A Synchronous Stream Cipher Generator Based on Quadratic Fields (SSCQF)

In this paper, we propose a new synchronous stream cipher called SSCQF whose secret-key is   N 1,...z z  S K where i z is a positive integer. Let N d d d ,..., , 2 1 be N positive integers in   1 2 ,..., 1 , 0  m such that i i z d  mod m 2 with m and 8  m . Our purpose is to combine a linear feedback shift registers LFSRs, the arithmetic of quadratic fields: more precisely the unit group of quadratic fields, and Boolean functions [14]. Encryption and decryption are done by XRO'ing the output pseudorandom number generator with the plaintext and ciphertext respectively. The basic ingredients of this proposal stream generator SSCQF rely on the three following processes: In process I , we constructed the initial vectors   N 1 X ,..., X  IV from the secret-key   N 1,...z z  S K by using the fundamental unit of Q   i d if i d is a square free integer otherwise by splitting i d , and in process II , we regenerate, from the vectors i X , the vectors i Y having the same length L, that is divisible by 8 (equations   2 and   3 ). In process III , for each i Y , we assign 8 / L linear feedback shift registers, each of length eight. We then obtain / 8 N L  linear feedback shift registers that are initialized by the binary sequence regenerated by process II , filtered by primitive polynomials, and the combine the binary sequence output with 8 / L Boolean functions. The keystream generator, denoted K , is a concatenation of the output binary sequences of all Boolean functions. Keywords—Synchronous stream cipher SSCQF; linear feedback shift registers LFSRs; arithmetic of quadratic fields; Boolean functions; pseudorandom number generator and keystream generator