Study the Linear Equivalent of the Binary Nonlinear Sequences

Linear orthogonal sequences, special M-Sequences, are used widely in the systems communication channels as in the forward links for mixing the information on connection and as in the backward links of these channels to sift this information which transmitted and the receivers get the information in a correct form. In current research trying to study the construction of the linear equivalent of a multiplication sequence and answering on the question "why the length of the linear equivalent of a multiplication sequence (on a linear M-sequence{an}), in some cases doesn't reach the maximum length rNh, special, when the multiplication is on three or more degrees of the basic sequence {an} The multiplication sequence has high complexity and the same period of the basic sequence, or if the multiplication sequence on two different basic sequences then the period of multiplication sequence is equal to multiplication the two periods of the basic sequences, and in the two cases the multiplication sequence is not an orthogonal sequence.


Introduction
The main obstacle to encoding and decoding is the complexity of decoding and decoding. For this reason, efforts have been made to design cryptographic and decoding methods in an easy way. The works of Hocquenghem in 1959, Reed Solomon 1960, Chaudhuri and Bose in 1960, BCH codes or Bose-Chaudhuri-Hocquenghem codes and others as Goppa, and Peterson 1961 were a new starting point for solving this issue. [1][2][3][4][5] In all stages of the encoding and the decoding, the orthogonal sequences play the main role in these processes, including the sequences with maximum period M-Sequences, Walsh sequences, Reed-Solomon sequences, and other sequences. [6][7][8][9][10][11][12] Sloane, N. J. A., discuses that the multiplication sequence {z n } on h degrees of {a n } which has the r complexity the complexity of {z n } can't be exceeded Orthogonal Sequences are used widely in the systems communication channels as in the forward links for mixing the information on connection and as in the backward links of these channels to sift this information which transmitted and the receivers get the information in a correct form. Especially in the pilot channels, the Sync channels, and the Traffic channel. [10][11][12] Shannon's classic articles, 1948-1949, were followed by many research papers on the question of finding successful ways to encode and successful decoding the media to allow it to be transmitted correctly through jammed channels. [6-8, [13-14].

Research Method and Materials
Is called the characteristic polynomial. In this study, we are limited to 0 1 λ = .
Definition 1. The ultimately sequence 0 1 , ,.... a a in F 2 with the smallest natural number r is called periodic with the period r iff: ; 0, 1, ... n r n a a n + = = [2][3][4][5][6] Definition 2. The linear register of a linear sequence is a linear feedback shift register with only addition circuits and the number in its output in the impulse n equal to the general term of the sequence {a n } and the register denoted as LFSR. [3] Definition Where x i + y i is computed mod 2. It is equal to the number of agreements components minus the number of disagreements corresponding to components or if , {1, 1} i i x y ∈ − (usually, replacing in binary vectors x and y each "1" by "-1" and each "0" by "1") then [2][3][4][5][6][7][8][9].  ( , ,..., ), , 0,1,..., 1 Let's 1 * =-1 and 0 * =1, The set G is said to be orthogonal if the following two conditions are satisfied: That is, the absolute value of "the number of agreements minus the number of disagreements" is less than or equals one. [6,9] Definition 7. (Euler function φ ). ( ) n φ is the number of the all-natural numbers that are relatively prime with n. [11][12][13][14] Definition 8. The linear equivalent of a multiplication sequence {z n }, on a binary linear sequence {a n } which generated with the linear register LFSR1 and the sequence{z n } is a multiplication on some terms of {a n } (that is a result of multiplication circuits over the LFSR1), is a linear shift register LFSR2 generates the same sequence {z n }. [2,3,8] Definition 9. The length of the linear equivalent of a multiplication sequence is the number of its complexity and equal to the degree of the characteristic polynomial which generates the same multiplication sequence, and the multiplication sequence can be generated through the linear equivalent. [8] Definition 10. The maximum length of a linear equivalent is the maximum length of the linear equivalent LFSR2 (it is the number of its complexity) which can be reached and the length of linear equivalent is always less than or equal the maximum length r h N .[2, 3, 8] Definition 11. Inverse problem: Finding the sequence {a n } which {z n } is a multiplication sequence on it and it is one of the issues at present and it requires a solution. [8] Theorem12.
i. If 0 1 , ,.... a a is a homogeneous linear recurring sequence of order k in 2 F , satisfies (1) then this sequence is periodic. ii. If this sequence is a homogeneous linear recurring sequence, periodic with the period r, and its characteristic polynomial ( ) f x then ( ) r ord f x . If the polynomial ( ) f x is primitive then the period of the recurring sequence which has f(x) as a characteristic polynomial is 2 1 k − , this sequence is called M-Sequence over 2 F , or briefly M-Sequences. [6,[11][12][13][14]. Lemma 13. (Fermat's theorem). If F is a finite field and has q elements then each element a of F satisfies the equation: [6,10] Theorem 14. If ( ) g x is a characteristic prime polynomial of the (H. L. R. S.) 0 1 , ,.... a a of degree k, and α is a root of ( ) g x in any splitting field of F p then the general term of this sequence is: [6,11] Theorem 15.
ii. If q F is a field of order 2 n q = then any subfield of it is of the order 2 m and m n , and by inverse if m n then in the field q F there is a subfield of order 2 m . [6,[10][11][12][13][14] Theorem 16. The number of irreducible polynomials in ( ) q F x of degree m and order e is ( ) / e m φ , if 2 e ≥ , when m is the order of q by mod e, and equal to 2 if m=e=1, and equal to zero elsewhere. [6,[10][11][12][13][14] * The study here is limited to the Galois Fields of the form 2 k F , then the period 2 1 k r = − .

