Abstract
In this work, we investigate the hardness of learning Burnside homomorphisms with noise (\(B_{n} \hbox {-}\mathsf {LHN}\)), a computational problem introduced in the recent work of Baumslag et al. This is a generalization of the learning with errors problem, instantiated with a particular family of non-abelian groups, known as free Burnside groups of exponent 3. In our main result, we demonstrate a random self-reducibility property for \(B_{n} \hbox {-}\mathsf {LHN}\). Along the way, we also prove a sequence of lemmas regarding homomorphisms of free Burnside groups of exponent 3 that may be of independent interest.
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Notes
This argument requires the existence of at least one \(g\) such that \(g\cdot s = t\); we are given such a \(g\) by transitivity.
We recall that the size of \(B_{r}\) is independent of the security parameter. Thus, enumerating all elements of \(B_{r}\) takes \({\mathcal {O}}(1)\) time.
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Acknowledgments
The authors are grateful to the anonymous reviewers for their detailed and thoughtful comments that helped improve the presentation of the results. Nelly Fazio’s research is sponsored in part by NSF CAREER award #1253927, by the U.S. Army Research Laboratory and the U.K. Ministry of Defence under Agreement Number W911NF-06-3-0001, and by PSC-CUNY award 64578-00 42 (jointly funded by The Professional Staff Congress and The City University of New York). Nelly Fazio and William E. Skeith are sponsored in part by NSF award #1117675 and CUNY Round 19 CIRG award. Antonio Nicolosi’s research is sponsored in part by NSF awards #1117679 and #1040784. Ludovic Perret’s research is supported in part by the French ANR under the Computer Algebra and Cryptography (CAC) project ANR-09-JCJCJ-0064-01.
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Communicated by D. Jungnickel.
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Fazio, N., Iga, K., Nicolosi, A.R. et al. Hardness of learning problems over Burnside groups of exponent 3. Des. Codes Cryptogr. 75, 59–70 (2015). https://doi.org/10.1007/s10623-013-9892-6
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DOI: https://doi.org/10.1007/s10623-013-9892-6
Keywords
- Learning with errors
- Post-quantum cryptography
- Non-commutative cryptography
- Burnside groups
- Random self-reducibility