Efficient Construction of Cross-Join Pairs in a Product of Primitive Polynomials of Pairwise-Coprime Degrees

We study the cross-join pairs in the cycles of linear feedback shift registers whose characteristic polynomials are of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x) = p_{1}(x)p_{2}(x)\cdots p_{k}(x)$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{i}(x), 1\leq i\leq k$ </tex-math></inline-formula> are primitive polynomials of coprime degrees. Firstly, we use Coppersmith <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.’s</i> generating function theory to derive the lower and upper bounds for the number of cross-join pairs. Then we design an algorithm to generate these cross-join pairs. The algorithm requires a preparatory phase which costs <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(2^{n'})$ </tex-math></inline-formula> time where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n'$ </tex-math></inline-formula> is the largest degree of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{i}(x), 1\leq i\leq k$ </tex-math></inline-formula> , and after that it costs only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n^{3})$ </tex-math></inline-formula> time to generate one cross-join pair where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the degree of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x)$ </tex-math></inline-formula> . We also consider a special class of cross-join pairs, for which the preparatory phase costs only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(2^{n''})$ </tex-math></inline-formula> time where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n''$ </tex-math></inline-formula> is the second-largest degree of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{i}(x), 1\leq i\leq k$ </tex-math></inline-formula> . The number of these special cross-join pairs is about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {1}{12}2^{2n-n'}$ </tex-math></inline-formula> . We present some experimental results, which validate our analysis and demonstrate the efficiencies of the algorithms. These cross-join pairs can be used in the cross-joining method to construct de Bruijn sequences.