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

The proposed stream cipher SSCQF is a binary addition stream cipher [14].In a binary addition stream cipher, the plaintext is given as a string ,... , 2 1 m m of elements of the finite field   1 , 0 2  k .The keystream ,... , 2 1 z z is a binary pseudorandom sequence [13].The sender encrypts the plaintext message according to the rule .The keystream generator denoted K , is a concatenation of the output binary sequences of all Boolean functions j R .
The output function of our stream cipher is parameterized only by the secret-key S K .As the keystream bits are produced independently of the plaintext, the proposed stream cipher SSCQF belongs to the category of synchronous stream ciphers.
In this section, we introduce the notations that will be used throughout this paper in TABLE 1. Stream cipher [14] is a secret-key cryptosystem constructed for improve secrecy of transmitted data.It is a lightweight and efficient cryptographic primitive for ensure confidentiality of transmitted data between two communicated pairs.It proves its robustness by its ability to resist against attacks [3][4] [7] [14].It has a wide application area especially in mobile devices and embedded systems.In this section we introduce the notation and terminology that will be used throughout the proposal.We use the symbol   . Theorem 2.1: Let X , Y and Z be three vectors of Let m be a positive integer.A binary feedback shift register (FSR) of length m is uniquely determined by its feedback function     , where If the feedback function F of an m -stage feedback shift register is linear, one speaks of a linear feedback shift registers (LFSR).Otherwise one speaks of a nonlinear feedback shift register (NLFSR).All feedback shift registers used in this paper are nonsingular and linear.In this case,   where the i a 's are either 0 or 1 for all and its linear recursion is of the form: [17].An alternative way to describe this recursion is to specify the th m degree binary characteristic polynomial [16] Definition 2.3 (see [12]): Let , each of positive degree.Definition 2.4 (see [12]): Let of degree less than N , is a field of order N 2 .Addition and multiplication are performed

III. A BRIEF DESCRIPTION OF SSCQF ALGORITHM
Stream cipher encrypts the plaintext by using a key stream generator.The latter can be a synchronous or an asynchronous stream cipher.This property is related to regenerate a nature of secret-key.A generator is qualified as a synchronous stream cipher if the regeneration of the secretkeys carries out independently of the plaintext and ciphertext messages.By contrast, an asynchronous stream cipher products the keystreams as a function of the input secret-key and previous ciphertexts [14].Our synchronous algorithm SSCQF can briefly be described as follows: It takes a secret-key constructed by a sequence of positive integers N z z ,..., 1 and let For each i d we assign them only two positive integers i n and i m as follows: d is a square free integer, then we assign only one fundamental unit i  of the quadratic field ℚ   i d [2] [5] where We then construct the initial vectors . Since the vectors i X do not have the same length, then we regenerate the vectors i Y , from the vectors i X , having the same length L .The number L is divisible by eight via equations

IV. DETAILED DESCRIPTION OF SSCQF ALGORITHM
The overall structure of the keystream generator SSCQF is depicted in the following figure.www.ijacsa.thesai.orgThe basic ingredients of the keystream generator SSCQF rely on the following three processes:

A. Process I
The main goal of this process is to generate the initial vectors , we get: for all 0 l x x for all 0 t L'-l x x for all 0 s 8 L' mod 8 . The keystream is obtained by concatenation of the output binary sequences of all Boolean functions.

V. BEHAVIORAL STUDY
After presenting and explaining the principle components of our SSCQF algorithm, in this section, we focus a behavioral study for all elements constituting our regenerator in order to highlight its internal characteristics.We begin by studying the complexity of the output binary sequences of all Boolean functions j R related to their lengths for a given password.Effectively, our goal, in this subsection, is to appear the cryptographic nature of the internal states of our regenerator of binary sequences.Then, we pass to analysis the keystream regenerated by our system after the minimal perturbations on the initial condition.Finally, we present an analytical study simulating the human system.

