# A Novel Hybrid Elementary Cellular Automata and Its Application in a Stream Cipher
## 4 Stream Cipher Based on HECA-M
### Description of Proposed Stream Cipher
We calculated the correlation coefficient between the secret key _SK_ and the corresponding keystream _KS_. The correlation coefficient between the secret key SK and the truly random sequence (TRS) is also acquired for comparison analysis. The above correlation coefficients are depicted in Figure 11 (512 bits) and Figure 12 (256 bits). The \(X\)-axis and \(Y\)-axis denote the lags and the correlation coefficient, respectively, in Figures 11 and 12. These two figures show that the correlation coefficient between the secret key _SK_ and the corresponding keystream _KS_ is similar to the correlation coefficient between _SK_ and the true random sequence _TRS_. In order to prove the similarity between the above two correlation coefficients further, the \(t\)-test is utilized to determine whether these two coefficients possess the same statistical characteristics. The significance level \(\alpha\) is set to 0.05, and the mean is zero. The test is operated five times with different _SK_s. The results of the \(t\)-test are listed in Table 9.

Figure 11: The correlation coefficients between (**a**) the secret key and corresponding keystream and (**b**) the secret key and true random sequence. The length of the secret key is 512 bits. Figure 12: The correlation coefficients between (**a**) the secret key and corresponding keystream and (**b**) the secret key and true random sequence. The length of the secret key is 256 bits. Figure 10: The correlation coefficients of the keystreams generated by the proposed stream cipher under the secret keys with 256 bits. (**a**) The cross-correlation coefficient between _KS_ and _KS_. (**b**) The cross-correlation coefficient between _KS_ and _\(\sim\)KS_. (**c**) The autocorrelation coefficient of _KS_. As shown in Table 9, \(p\)-values are always above 0.9. Hence, we accept the null hypothesis that the correlation coefficient between \(\mathcal{S}\)K and \(\mathcal{K}\)S is indistinguishable from \(\mathcal{S}\)K and \(\mathcal{T}\)RS no matter how long the secret key \(\mathcal{S}\)K is. It means that the correlation analysis is useless for the proposed stream cipher. 6. Time-memory-data tradeoff

Time-memory-data tradeoff attack is a novel cryptanalysis method. The core of this method is increasing space consumption to reduce time consumption for cryptanalysis. In this attack, a lookup table (LUT), including the state transition of the keystream generator, is pre-generated. The enemy can search the LUT to find the initial state of the key generator according to the current state. In the proposed stream cipher, the keystream is generated by the driving part of the HECA-M. We assign the current state of the driving part as \(\mathcal{S}_{n}\) acquired by the keystream. It should be noted that only half of \(\mathcal{S}_{n}\) can be obtained directly because the keystream is generated by the even or odd cells in the driving part. Another half of \(\mathcal{S}_{n}\) has to be speculated. Thus, there are \(2^{128}\) possible current states. Moreover, the state transition of the driving part is determined by the mask part with 256 cells. It means that there are \(2^{256}\) different initial values in the mask part. Furthermore, there are 27 rule groups in the LUT (Table 5). As for \(2^{256}\) initial values in the driving part, the total number of state transitions can reach \(2^{256\times 2}\times 27\). It is an unacceptable quantity for the enemy. In addition, the probability of finding possible current states in these state transitions is \(1/(2^{384}\times 27)\), below \(10^{-117}\). Meanwhile, the initial iteration \(n_{0}\) is determined by the secret key unknown to the enemy. Hence, the enemy has to find which one is the real initial value in these initial iterations. It increases the difficulty of the time-memory-data tradeoff attack further. In summary, the proposed stream cipher can resist the time-memory-data tradeoff attack. 7. Guess-and-determine attack

The guess-and-determine attack (GDA) is proposed to analyze the stream cipher RC4 by [PERSON] in ASIACRYPT 1998.