Solving systems of diagonal polynomial equations over finite fields

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• 作者：Gá ; bor Ivanyos^a ; ^{Gabor.Ivanyos@sztaki.mta.hu} ; Miklos Santha^b ; ^c ; ^{miklos.santha@gmail.com}
• 关键词：Algorithm ; Polynomial equations ; Finite fields ; Chevalley&ndash ; Warning theorem ; Quantum computing
• 刊名：Theoretical Computer Science
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：657
• 期：part_PA
• 页码：73-85
• 全文大小：397 K
• 卷排序：657

文摘

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations. Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the polynomial equations is constant. As a consequence we design polynomial time quantum algorithms for two algebraic hidden structure problems: for the hidden subgroup problem in certain semidirect product p-groups of constant nilpotency class, and for the multi-dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant. 1