Studying Multiplication Sequences on the Binary Recurring M-Sequences
Suppose the binary recurring M-Sequence {a n } with the complexity r and α 1 , α 2 ,…, α r are its different linear independent roots of the characteristic equation of the sequence then the general term of the sequence is given through the relation; 1

The Sequence {z n } Is a Multiplication on Two Degrees of the Sequence {a n }
Suppose the multiplication sequence {z n } as multiplication on two different degrees of {a n } as following; (1) The first degree is a n (in another case we can make a shift to the first term).
Thus we have the following properties; P1. Each term of the first sum is not equal to zero and the number of these terms is equal to 1 P2. Also, a term of the second sum is equal to zero if and only if;  Thus, for the first condition each of the i, j, k can't be one, and if there is be such other values will be as; 1 2 k i − + and 2 j i − are relatively primes) and it is contradiction then the sum of one term from the first sum with one term from the second sum can't be equal to zero and the linear equivalent reached the maximum length ( 1) The sequence {z n } is periodic with the same period of the sequence {a n }.  11  3  2  12  3  2  13  3  2  14  3  15 , 1, 1, 1, 1} α α α α α α α α α α α α α α = + + = + + + = + + = + = (9) The general term of the sequence is; The sequence is periodic with the period 2 4 -1=15 and; Solving this system of equation we have; 14 13 Thus, the general term of the sequence is; Suppose the sequence {z n }is a multiplication sequence on two degrees (a n , and a n+1 ) of the sequence {a n } as is showing in figure 1 The zeros of the characteristic polynomial of the sequence {z n }are; The coefficient of x 10 and the constant also each of them is equal to one, the other coefficients except; x 5 , x 4 , and x 3 are equal to zero and the characteristic equation is; 10 5 4 The subsequence as result of x 3 + x + 1=0 is periodic with the period 2 3 -1=8, x 7 + x 5 + x 4 + x 3 + x 2 + x +1 is a irreducible polynomial. The sequence {z n }, as a result of the characteristic equation x 10 + x 5 + x 4 + x 3 +1=0 is periodic with the period 2 4 -1=15, the same period of the sequence {a n }, and the sequence {z n } defined by the recurring formula Thus, the sequence reached its maximum length; And one period, with its cyclic permutations, don't form an orthogonal set. Figure 2 showing the linear feedback shift register which generates {z n }.