A. Correlation and normalized distance of periodic binary strings
For the binary sequences, we must exploit the Hamming principle to make sure their nature distribution.It aids in estimating the complexity of binary strings that have the same period.However, the testing of the keystreams regenerated by our regenerator show that not necessarily of the same period.Hence, we should use an extension of a Hamming distance as we defined in [1] [21]: Let S and S' be two elements of  of periods k and k' respectively and ( , ') Also in [21], we defined another interesting property allowing to more ensure the nature of binary sequences: uncorrelation of the binary strings.Thus, for all and ' SS in  , we say that two binary strings are weakly correlated if: '( , ') 0.5 D S S (5) This property allows us to prove the complexity of the binary sequences not necessarily of the same period.More precisely, the obtained values of a normalized distance are used to make sure about the uncorrelation or the correlation of the sets of periodic binary strings.

B. Impact of the lengths on the output binary sequences of all Boolean functions
Firstly, we propose an analysis study of each output binary sequences of all Boolean functions j R related to their lengths for a given password.In this case, we change the length of output binary sequences of all Boolean functions j R in order to ensure the internal nature of our regenerator.For this object, we propose a fixed secret-key
 In second case (Fig. 3), the length of a binary sequence is: L B2 =4005 bits.
 In third case (Fig. 4), the length of a binary sequence is: L B3 =6005 bits.
From [14], we say the binary sequences 1 ,..., N XX of same lengths are independent if each taking on the values 0 or 1 with probability 1 2 . Then, we talk about the unpredictable and uncorrelated primitive signals if the distribution of hamming distance accumulates near to half-length (L 1/2Bi ) of this binary sequence.This means that almost half the bits in same position of two set of the binary sequence are different.
 L 1/2B3 3002 bits.From these histograms, we notice, for a same secret-key, the distribution of hamming distances in these three cases accumulates in the vicinity of half-length of each output binary sequences of all Boolean functions.In addition, the obtain results are almost identical in all three histograms.In two first cases, we have three accumulations regions nearest to halflength.But, in third case, we have only a peak nearest to halflength.Accordingly, the cryptographic nature of each primitive signal in any internal state is not only related to the length of the regenerated a binary sequence.Effectively, these results are strongly linked to Boolean Functions and linear feedback shift registers filtered by the primitive polynomials of degree eight integrated in our system.Hence, our purpose has unpredictable internal characteristics [1][21], which is recommended in order to resist against attack periodic sequences [5] [10].This enables us to ensure the cryptographic nature of SSCQF algorithm.Finally, for each internal state, we can summarize these features as follows:  The length of each block regenerated has a positive effect on the cryptographic quality of the regenerated primitive signals.
 The distribution of lengths and periods are random.
 The primitive signals are unpredictable or cryptographically strong.
 When we increase the period length of the internal states, their regenerated the primitive signals became more uncorrelated.Then, long period has a positive impact on the cryptographic nature of internal primitive signs.This property is more desirable for an efficient stream cipher generator.
 The cryptographic quality of each regenerated primitive signals is strongly related to Boolean Functions and linear feedback shift registers filtered by the primitive polynomials of degree eight integrated in our system.

C. Impact of Minimal Perturbations
After introducing an analytical study of the internal states of our system, in this subsection, we concentrate to the behavioral study of external states Keystream of our system.The benefit is to interpret the responses of our proposed system in the minimal conditions.Objectively, for each iterations, we choose the secret-keys the same length .Also, we perform the minimal perturbations on the input secret-key in order to examine their impact on the lengths and the nature of primitive signals of the associated keystreams.We increment, in each iteration, an integer number i z of input secret-key in a given position progressively.The importance is to show if the linearity of input secret keys has an effect on the cryptographic quality of output secret-keys.www.ijacsa.thesai.orgFrom this histogram (Fig. 5), we observe, for the minimal perturbations, that the lengths distribution of primitive signals does not admit a probabilistic law.That means, it hard to an attack to infer the input length according to the lengths of output secret-keys.Its period represents an important benefit to distinguish a good stream cipher regenerator.This dynamite confirms another robustness factor of our regenerator of binary sequences.In this histogram (Fig. 6), it appears clearly the accumulation of normalized distances nearest to 0.5 followed by small peaks and a large peak exactly in 0.5.This result of normalized distances reassures another significant property filled by our proposed system: unpredictable of each binary sequence.Therefore, we confirm the uncorrelation of generated primitive signals able to withstand the collision and correlation attacks [5]

