Abstract
This paper studies the distinctness of modular reductions of primitive sequences over \({\mathbf{Z}/(2^{32}-1)}\) . Let f(x) be a primitive polynomial of degree n over \({\mathbf{Z}/(2^{32}-1)}\) and H a positive integer with a prime factor coprime with 232−1. Under the assumption that every element in \({\mathbf{Z}/(2^{32}-1)}\) occurs in a primitive sequence of order n over \({\mathbf{Z}/(2^{32}-1)}\) , it is proved that for two primitive sequences \({\underline{a}=(a(t))_{t\geq 0}}\) and \({\underline{b}=(b(t))_{t\geq 0}}\) generated by f(x) over \({\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}\) if and only if \({a\left( t\right) \equiv b\left( t\right) \bmod{H}}\) for all t ≥ 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.
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Zheng, QX., Qi, WF. & Tian, T. On the distinctness of modular reductions of primitive sequences over Z/(232−1). Des. Codes Cryptogr. 70, 359–368 (2014). https://doi.org/10.1007/s10623-012-9698-y
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DOI: https://doi.org/10.1007/s10623-012-9698-y