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

Abstract

In this paper, we propose a new synchronous stream cipher called SSCQF whose secret-key is Ks=(z1,...zn) where zi is a positive integer. Let d1, d2,..., dN be N positive integers in {0,1,...2m -1} such that di=zi mod2m with m and m>=8. 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 IV={X1,...,Xn} from the secret-key Ks=(z1,...zn) by using the fundamental unit of Q( Nvdi) if di is a square free integer otherwise by splitting di, and in process II, we regenerate, from the vectors Xi, the vectors Yi having the same length L, that is divisible by 8 (equations (2) and (3) ). In process III , for each Yi , we assign L/8 linear feedback shift registers, each of length eight. We then obtain N x L/8 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 L/8 Boolean functions. The keystream generator, denoted K , is a concatenation of the output binary sequences of all Boolean functions.