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

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 ${2^{(1-\mathcal{O}(\log n/n))n}}$ .