A chaos theory inspired, asynchronous two-way encryption mechanism for cloud computing

Lemma 1

The generation of the keys for encryption and decryption for an infinite timeline is not unique.

Proof

Assuming that the key set, K[], is collecting the individual key items, k_i, which is again an outcome from the time-dependent function. Thus, the primary relation, for a total of n number of observations, can be formulated as, (2)${\displaystyle K\left[\right]=\sum _{i=1}^n{k}_i}$

Further, as the generated keys are time-dependent thus, each key can be presented as, (3)${\displaystyle k_i\leftarrow \Phi \left(t\right)}$

The two different keys are generated as k1_i and k2_i on the time instances as t₁ and t₂. Thus this relation can be formulated as, (4)${\displaystyle k{1}_i\leftarrow \Phi \left(t_1\right)}$

And, (5)${\displaystyle k{2}_i\leftarrow \Phi \left(t_2\right)}$

These time instances are not the same, hence, (6)${\displaystyle t_1\ne t_2}$

Nevertheless, the random value generation function, which is time-dependant, is bound to have one upper limit λ, during the key generation, as mentioned below: (7)${\displaystyle {\int }_1^{\lambda }\Phi \to \Phi \left(t\right)}$

Eventually, the upper limit λ is bound to be reached for a longer time duration. For every t_n time instance, the upper limit is reached for this random key generation function, which again can be presented as, (8)${\displaystyle \Phi \left(t\right)=\Phi \left(t_n\right)}$

Thus, as the t₂ time instance reaches the t_n time instance, the uniqueness and the randomness of the generated keys must repeat. The complete key generation process starts with generating the non-unique keys. As (9)${\displaystyle \Phi \left(t\right):\left\{\begin{array}{c}{t}_1\to t_2\to t_n\hfill \\ k{1}_i\approx k{2}_i\hfill \end{array}\right.}$

Henceforth, the uniqueness of the key generation process is violated and makes the key sets vulnerable to attacks.

Thus, this problem must be solved.

Lemma 2

Validation of the randomness of the key generation process is coexisting on the actual key generation algorithm. Thus, the validation is invalid.

Proof

Assuming that the key set, K[], is collecting the individual key items, k_i, which is again an outcome from the time-dependent function. Thus, the primary relation, for a total of n number of observations, can be formulated as, (10)${\displaystyle K\left[\right]=\sum _{i=1}^n{k}_i}$

Further, as the generated keys are time-dependent thus, each key can be presented as, (11)${\displaystyle k_i\leftarrow \Phi \left(t\right)}$

Assuming that the key generation’s primary function is Φ₁(t) andfor randomness validation, the used function is Φ₂(t).

 Here, as both the functions have the same limits and terminating conditions with the same characteristics, it is safe to state that the functions are the same. (12)${\displaystyle {\Phi }_1\left(t\right)={\Phi }_2\left(t\right)}$

Again, assuming that each function generates a set of keys as K1[] and K2[]. Thus, due to the similar characteristics of the functions, the generated random key sets also must be similar. Thus, (13)${\displaystyle K1\left[\right]=K2\left[\right]}$

Henceforth, following Eq. (10), the following can be stated: each set must contain a similar set of values. (14)${\displaystyle \prod _{i=n}{k}_i=\prod _{j=n}{k}_j}$

Thus, the random key generated pattern will be similar. The key randomness validation method will justify the primary key generation method. This validation will generate a confirmation for validation each time irrespective of the actual randomness of the elements. Hence, this problem also needs to be addressed and solved. Furthermore, with a detailed understanding of the problems of the mathematical models, the solution is realized using mathematical formulations in the next section of this work.

Lemma 3

Generation of the keys for encryption and decryption using a two-phase chaos method terminating condition is highly random.

Proof

Assuming that the key set, K[], collects the individual essential items, k_i, results from the time-dependent function. Thus, the primary relation, for a total of n number of observations, can be formulated as, (15)${\displaystyle K\left[\right]=\sum _{i=1}^n{k}_i}$

Depending on the basic foundation of the chaos theory, the function responsible for generating the random keys, φ, can be further formulated as, (16)${\displaystyle k_i\left(t\right)=\phi \left(t\right).\left[1-k_i\left(t-1\right)\right]}$

The randomness for the key generation function is again a time-dependent function, and further can be formulated as, (17)${\displaystyle \phi \left(t\right)\leftarrow \prod _{time=t}r\left[\right]}$

Nonetheless, like any other randomness generation function, this function φ(t)must have one upper limit. Beyond that specified upper limit, the function shall produce the repetition of the random values, which will make the generated keys vulnerable to attacks. As the random value generation function, which is time-dependant is bound to have one upper limit, λ, during the key generation, as, (18)${\displaystyle {\int }_1^{\lambda }\phi \to \phi \left(t\right)}$

Henceforth, stopping the function for random value generation before the upper limit is a must to ensure randomness. Assuming that the upper limit, λ, shall be reached in a time limit of t _n, thus, the function must be stopped within the t _n time limit. It can be formulated as, (19)${\displaystyle k_i\left(t\right)=\left[\prod _{\underset{t\to t_n}{lim}}r\left[\right]\right].\left[1-k_i\left(t-1\right)\right]}$

Thus, the generated key, k_i(t), is ensured to have no repetitions and can be further used for public and private key component generation. It is Phase-I of the proposed key generation method.

Further, in Phase-II, the public key and the private key must be generated as Key _Pub and Key _Pri respectively. It can be formulated, using two instances of the k_i(t) and k_i(t + 1) at any random instance ∂, as, (20)${\displaystyle Ke{y}_{Pub}=!\prod _{Random=\partial }{k}_i\left(t\right)\wedge k_i\left(t+1\right)}$

And, (21)${\displaystyle Ke{y}_{Pri}=!\left[\prod _{Random=\partial }{k}_i\left(t\right)\wedge k_i\left(t+1\right)\right]}$

Henceforth, the generation of the keys is completed and ensured to be highly random. This process is Phase-II of the fundamental generation mechanisms.

Lemma 4

Validation of the uniqueness of generated keys by introducing a control set can result in perfect validation.

Proof

Assuming that the key set, K[], is a collection of the individual key items, k_i, which is again an outcome from the time-dependent function.

 Thus, the primary relation, for a total of n number of observations, can be formulated as, (22)${\displaystyle K\left[\right]=\sum _{i=1}^n{k}_i}$

Simultaneously, assuming that the key set, K1[], is a collection of the individual key items, k1_i, which is again an outcome from the time-dependent function.

Thus, the primary relation, for a total of n number of observations, can be formulated as, (23)${\displaystyle K1\left[\right]=\sum _{i=1}^{n}k{1}_i}$

Further, to validate the randomness of the generated keys, the standard deviation must be calculated for the generated key set and the control key τ₁τ₂. It can be formulated as, (24)${\displaystyle \tau =\frac{dK\left[\right]}{dn}}$

And, (25)${\displaystyle \tau 1=\frac{dK1\left[\right]}{dn}}$

Thus, the similarity measure, X, can be formulated as, (26)${\displaystyle \mathrm{P}=\frac{{\left(K\left[\right].{\tau }_2\right)}^2-{\left(K1\left[\right].{\tau }_1\right)}^2}{{\tau }_1.{\tau }_2}}$

Hence, based on general correlation theory, if the P-value is less than 0.05, it is safe. To state that the generated keys are highly random and do not control the set keys.

 Henceforth, this section of the work formulates the random key generation’s mathematical models and randomness validation. In the next section of this work, based on the mathematical foundations, the proposed algorithms are furnished and discussed.