The Sequence {z n } Is a Multiplication on Three Degrees of the Sequence {a n }
Suppose the new product sequence {z n } as a product of three different degrees in {a n } as following; 1) The first degree is a n (in another case we can make a shift until to the first term) as in part1.
2) The second degree is n n b a δ + = (as a result of shift n by δ ).
3) The third degree is c n =a n + γ (as a result of shift the first by γ and & r r δ γ ≤ ≤ ) then: Thus, we have the following properties; P1. Each term of the first sum, not equal to zero. P2. For one term of the second sum is equal to zero is equivalent to; we have the previous condition is equivalent to; P3. For one term of the third sum is equal to zero necessary and sufficient; This equation is symmetric and can write it as follows; ( P4. Sum, one term of the first sum with one term of the second sum, is equal to zero necessary and sufficient the two conditions; The second condition can be written as; P5. Sum, one term of the first sum with one term of the third sum, is equal to zero necessary and sufficient the two conditions; Where, no two between the indexes i, j, k, and m are equal, and; 2).
( ) Or, by division on i δ γ α + , the second condition can be written as; Or; P6. Sum, one term of the second sum with one term of the third sum, is equal to zero necessary and sufficient the two conditions; Where, no two between the indexes i, j, k, m and l are equal, and P7. Sum, one term of the first sum, with one term of the second sum, and with one term of the third sum, is equal to zero, necessary and sufficient the two conditions; where, no two indexes between i, j, k, h, l, and m are equal Each of P4,…., and P7 properties leads to decrease the length of linear equivalent by one (for each case) relatively the maximum length r h N .
The complexity of the sequence is 10 and doesn't achieve its maximum length 4 3 14 N = , and the set of all cyclic permutations of one period of {z n } is not orthogonal set.
From P2 of Part2, For the one term of the second sum is equal to zero necessary?
The same is true for the other properties, and thus, the length of the linear equivalent generated {z n } is; The linear equivalent of {z n } is showing in figure 4;

Suppose, the Sequence {z n } Is a Result of Multiplication on Four Terms from the Sequence {a n }
As the following; the first term from {a n } is a n (in another case, we can shift the terms to the first term) second term is   , , ) , , , 1, , , , Where (j, k, l) is the set of permutations of {j, k, l}, Thus, we have the following properties; P1. Each term of the first sum is not equal to zero. P2. For one term of the second sum is equal to zero to necessary and sufficient; β µ γ β γ µ µ γ β β µ γ α α α α α α α Or (by division on i β µ γ α + + and arranging the order of the terms); P3. For one term of the third sum is equal to zero is necessary and sufficient realized; Or; (by division on i β µ γ α + + and arrange the terms as necessary); And it is a symmetric equation and can writ it as following; ( α α α α α α α α α α α α α α α α α α + + + + + + + + = By division on i β µ γ α + + then the latest equation can be written as; then the latest equation can be written as; (1 )( P5. For the sum of one term from the first sum with one term from the second sum is equal to zero is necessary and sufficient the following two conditions; 1 ( For the sum of one term from the first sum with one term from the third sum is equal to zero is necessary and sufficient the two conditions; 4 2 n n n n l i j k α α α α = Where i, j, k, and l are different (47) Thus, as the same, we have the corresponding relations for the following cases; 7) Sum one term from the second sum with one term from the third sum is equal to zero. 8) Sum one term from the second sum with one term from the forth sum is equal to zero. 9) Sum one term from the third sum with one term from the forth sum is equal to zero.
10) The sum of three terms from the different four sums is equal to zero.
11) The sum of four terms from the different four sums is equal to zero. Suppose the recurrent binary sequence {a n } which has the complexity r and α is a primitive root of its characteristic equation then this sequence is periodic and its period is 2 r -1,{b n }is other binary sequence which has the complexity s and β is a primitive root of its characteristic equation then this sequence is periodic and its period is 2 s -1, and suppose, for easily, r and s are relatively prime then the roots of the characteristic equation of the binary sequence {z n }={a n .b n } are in the field . 2 r s F and this sequence has the period 2 r.s -1. Example 3. Suppose, the linearly binary recurring sequence {a n } defining through the recurring formula a n+2 + a n+1 + a n =0 or a n+2 =a n+1 + a n and the linearly binary recurring sequence {b n } defining through the recurring formula b n+3 + b n+1 + b n =0 or b n+3 =b n+1 + b n and {z n }={a n . b n } as in the following figure 5, which shows the feedback shift registers for the sequences {a n }, {b n }, and the product sequence {z n }.

Multiplication Sequence on Two Linear Sequences Generated by Different LFSR
The characteristic equation of the sequence {a n } is 2 1 0 x x + + = and its characteristic polynomial is the prime   think, if r and s are not relatively prime then the period of the sequence {z n }is (2 lcm(r,s) -1) and its complexity is lcm(r,s). 6) Using multiplication operation on different sequences operation leads to getting sequences with high complexity and with a high period but not orthogonal. The method for reading a page which has a block will be according to the following direction as in figure 1.