D. Simulating a human system
In reality, Man has a chaotic mind.It is hard to control an user during the choice its input secret-key s K .But, we can - have an impact on the balancing results.Wherefore, our system inspires its robustness.This outcome (Fig. 8) is identical to the result obtained in figure 6.It proves, in the minimal conditions, the cryptographic nature of SSCQF algorithm [21].In effect, our algorithm is efficient and able to resist against attack periodic sequences

A. Implementation of process I
The first aim of this process is to generate the integer numbers i d , i n and i m for each element i z of a secret-key www.ijacsa.thesai.orgs K , then, their binary representations.In each iteration, the binary representations of i d , i n and i m will be combined in order to create an initial vector as follows From this figure (Fig. 9), we show that the values of i n and i m don't depend on the values of i d , but, these are strongly related to its quadratic structure.In reality, it gives more complexity and dynamite of our proposed system.It suffices to behold here that any added bit has an impact on the balancing results of initial binary vectors i X .From this outcome (Fig. 10), the binary representations of each initial vector i X don't have the same length.But, in our proposal, we want to get the binary sequences which have the same length L divisible by eight.This is the object of the following process.

B. Implementation of process II
As we have previously explained, we dedicate this process to balancing the binary sequences generated in previous process.The aim is to obtain initial binary vectors i X that have a length multiple to eight.Because, in these situation, we use a linear feedback shift register filtered by the primitive polynomial of degree eight.So, if we change the degree of primitive polynomial, in this case, we should adapt this process to regenerate the initial vectors that have a length of its degree.The results of this process are presented in following figure (Fig. 11).In this work, we innovate a quick, dynamic and complex generator of the binary sequences.We are combined a large theory concept for product a pseudorandom stream cipher.It will be used as a symmetric key cipher for avoid the serious security problems.This synchronous generator products primitive signals uncorrelated, unpredictable and independents of the same input secret-keys lengths.Moreover, it ensures the cryptographic quality of internals states in order to avoid correlation attacks [9][14] [18] [19].

VII. CONCLUSION
We introduced, in this paper, a new synchronous stream generator cipher named SSCQF.Our proposed symmetric key system is founded on quadratic fields.We aim by this work to improve the confidentiality of transmitted data between two communicated pairs.A behavioral study, in the minimal conditions, appears the cryptographic nature of our construction.It also confirms the concrete security of the internal and external states, more, its ability to conserve the unpredictable nature of each regenerated primitive signals.In addition, the output secret-key length is not related to the input secret-key length, but, is strongly linked to quadratic nature of each element constructing an input secret-key.Idem, these dynamite and robustness are clearly proved in implementation section.
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 of length eight.We then obtain /8 NL  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 , INTRODUCTION

Definition 2 . 6 :
We call a Boolean function upon   N

2 and 3 .
Each binary standard sequence is subdivided into 8 / L binary sequences of length eight, each of them initializes one linear feedback shift register of length eight.by primitive polynomials of degree eight.And we combine the output binary sequence of all ij digit is obtained by concatenation of the output binary sequences of all Boolean functions j R .
Fig. 9. Regeneration of the i d , i n and i m for a secret-key

Fig. 10 .
Fig. 10.Binary representation of each initial vector i X

TABLE I
,  2  where 1  i r are not necessarily of the same length.The goal of this process is to balancing those vectors.For that, we then choose a vector of a maximal length, for example k 1  generated in the process II , are of the same length L divisible by eight.We