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On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators

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  • Uniformity in binary tuples of various lengths in a pseudorandom sequence is an important randomness property. We consider ideal $ t $-tuple distribution of a filtering de Bruijn generator consisting of a de Bruijn sequence of period $ 2^n $ and a filtering function in $ m $ variables. We restrict ourselves to the family of orthogonal functions, that correspond to binary sequences with ideal 2-level autocorrelation, used as filtering functions. After the twenty years of discovery of Welch-Gong (WG) transformations, there are no much significant results on randomness of WG transformation sequences. In this article, we present new results on uniformity of the WG transform of orthogonal functions on de Bruijn sequences. First, we introduce a new property, called invariant under the WG transform, of Boolean functions. We have found that there are only two classes of orthogonal functions whose WG transformations preserve $ t $-tuple uniformity in output sequences, up to $ t = (n-m+1) $. The conjecture of Mandal et al. in [29] about the ideal tuple distribution on the WG transformation is proved. It is also shown that the Gold functions and quadratic functions can guarantee $ (n-m+1) $-tuple distributions. A connection between the ideal tuple distribution and the invariance under WG transform property is established.

    Mathematics Subject Classification: Primary: 94A60, 94A55; Secondary: 94A15.


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  • Figure 1.  Block diagram of a filtering de Bruijn generator (FDBG)

    Figure 2.  Relations among Hadamard Transform, WG-inv, $ x_0 $-independence and ideal tuple distribution. $ g(\cdot) $ is independent of $ x_0 $. $ \leftrightarrow $ denotes if and only if condition, $ \rightarrow $ denotes if condition and $ n $ is the NLFSR length

    Table 1.  A summary of the Hadamard transform values of three-term functions

    Functions Hadamard Transform values Ref.
    $ w(x) = T3(x^{2^k+1}) = {\rm Tr}(x + x^r + x^{r^2}) $ $ 3 $-valued [11]
    $ g(x) = T3(x^{2^k-1}) = {\rm Tr}(x + x^{q_2} + x^{q_2^2}) $ at most $ 5 $-valued [39]
    $ WG_{w}(x) = {\rm Tr}(x + (x+1)^{r} + (x+1)^{r^2}) $ $ 3 $-valued Lemma 3.7
    $ WG_{g}(x) = {\rm Tr}(x + (x+1)^{q_2} + (x+1)^{q_2^2}) $ at most $ 5 $-valued Lemma 3.7
    $ WG_{T3}(x^d) = {\rm Tr}(x^d + (x^d+1)^{q1} + (x^d+1)^{q_2}) $ at most $ 5 $-valued Theorem 3.8
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    Table 2.  Experimental results for ideal $ t $-tuple distributions of the WG transform of the Kasami power functions for $ k' = 3, 4 $ and $ 5 $. When $ k' = 3 $, $ WG_{R_3}(x^d) $ over $ {\mathbb F}_{2^m} $

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