A Mathematical Development of Gordon, Mills and Welch Generator Using Galois Field and Trace Polynomials

The paper presents a newly developed method depending on trace polynomial with new algorithm which has been written in Matlab language to modify the Gordon, Mills and Welch generator by increasing the complexity of the output sequence to increase the security of this generator concerning the designed feature that limits the ability of anti-jammer. Good results are obtained compared with Gordon, Mills and Welch generator using Berlekamp-Massey method for determining the complexity of the output sequences. Moreover, this paper has the useful properties of the trace function. Some examples are given of show conciliated the results of this proposed method.


Introduction
A pseudo-noise (PN) generator is a mechanism for generating a PN-sequence of binary or real digits.Pseudo-Noise generators generate (PN) sequences which are used as spectrum-spreading modulations for direct sequence spread spectrum design for digital communication system and as a key in cipher systems [1,2].The resulting sequence is called pseudo-noise sequence since it is periodic and there is no algorithm using a finite state machine which can produce a truly random sequence [2].PN-sequences are characterized by three properties; namely: period, complexity and randomness.These properties define the measure of security for the sequences [3,4].The complexity of the sequence is one of important properties for security of the information from unauthorized person.
Gordon, Mills and Welch generator is one of these PN generators.It produces a binary sequence with good period and randomness properties but with relatively small complexity.The idea of this paper is to increase the complexity of the produced sequence but with remaining the same period.
The new generator has a sequence with high complexity and good randomness properties.Hence, it has enough ability for the security of the information from interceptor and jammer.

The Complexity of The Periodic Sequence :
The complexity of the sequence (or generator) in the cipher and communication systems is the length of the minimum linear feedback shift register (LFSR) that can generate the sequence [5,6].We can characterize the LFSR of length (n) by the characteristic polynomial f(x) as: where c 0 ,c 1 , . ..,c n-1 are 0 or 1.
For a sequence with a known complexity (L), we need (2L) consecutive bits to deduce the entire sequence since if (2L) consecutive bits are given, then we can write a system of L-equations in the L unknown variables and find its unique solution.This gives the characteristic polynomial of the minimal LFSR that can generate the given sequence.Hence for acceptable security, we need to have a sequence or generator with high complexity [2,7].There are several methods to determine the complexity of the sequences like Berlekamp-Massey method which is given in the following section, Ztransform and matrices techniques [7].

The Berlekamp-Massey Method
Berlekamp-Massey (BM) technique [3,5,6] uses the description based on the synthesis of a shift register where it is used to determine the complexity and the minimal characteristic polynomial that can generate the given sequence .
2. The linear equivalence or the complexity (L) of the sequence .
BM technique is explained in the following algorithm :

Definition(1):
A finite field F is called a Galois field denoted by GF(q m ) if the number of elements of F is q m (i.e. the order of F is q m ), where q is a prime number and m is a natural number.q is called the characteristic of the field GF(q m ) .

Definition(2):
It is known that every nonzero element α of GF(q m ) which satisfies the equation : α is said to be a primitive element of GF(q m ) if all the powers of α less than (q m -1 ) are different .Thus , if α is a primitive element then α i ≠ 1 , for 0 < i < q m -1.
The powers of α were reduced to polynomials of degree (3) or less in α.Table (1) shows the representation elements of GF(16).

Trace polynomial :
A fundamental mathematical tool used in investigation of PN generator is a linear mapping from a finite field onto a subfield.This mapping is called the "Trace polynomial" or the The Trace polynomial from GF(q n ) to GF(q) where ( q>1) and ( n≥2) is defined as [1]:

Power
has values in GF(q) .
Proof : r Τ is a field element whose q-th power equals itself .Only elements of GF(q) have this property .

Property 2 :
Conjugate field elements which have the same trace value.
The proof of this property follows from the definition of Trace polynomial and the previous property.
Property 3 : The Trace is linear : for a,b ∈GF(q) and ) ( ,

Gordon, Mills and Welch Generator :
Gordon, Mills and Welch (GMW) [1,8,9] suggested the creation of a binary generator depending on "Trace polynomial".When the integer m is a composite, i.e. m = j× k … (2) We define a GMW sequence generator of period (2 m -1) as : ( ) from GF(q d ) to GF(q) and r is any integer relatively prime to 2 j -1 (i.e.
where the t i 's are distinct integers in the range 0 ≤ t i < j for all i and w is the number of ones in the base-2 representation of r.Hence, Since the inner trace is a sum of elements from a field of characteristic (2) (i.e. the operations over GF(2) ), all cross-product terms disappear when the trace is squared and Eq.( 5) reduces to: where k is defined in Eq.( 2) and When expanding the outer Trace polynomial in Eq.( 6) , cross-product terms again must disappear, and the GMW generator becomes : The GMW generator producing a binary sequence of period (2 m -1) is computed by the following algorithm:
(2) The values of j, k and r .

Example :
A GMW sequence of period ( 63) is:

Proposed Method for Developing
The Gordon, Mills and Welch (GMW) Generator Using Galois Field and Trace polynomial : The GMW in previous section is a nonlinear generator which produces a pseudo-noise sequence of period (2 m -1) over GF(2), where m is a composite integer as well as the degree of the minimum polynomial We are developing the GMW generator by using "Trace polynomial" from GF(q n ) to GF(q) to produce a sequence which has the same period but with complexity higher than GMW generator and it is denoted by (DGMW).
Hence, the DGMW generator is defined as : is the "Trace polynomial" in eq.( 1) from GF(q d ) to GF(q) which is defined as: The following algorithm summarizes the steps for finding the binary sequence of period (2 m -1) of DGMW generator.

Illustrative Examples : Example (1) :
Consider the following GMW generator : The following table presents the results obtained by applying DGMW algorithm.

BM Method The Generators
The Complexity (L) GMW in Eq.( 11)  12) It is obvious from Table (3) that the complexity of DGMW sequence (or generator) is higher than GMW sequence.So, the security of DGMW generator is more than GMW, since in DGMW sequence we need (144) consecutive bits to find the entire sequence while in GMW sequence we need only (32) consecutive bits to find the entire sequence (see the definition of the complexity in section(2)).

The Generators
The Complexity (L) GMW in Eq.( 9)  5) that the complexity of DGMW sequence (or generator) is higher than GMW sequence.So, the security of DGMW generator is more than GMW.

Example (3) :
Consider the following GMW generator :  6) presents the sequences of GMW and DGMW generators by applying their algorithms respectively and shows their complexities by using BM algorithm.

Table (6) The sequences of GMW and DGMW
generators with their complexities.
Technology, Vol.25, No.8, 2007A Mathematical Development of Gordon, Mills and Welch Generator Using Galois Field and Trace Polynomials 964 2 and it is defined as :

Table ( 4
) presents the results by applying DGMW algorithm.