For a positive integer ^{n}

So far, researchers have not given sufficient conditions for NFSRs to directly generate dBSs. In this case, a relatively weak problem is to find the necessary conditions for NFSRs to construct dBSs, which is considered to be important. Several necessary conditions for feedback functions (FFs) of NFSRs to produce dBSs have been given successively by some researchers.

Let _{i}

The rest of this paper is organised as follows. In Section 2, we present a simple introduction of Boolean functions and NFSRs. We also list the known necessary conditions for FFs of NFSRs that can produce dBSs. Section 3 gives some new necessary conditions for FFs of dBSs. Section 4 concludes the work.

Throughout this paper,

In the following, let us recall some results on feedback shift registers and Boolean functions.

A mapping from the vectorspace

For any _{0} is referred to as the _{0} denote the constant term of _{i}

If

From a Boolean function

Otherwise, we call it a ^{n}

The smallest integer

The period of output sequence from each ^{n}

We review the known necessary conditions for FFs of NFSRs, which can produce dBSs. We copy their list into the following theorem.

_{i}_{i}_{i}

_{1}_{2}, then

for positive integers

With the above preparation, we will investigate the necessary conditions for FFs of NFSRs corresponding to dBS in the next section.

In this section, we consider some new necessary conditions for FFs of NFSRs that can produce dBSs.

Suppose that

It is worth noting that the new necessary conditions in Theorem 3 cannot be inferred from the conditions in Theorem 2. This fact is demonstrated in the following Example 4.

_{2} contains all the terms in

By

Using the same method we can obtain the case (2), and thus we omit the proof here.

Note that the new necessary conditions in Theorem 5 cannot be inferred from the conditions in Theorems 2 and 3. This fact can be demonstrated in the following Example 6.

_{1} does not contain

_{2} contains terms

_{1}_{2} and

By

From the following Example 8, we can indicate that the conditions in Theorem 2 cannot deduce the necessary condition in Theorem 7.

By

The following Example 10 indicates that the new necessary condition in Theorem 9 cannot be inferred from the conditions in Theorem 2.

The following Example 12 can show that the conditions in Theorem 2 cannot deduce the new necessary condition in Theorem 11.

The following Example 14 can indicate that the conditions in Theorem 2 cannot deduce the new necessary condition in Theorem 13.

The following Example 16 indicates that the conditions in Theorem 2 cannot deduce the new necessary condition in Theorem 15.

The main contribution of this paper is that some new necessary conditions for FFs of NFSRs to produce dBSs are presented from the relationship among linear terms, quadratic terms and cubic terms in the FFs of dBSs. The main difference between our proposed necessary conditions and previous necessary conditions lie in the nonlinear terms of the ANF of the FFs of NFSRs. The previous results about the terms of the ANF of the FFs of

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