Abstract
We present a new formulation and its simpler analysis of the lattice-based attack of Boneh and Durfee for the RSA cryptography [1]. We follow the same approach as theirs, however, we propose a new way of defining a lattice with which we can achieve the same solvable key bound \(d<N^{0.292}\). Our lattice is represented as a lower triangle matrix, which makes its analysis much simpler than that of [1]. We think that our analysis technique would be useful for considering applications/generalizations of this approach.
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Notes
- 1.
Our \(g_{i,j}(x,y)\) for \(i\ge j\) and for \(i<j\) correspond to their \(g_{i,j}(x,y)\) and \(h_{i,j}(x,y)\) respectively.
- 2.
For example, for \(m=3\) and \(\delta =0.25\); we have \(\mathbf {I}_1 = ((0,0),(1,0),(1,1),(2,0),\) \((2,1),(2,2),(3,0),(3,1),(3,2),(3,3))\) and \(\mathbf {I}_2 = ((2,3),(3,4))\). By them, we have the monomial sequence \(\mathbf {K}_1 = (1,x,z,x^2,xz,z^2,x^3,x^2z,xz^2,z^3)\) and \(\mathbf {K}_2 = (yz^2,yz^3)\).
- 3.
This \(\delta \) is L\(^2\) algorithm’s parameter and different from \(\delta \) used for defining RSA instance. See the original paper [9] about this \(\delta \).
- 4.
TSUBAME is a grid type supercomputer at Tokyo Inst. of Tech. A node of the supercomputer which we used contains eight Opteron Dual Core model 880 processors of 2.4 GHz and 32 GB RAM. Note, however, we have not been able to make a parallel version of our algorithm; it was used only for reducing the total experiment time.
References
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Acknowledgments
I am grateful to Osamu Watanabe for his advice, careful reading, and for correcting some expressions. The author and this research was supported in part by the JSPS Global COE program “Computationism as a Foundation for the Sciences.”
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Aono, Y. (2014). Simplification of the Lattice Based Attack of Boneh and Durfee for RSA Cryptoanalysis. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_3
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DOI: https://doi.org/10.1007/978-3-662-43799-5_3
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