The Quartic Public Key Transformation

This paper presents the quartic public key transformation which can be used for public key applications if side information is also used. This extends an earlier work where the cubic transformation was similarly used. Such a transformation can be used in multiparty communications protocols. Introduction Cryptographic transformations may be visualized in locked box or a piggy-bank modes [1], or alternatively as one-way functions or puzzles [2],[3],[4], or as unitary transformations in quantum states [5],[6]. All these methods are variations on one-to-one mappings of different kinds. In a method that is a departure from these techniques, a three-to-one mapping for cryptography was proposed [7] with the provision that additional side-information will be provided to determine the correct message. This method was a variant of the RSA transformation where the value of the encrypting exponent was taken as 3 and φ(n) (the totient function of the composite number n = pq) was divisible by 3, but not by 9. Clearly, the use of non-one-to-one mappings can make the task of the eavesdropper harder than it would be otherwise. In the RSA transformation the decryption key is calculated by the formula ed=1 mod φ(n). When Alice wants to send a message to Bob, the message to be sent is encrypted by c=me mod n. Bob decrypts the ciphertext using his private key and gets the original message by m=cd mod n=med mod n=m. The cubic public key transformation shows how the cube roots of 1 for a prime p and composite number n = pq is calculated and how the three different values of messages m produces the same cipher c. In some cases, the cube roots are obtained by applying Chinese Remainder Theorem (CRT) although congruences could also be solved by the Aryabhata Algorithm [8],[9]. Cryptography based on extraction of cubic roots from Gaussian Integers is explained in [10]. In this paper, we study quartic public key transformation as an extension of the cubic transformation. We are separately presenting additional results for general power transformations that are not one-to-one. The properties of the quartic transformation We first consider quartic mapping modulo a prime number. For quartic transformation c = m4 mod p, four different values of message m would give the same cipher c, where the value of prime p is given by p = 4k + 1. The quartic roots of 1 would be 1, α, α2 and α3 and they are calculated by solving the equation α4 − 1 = 0 (1) (α − 1)(α3 + α2 + α1 + 1)=0