崩逻辑公式的真度之集为H1={k/2n-12n-3≤k≤3×2n-3;n=3,4,…},在此基础上,通过引入函数ξ建立了n(n≥3)元雪崩布尔函数个数的表达式,给出了雪崩逻辑公式的构造方法。最后,研究了反射变换下k阶雪崩逻辑公式的性质。

The concept of Boolean functions satisfying strict avalanche criterion in cryptology is introduced into Quantitative Logic.The concept of avalanche logic formulae is proposed. The truth degree of avalanche logic formulae and their properties are studied.It is proved that the set of truth degree of avalanche logic formulae which contain at least three atom formulae is H1={k/2^n-1｜2^n-3≤k≤3×2^n-3;n=3,4,…}.Then,a formula for calculating the total number of avalanche Boolean functions of arity n（n≥3） is established by means of a newly introduced function ξ, and a method of construction of avalanche logic formulae is given.Finally, the properties of avalanche logic formulae of order k under reflexive transformation are studied.