An artificial neural network approach to finding the key length of the Vigenère cipher

Figures & data

Table 1. Model features and accuracy rates.

Model	Input features	Number of Features	NN Accuracy on unseen test data
Model 1	All features	114	88.8%
Model 2	All features except 12 and 13	112	88%
Model 3	All features except 14	113	88.2%
Model 4	All features except 2 and 14	112	87.6%
Model 5	All features except 14, 3 and 4	111	87.4%
Model 6	All features except 14, 5	112	87.7%
Model 7	All features except 14, 5, 10 and 11	102	88.3%
Model 8	All features except 14, 5, 10, 11 and 9	79	88.7%
Model 9	All features except 14, 5, 9, 10, 11 and 6	54	88.9%
Model 10	All features except 14, 5, 9, 10, 11, 6 and 7	30	87.6%
Model 11	All features except 14, 5, 9, 10, 11, 6 and 8	30	87.2%

Figure 1. Example of a feedforward neural network.

Figure 2. Accuracy rates by key length finding method.

Figure 3. Accuracy rate by key length for $200\le N\le 299.$

Table 2. Features considered.

Length of ciphertext Has repeated sequences Index of coincidence of the ciphertext Index of coincidence of English (constant, which is 0.066) The quotient q from $N=12q+r$ where N is the length of the ciphertext. The twist indices: $T(\mathcal{M},m)$ for $1\le m\le 25$ The twist ${}^{+}$ indices: $T^{+}(\mathcal{M},m)$ for $2\le i\le 25$ The twist ${}^{++}$ indices: $T^{++}(\mathcal{M},m)$ for $2\le i\le 25$ The average index of coincidence for cosets m where $3\le m\le 25$ The 5 most common distances that occurred between repeated sequences of length 3 or 4. The number of times each of the top 5 most common distances appeared. H $\Delta$ First order entropy

Table 3. Feature importance.

Model amended	Feature removed	Effect on accuracy rate after removal
Model 1	H and $\Delta$	−0.8%
Model 2	First order entropy	+0.2%
Model 3	Has repeated sequences	−0.6%
Model 4	Index of coincidence of the ciphertext and English	−0.2%
Model 5	The quotient q from $N=12q+r$ where N is the length of the ciphertext.	+0.3%
Model 6	10 & 11 (Babbage-Kasiski features)	+0.6%
Model 7	The average index of coincidence for cosets m where $3\le m\le 25$	+0.4%
Model 8	Twist indices: $T(\mathcal{M},m)$ for $1\le m\le 25$	+0.2%
Model 9	Twist ${}^{+}$ indices: $T^{+}(\mathcal{M},m)$ for $2\le i\le 25$	−1.3%
Model 10	Twist ${}^{++}$ indices: $T^{++}(\mathcal{M},m)$ for $2\le i\le 25$	−0.4%

Table 4. Final model features.

Length of ciphertext Has repeated sequences Index of coincidence of the ciphertext Index of coincidence of English (constant, which is 0.066) The twist ${}^{+}$ indices: $T^{+}(\mathcal{M},m)$ for $2\le i\le 25$ The twist ${}^{++}$ indices: $T^{++}(\mathcal{M},m)$ for $2\le i\le 25$ The average index of coincidence for cosets m where $3\le m\le 25$ H $\Delta$

Table 5. Accuracy rates by key length finding method.

Text-Length/Method	IC	Twist Index	$T^{+}$	$T^{++}$	Neural network
200–500 (Overall accuracy)	7.4%	22.4%	66.2%	63.3%	89.2%
200–299	6.8%	16.1%	47.3%	42.3%	75.1%
300–399	7.6%	20.3%	71.7%	65%	94.6%
400–500	7.7%	30.6%	79.4%	82.2%	97.9%

Figure 4. Accuracy rate by key length for $300\le N\le 399.$

Figure 5. Accuracy rate by key length for $400\le N\le 499.$

Data availability statement

The data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.10363051.