This paper proposes three public key exponent attacks of breaking the security of the prime power modulus 𝑁=𝑝2𝑞2 where 𝑝 and 𝑞 are distinct prime numbers of the same bit size. The first approach shows that the RSA prime power modulus 𝑁=𝑝2𝑞2 for q<𝑝<2q using key equation 𝑒𝑑−𝑘𝜙(𝑁)=1 where 𝜙(𝑁)= 𝑝2𝑞2(𝑝−1)(𝑞−1) can be broken by recovering the secret keys 𝑘 /𝑑 from the convergents of the continued fraction expansion of e/𝑁−2𝑁3/4 +𝑁1/2 . The paper also reports the second and third approaches of factoring 𝑛 multi-prime power moduli 𝑁𝑖=𝑝𝑖2 𝑞𝑖2 simultaneously through exploiting generalized system of equations 𝑒𝑖𝑑−𝑘𝑖𝜑(𝑁𝑖)=1 and 𝑒𝑖𝑑𝑖−𝑘𝜑(𝑁𝑖)=1 respectively. This can be achieved in polynomial time through utilizing Lenstra Lenstra Lovasz (LLL) algorithm and simultaneous Diophantine approximations method for 𝑖=1,2,…,𝑛.

The technical details of RSA works on the idea that it is easy to generate the modulus by multiplying two sufficiently large prime numbers together, but factorizing that number back into the original prime numbers is extremely difficult. Suppose that \(N=p^r q^s\) are RSA modulus, where \(p\) and \(q\) are product of two large unknown of unbalance primes for \(2 \leq s<r\). The paper proves that using an approximation of \(\phi(N) \approx\) \(N-N^{\frac{r+8-1}{2 r}}\left(\lambda^{\frac{1-8}{2 r}}+\lambda^{\frac{-8}{2 r}}\right)+N^{\frac{r+8-2}{2 T}} \lambda^{\frac{1-8}{2 r}}\), private keys \(\frac{x^2}{y^2}\) can be found from the convergents of the continued fractions expansion of\[\left|\frac{e}{N-N^{\frac{r+8-1}{2 r}}\left(\lambda^{\frac{1-8}{2 r}}+\lambda^{\frac{-8}{2 r}}\right)+N^{\frac{r+8-2}{2 r}} \lambda^{\frac{1-8}{2 r}}}-\frac{y^2}{x^2}\right|<\frac{1}{2 x^4}\] which leads to the factorization of the moduli \(N=p^r q^s\) into unbalance prime factors p and q in polynomial time. The second part of this reseach report further, how to generalized two system of equations of the form \(e_ux^2\) - \(y^2_u\phi(N_u)\) = \(z_u\) and \(e_ux^2_u\) - \(y^2\phi(N_u)\) = \(z_u\) using simultaneous Diophantine approximation method and LLL algorithm to and the values of the unknown integers \(x,y_u\),\(\phi(N_u)\) and \(x_u\),y,\(\phi(N_u)\) respectively, which yeild to successful factorization of k moduli \(N_u=p^r_uq^s_u\) for u = 1,2, ... k in polynomial time.

It was shown in Sadiq (2013) that succession parameters under the Aunu permutation patterns can be used as vertices of the graph model resulting from different transition of the automata scheme employed. This paper generates a graph model using the Aunu permutation patterns governed by some properties as embedded in method of construction of a typical game of chance scheme. A finite automata model was constructed from the game of chance using the (123) avoiding class of the Aunu permutation patterns.Furthermore, the paper illustrated some useful relationship between the field of automata theory and Combinatorics; it also highlights some important applications of the Aunu Permutation Patterns in graph theory.

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The importance of keeping information secret cannot be overemphasized especially in today,s digital world where eavesdroppers are rampant in our chanels of communication. This made the use of strong encryption schemes inevitable in order to safeguard the security of our system. RSA cryptosystem and its variants have been designed to provide confidentiality and integrity of data in our medium of communication. This paper reports new short decryption exponent attack on prime power with modulus $N=p^rq$ for $r\geq 2$ using continued fraction method which makes it vulnerable to Diophantine attack and breaks the security of the cryptosystem by factoring the modulus into its prime factors since the hardness relies on the integer factorization problem. The paper also shows that if the short decryption exponent $d