Abstract
The notion of confusion coefficient (CC) is a property that attempts to characterize the confusion property of cryptographic algorithms against differential power analysis. In this article, we establish a relationship between CC and the transparency order (TO) for any Boolean function and deduce some relationships between the sumofsquares of CC, signaltonoise ratio, and TO. We also give a tight upper bound and a tight lower bound on the sumofsquares of CC for balanced splateaued functions. Finally, the results generalized a lower bound on the sumofsquares of CC of Boolean functions with the Hamming weight k.
1 Introduction
Sidechannel analysis (SCA) is a very powerful technique for block ciphers [1]. Differential power analysis (DPA) is one of the effective methods of SCA. To improve the resistance of a block cipher to DPA, the substitution boxes (
(1) Signaltonoise ratio (SNR) following [2] was proposed by Guilley at CARDIS conference in 2004. First, they built a complete model of information leakage based on the framework of traditional cryptographic analysis, so that the attacker could obtain the autocorrelation value of Hamming weight of the guessed key value.
(2) In 2005, transparency order (TO) was introduced for
(3) In 2012, confusion coefficient (CC) was presented when they studied the confusion property of cryptographic algorithms in the study by Fei et al. [5]. Based on the results of the study by Fei et al. [5], Picek et al. [6] calculated the nonlinearity of Sboxes of different sizes in 2014 and obtained the variance of CC. In the same year, Qiu et al. [7] revised the original CC and gave a new definition of CC in order to reduce the dimension and the number of CC.
The organization of this article is as follows. In Section 2, the basic concepts and notions are presented. In Section 3, we deduce the relationship between TO and CC. In Section 4, we derive the lower bound on the sum of squares of CC from TO and sum of squares of Boolean functions and give the relationships between CC, SNR and TO. We also investigate the upper bound and lower bound on the sumofsquares of CC for a splateaued function and discuss the lower bound on the sumofsquares of CC of Boolean function with the Hamming weight
2 Preliminaries
Let
For any function
where
The Hamming distance between two functions
Any
where
The nonlinearity of an
Let
If
The two indicators
Let
TO of
where
is the crosscorrelation between
This article only focus on the case when
The next definition gives the distribution of the Walsh spectra for a threevalued Boolean function.
Let
The SNR of
Let
where
Carlet et al. [8] studied the intrinsic resiliency of Sboxes against SCA and further gave the concrete form of CC for a Boolean function
where
3 Relationship between TO and CC
We first discuss the relationship between TO and CC.
Lemma 1
[9] Let
According to Lemmas 1 and 2, we obtain Theorem 1.
Corollary 1
Let
Proof
By Lemma 1, we have
and from Lemma 2, we have
According to Corollary 1, we can find that the smaller CC of a Boolean function is, the smaller the upper bound of TO is.
4 Some research results of sumofsquares of CC
4.1 Bounds on the sumofsquares of CC of one Boolean function
For the convenience, for a given
Lemma 3
[12] Let
Theorem 1
Let
Proof
We know the Walsh spectrum of
From the definition of TO
Based on Lemma 3,
Thus,
According to Theorem 1, we can find that the bigger the TO and the
4.2 Relationships between
K
f
(
k
*
)
, SNR, and TO
In this section, we give the relationships between the
Lemma 4
[12] Let
Theorem 2
Let
Proof
By Lemma 4,
Clearly,
Therefore,
Hence,
Based on Theorem 2, we know that the lower bound of sumofsquares of CC is directly proportional to TO and inversely proportional to SNR for a Boolean function; thus, these indicators cannot be the best at the same time.
4.3 Bounds on the sumofsquares of CC of splateaued function
Further, recall that
Lemma 6
[12] Let
Theorem 3
Let
Proof
By Lemma 4, we know that
According to the condition and Lemma 5, we know that
Based on Lemma 2, Lemma 6, and the condition, we have
Thus, this result is proved.□
Example 1
If
n  Lower bound on

Upper bound on


1  0.0625  0.0625 
3  0.1328  0.1563 
5  0.5020  0.5313 

Lower bound on

Upper bound on


2  0.125  0.125 
4  0.2656  0.3125 
6  1.0039  1.0625 
4.4 Bounds on the sumofsquares of CC of Boolean function with the Hamming weight
k
Finally, we discuss some properties of CC of Boolean function with the hamming weight
Lemma 7
[11] Let
Theorem 4
Let
Proof
By Lemma 7, we know that:
Example 2
We can deduce that

Lower bound on


3  0.1563 
4  0.2734 
5  0.5196 
6  1.0176 
7  2.0167 
5 Conclusion
In this article, we give the relationship between CC and TO. And we also give the relationships between sumofsquares of CC, TO, and SNR of Boolean function. Furthermore, we give the upper and lower bound on the sumofsquares of CC of splateaued function and the lower bound on sumofsquares of CC of Boolean function with the Hamming weight

Funding information: This study was supported by the Natural Science Foundation of Anhui Higher Education institutions of China (No. KJ2020ZD008) and Graduate Innovation Fund of Huaibei Normal University (No. yc2021022).

Conflict of interest: Authors state no conflict of interest.
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