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Computing discrete logarithms using 𝒪((log q) 2 ) operations from {+,-,×,÷,&}

Computing discrete logarithms using 𝒪((log q) 2 ) operations from {+,-,×,÷,&} Abstract Given a computational model with registers of unlimited size that is equipped with the set { + , - , × , ÷ , & } = : 𝖮𝖯 ${\{+,-,\times,\div,\&\}=:\mathsf{OP}}$ of unit cost operations, and given a safe prime number q , we present the first explicit algorithm that computes discrete logarithms in ℤ q * ${\mathbb{Z}^{*}_{q}}$ to a base g using only 𝒪 ⁢ ( ( log ⁡ q ) 2 ) ${\mathcal{O}((\log q)^{2})}$ operations from 𝖮𝖯 ${\mathsf{OP}}$ . For a random n -bit prime number q , the algorithm is successful as long as the subgroup of ℤ q * ${\mathbb{Z}^{*}_{q}}$ generated by g and the subgroup generated by the element p = 2 ⌊ log 2 ⁡ ( q ) ⌋ ${p=2^{\lfloor\log_{2}(q)\rfloor}}$ share a subgroup of size at least 2 ( 1 - 𝒪 ⁢ ( log ⁡ n / n ) ) ⁢ n ${2^{(1-\mathcal{O}(\log n/n))n}}$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Computing discrete logarithms using 𝒪((log q) 2 ) operations from {+,-,×,÷,&}

Groups Complexity Cryptology , Volume 8 (2) – Nov 1, 2016

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Publisher
de Gruyter
Copyright
Copyright © 2016 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc-2016-0009
Publisher site
See Article on Publisher Site

Abstract

Abstract Given a computational model with registers of unlimited size that is equipped with the set { + , - , × , ÷ , & } = : 𝖮𝖯 ${\{+,-,\times,\div,\&\}=:\mathsf{OP}}$ of unit cost operations, and given a safe prime number q , we present the first explicit algorithm that computes discrete logarithms in ℤ q * ${\mathbb{Z}^{*}_{q}}$ to a base g using only 𝒪 ⁢ ( ( log ⁡ q ) 2 ) ${\mathcal{O}((\log q)^{2})}$ operations from 𝖮𝖯 ${\mathsf{OP}}$ . For a random n -bit prime number q , the algorithm is successful as long as the subgroup of ℤ q * ${\mathbb{Z}^{*}_{q}}$ generated by g and the subgroup generated by the element p = 2 ⌊ log 2 ⁡ ( q ) ⌋ ${p=2^{\lfloor\log_{2}(q)\rfloor}}$ share a subgroup of size at least 2 ( 1 - 𝒪 ⁢ ( log ⁡ n / n ) ) ⁢ n ${2^{(1-\mathcal{O}(\log n/n))n}}$ .

Journal

Groups Complexity Cryptologyde Gruyter

Published: Nov 1, 2016

References