A RSA-type cryptosystem based on quartic polynomials

RSA cryptosystem was introduced by Rivest. Shamir, and Adlernan 111 1978. Common users of RSA cryptosystem are currently using 1024-bit keys. They are recommended to use 2048-bit keys in 20 I I and 3072-bit key in 2031. However, increasing the bit of keys will decrease the efficiency and increase the cost. Therefore, the aim of this study is to analyze and implement a new cryptosystem which is more secure than RSA, LUC and LUC3 cryptosystem for same bit of keys. This cryptosystem which is called LUC.j() cryptosystem is derived 11'0111 a fourth and sixth order Lucas sequence and is based on quartic polynomial. In this research, numerous mathematical attacks will be analyzed with the cryptosystem and compared with RSA. LUC. and LUC3 cryptosystems, The numerous mathematical attacks arc Has tads attack. GCD attack, garbage-man-in the- middle (I) attack. chosen plaintext attack. garbage-man-in-the-middle (II) attack. common modulus attack. Wiener's attack. Lentras attack and faults based attack. Most of these attacks have shown that the LUCU1 cryptosystem is secure than RSA, LUe and LUC3 cryptosystems. The other attacks have shown that they are in the same security level. This is because these attacks do not result from a weakness of cryptosystem but rather from a bad implementation. The efficiency of the cryptosystem is the ability to compute e-th term of the fourth and sixth order of Lucas sequence in a reasonable period of time. Therefore. instate of computing Ve sequentially. a new algorithm will be presented to compute Vc in less away of time by omitting some terms in the calculations. By using this algorithm, the time for computations will be decreased. As a conclusion, this cryptosystem has the potential to replace the RSA cryptosystem in the future.

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