Reference : Cubic Sieve Congruence of the Discrete Logarithm Problem, and fractional part sequences
 Document type : Scientific journals : Article Discipline(s) : Engineering, computing & technology : Computer science To cite this reference: http://hdl.handle.net/10993/21828
 Title : Cubic Sieve Congruence of the Discrete Logarithm Problem, and fractional part sequences Language : English Author, co-author : Venkatesh, Srinivas Vivek [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC) >] C. E. Veni Madhavan [Indian Institute of Science, Bangalore, India > Department of Computer Science and Automation] Publication date : 2014 Journal title : Journal of Symbolic Computation Publisher : Academic Press Volume : 64 Pages : 22-34 Peer reviewed : Yes (verified by ORBilu) Audience : International ISSN : 0747-7171 e-ISSN : 1095-855X Keywords : [en] Computational number theory ; Cryptanalysis ; Diophantine equation ; Discrete Logarithm Problem ; Fractional part sequence Abstract : [en] The Cubic Sieve Method for solving the Discrete Logarithm Problem in prime fields requires a nontrivial solution to the Cubic Sieve Congruence (CSC) $x^3 \equiv y^2 z (mod p)$, where 'p' is a given prime number. A nontrivial solution must also satisfy $x^3 \neq y^2 z$ and $1 <= x,y,z < p^a$, where 'a' is a given real number such that $1/3 < a <= 1/2$. The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. CSC can be parametrized as $x \equiv v^2 z (mod p)$ and $y \equiv v^3 z (mod p)$. In this paper, we give a deterministic polynomial-time ($O(ln^3 p)$ bit-operations) algorithm to determine, for a given 'v', a nontrivial solution to CSC, if one exists. Previously it took $õ(p^a)$ time in the worst case to determine this. We relate the CSC problem to the gap problem of fractional part sequences, where we need to determine the non-negative integers 'N' satisfying the fractional part inequality ${\theta N} < \phi$ ($\theta$ and $\phi$ are given real numbers). The correspondence between the CSC problem and the gap problem is that determining the parameter 'z' in the former problem corresponds to determining 'N' in the latter problem. We also show in the $a = 1/2$ case of CSC that for a certain class of primes the CSC problem can be solved deterministically in $õ(p^{1/3})$ time compared to the previous best of $õ(p^{1/2})$. It is empirically observed that about one out of three primes is covered by the above class. Target : Researchers ; Professionals ; Students Permalink : http://hdl.handle.net/10993/21828 DOI : 10.1016/j.jsc.2013.12.004 Mentions required by the publisher for OA : The final publication is available at http://www.sciencedirect.com/science/article/pii/S0747717113001703.

File(s) associated to this reference

Fulltext file(s):

FileCommentaryVersionSizeAccess
Open access
csc_fps_jsc-final.pdfAuthor postprint470.87 kBView/Open

All documents in ORBilu are protected by a user license.