, Volume 26, Issue 2, pp 225245
First online:
On the Analysis of Cryptographic Assumptions in the Generic Ring Model
 Tibor JagerAffiliated withInstitut für Kryptographie und Sicherheit, Karlsruhe Institute of Technology Email author
 , Jörg SchwenkAffiliated withHorst Görtz Institute for IT Security, RuhrUniversity Bochum
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The generic ring model considers algorithms that operate on elements of an algebraic ring by performing only the ring operations and without exploiting properties of a given representation of ring elements. It is used to analyze the hardness of computational problems defined over rings. For instance, it is known that breaking RSA is equivalent to factoring in the generic ring model (Aggarwal and Maurer, Eurocrypt 2009). Do hardness results in the generic ring model support the conjecture that solving the considered problem is also hard in the standard model, where elements of ℤ_{ n } are represented by integers modulo n?
We prove in the generic ring model that computing the Jacobi symbol of an integer modulo n is equivalent to factoring. Since there are simple and efficient nongeneric algorithms which compute the Jacobi symbol, this provides an example of a natural computational problem which is hard in the generic ring model, but easy to solve if elements of ℤ_{ n } are given in their standard representation as integers. Thus, a proof in the generic ring model is unfortunately not a very strong indicator for the hardness of a computational problem in the standard model.
Despite this negative result, generic hardness results still provide a lower complexity bound for a large class of algorithms, namely all algorithms solving a computational problem independent of a given representation of ring elements. From this point of view, results in the generic ring model are still interesting. Motivated by this fact, we also show that solving the quadratic residuosity problem generically is equivalent to factoring.
Key words
Generic ring model Jacobi symbol Subset membership problems Idealized models of computation Quadratic residuosity assumption Title
 On the Analysis of Cryptographic Assumptions in the Generic Ring Model
 Journal

Journal of Cryptology
Volume 26, Issue 2 , pp 225245
 Cover Date
 201304
 DOI
 10.1007/s001450129120y
 Print ISSN
 09332790
 Online ISSN
 14321378
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Generic ring model
 Jacobi symbol
 Subset membership problems
 Idealized models of computation
 Quadratic residuosity assumption
 Industry Sectors
 Authors

 Tibor Jager ^{(1)}
 Jörg Schwenk ^{(2)}
 Author Affiliations

 1. Institut für Kryptographie und Sicherheit, Karlsruhe Institute of Technology, Karlsruhe, Germany
 2. Horst Görtz Institute for IT Security, RuhrUniversity Bochum, Bochum, Germany