Public key cryptography has received great attention in the field of information exchange through insecure channels. In this paper, we combine the Dependent-RSA (DRSA) and chaotic maps (CM) to get a new secure cryptosystem, which depends on both integer factorization and chaotic maps discrete logarithm (CMDL). Using this new system, the scammer has to go through two levels of reverse engineering, concurrently, so as to perform the recovery of original text from the cipher-text has been received. Thus, this new system is supposed to be more sophisticated and more secure than other systems. We prove that our new cryptosystem does not increase the overhead in performing the encryption process or the decryption process considering that it requires minimum operations in both. We show that this new cryptosystem is more efficient in terms of performance compared with other encryption systems, which makes it more suitable for nodes with limited computational ability.

Using a previously obtained structure theorem of Gelfand-Shilov spaces $\Sigma _{\alpha }^{\beta }$ of Beurling type of ultradistributions, we prove that these ultradistributions can be represented as an initial values of solutions of the heat equation by describing the action of the Gauss-Weierstrass semigroup on the dual space $(\Sigma _{\alpha }^{\beta})^{\prime }.$

A new fast public key cryptosystem is proposed, which is based on two dissimilar number-theoretic hard problems, namely the simultaneous chaotic maps (CM) problem and quadratic residue (QR) problem. The adversary has to solve the two hard problems simultaneously to recover the plaintext according to their knowledge about the public keys and the cipher-text. Cryptographic quadratic residue and chaotic system are employed to enhance the security of our cryptosystem scheme. The encryption, and decryption are discussed in details. Several security attacks are proposed to illustrate the system shield through chaotic maps and quadratic residue problems. The performance analysis of the proposed scheme show a much improved performance over existing techniques.