A quantum algorithm for testing and learning resiliency of a Boolean function

Abstract

A quantum algorithm to evaluate the resiliency of a Boolean function is explored. Recently, Chakraborty and Maitra (Cryptogr Commun 8(3):401–413, 2016) have provided quantum algorithms to check the non-resiliency of a Boolean function. However, the shortage of their algorithms is that they just output YES or NO. Refining one of the algorithms, a quantum algorithm is proposed here, which can describe the extent of the non-resiliency by $\epsilon$-almost resiliency under the condition NO.

Appendices

Appendix A

Proof of Lemma 1

Let $x'\in \{x_1,\ldots , x_n\}-\{x_{i_1},\ldots , x_{i_t}\}$, then

$$\begin{aligned} S_f(w_{I})= & {} \frac{1}{2^n}\sum _{x\in F^n_2}(-1)^{f(x)+w\cdot x}\nonumber \\= & {} \frac{1}{2^n}\sum _{x_{i_1}\ldots x_{i_t}}\sum _{x'}(-1)^{f(x)+w\cdot x}\nonumber \\= & {} \frac{1}{2^n}\sum _{x_{i_1}\ldots x_{i_t}}\sum _{x'}(-1)^{f(x)+ w_{i_1}x_{i_1}+\cdot +w_{i_t}x_{i_t}}\\= & {} \frac{1}{2^n}\sum _{x_{i_1}\ldots x_{i_t}}(-1)^ {w_{i_1}x_{i_1}+\cdots +w_{i_t}x_{i_t}} \sum _{x'}(-1)^{f(x)}.\nonumber \end{aligned}$$

(23)

Taking $x_{i_1}\ldots x_{i_t}=\alpha _{i_1}\ldots \alpha _{i_t}$, we have

$$\begin{aligned} \begin{aligned} \sum _{x'}(-1)^{f(x)}&=|\{f(x)=0\,|\,x_{i_1}\ldots x_{i_t}=\alpha \}|- |\{f(x)=1\,|\,x_{i_1}\ldots x_{i_t}=\alpha \}|\\&=(P_{I,\alpha }^{0}-P_{I,\alpha }^{1})\cdot 2^{n-t}. \end{aligned} \end{aligned}$$

(24)

And hence,

$$\begin{aligned} S_f(w_{I})=\frac{1}{2^n}\sum _{\alpha \in F_2^t}(-1)^ {w_{i_1}\alpha _{i_1}+\cdots +w_{i_t}\alpha _{i_t}}\cdot (P_{I,\alpha }^{0}-P_{I,\alpha }^{1})\cdot 2^{n-t}. \end{aligned}$$

(25)

Substitute (9) into (25), then we have

$$\begin{aligned} \begin{aligned} S_f(w_{I})&=\frac{1}{2^t}\sum _{\alpha \in F_2^t}(-1)^ {w_{i_1}\alpha _{i_1}+\cdots +w_{i_t}\alpha _{i_t}}\cdot 2\epsilon _{I,\alpha }\\&=\frac{1}{2^t}\sum _{\alpha \in F_2^t}(-1)^ {\eta \cdot \alpha }\cdot 2\epsilon _{I,\alpha }. \end{aligned} \end{aligned}$$

$\square$

Appendix B

Proof of Theorem 1

$$\begin{aligned} \begin{aligned} \sum _{{w}\in W_{I}}[S_f(w)]^2 =&\sum _{\eta \in F_2^t}\frac{1}{2^{2t}} \sum _{\alpha \in F_2^t}(-1)^{\eta \cdot \alpha }\cdot 2\epsilon _{I,\alpha } \sum _{\beta \in F_2^t}(-1)^{\eta \cdot \beta }\cdot 2\epsilon _{I,\beta }\\ =&\sum _{\alpha \in F_2^t}\sum _{\beta \in F_2^t}\frac{1}{2^{2t}} \sum _{\eta \in F_2^t}(-1)^{\eta \cdot (\alpha +\beta )} \cdot 2\epsilon _{I,\alpha }\cdot 2\epsilon _{I,\beta }. \end{aligned} \end{aligned}$$

(26)

In addition, we have

$$\begin{aligned} \sum _{\eta \in F_2^t}(-1)^{\eta \cdot (\alpha +\beta )} \cdot 2\epsilon _{I,\alpha }\cdot 2\epsilon _{I,\beta }= {\left\{ \begin{array}{ll} 0 &{} \alpha \ne \beta ,\\ {2^{t}}(2\epsilon _{I,\alpha })^2 &{} \alpha =\beta . \end{array}\right. } \end{aligned}$$

(27)

Combined (26) with (27), we obtain

$$\begin{aligned} \sum _{{w}\in W_{I}}[S_f(w)]^2 =\sum _{\alpha \in F_2^t}{2^{2-t}}(\epsilon _{I,\alpha })^2. \end{aligned}$$

(28)

$\square$