This week's study group was given by Emmanuela on the subject of digital signature schemes based on lattices. Because not everyone likes lattices as much as I do, Emmanuela decided to first convey some of the history of lattice signatures to the audience. The seminal work that everyone refers to when they talk about any sort of lattice-based cryptography is of course Ajtai's paper from '96 on the connection between the average and worst cases of certain lattice problems. Around the same time, the NTRU and GGH cryptosystems were proposed which provided both encryption and digital signature schemes. However, both schemes had their initial issues, and it turned out that especially the security of the digital signature schemes proved to be problematic. In hindsight it is pretty clear that the problem lies with the 'noise' distribution and that signatures leak information on the secret key.

The next steps towards the security of lattice-based signature schemes were taken in '08, when two independent works described schemes that are provably secure based on the hardness of standard lattice problems. The first is by Gentry, Peikert and Vaikuntanathan, which follows the hash-and-sign paradigm and includes a useful method to sample from the discrete Gaussian distribution, which is used in most modern lattice-based crypto schemes. As the word 'hash' implies, the security proof for this scheme is in the random oracle model. The second scheme is by Lyubashevsky and Micciancio in the standard model, but it only provides one-time secure signatures. These can be converted into fully secure signatures using a tree construction, which requires a logarithmic number (in the security parameter) of applications of the one-time scheme.

These two works inspired two different lines of research. One line focuses on getting the best efficiency in the random oracle model, whereas the other focuses on getting security in the standard model while maintaining a good asymptotic efficiency as well. The focus paper of the study group was in this second line of research: Improved Short Lattice Signatures in the Standard Model by Ducas and Micciancio from this year's Crypto. It combines the 'vanishing trapdoor' technique due to Boyen and the 'confined guessing' method due to Böhl et al. For their lattice trapdoors, they use the work of Micciancio and Peikert from Eurocrypt '12. This non-trivial combination leads to a scheme where the signatures are short (consisting of only one vector) at the cost of having keys consisting of a logarithmic number of vectors. They also propose a stateful scheme which reduces the key sizes to a log log number of vectors and also tightens the reduction, removing a factor introduced by the confined guessing stage as well as tightening the approximation factor of the underlying lattice problem. Interestingly, the schemes by Ducas and Micciancio require the additional algebraic structure of ideal lattices, whereas previous works only use this structure for the efficiency improvement.

In conclusion, the result is a new scheme that compares favourably to previous schemes by either allowing for smaller signatures or smaller keys. But things move fast in the lattice world, as there is already a new paper on the ePrint archive that reduces the keys to a constant number of vectors, at the cost of a bigger approximation factor in the underlying lattice problem. It is still possible to choose parameters such that this approximation is polynomial, but it is also possible to pick them less conservatively, resulting in a subexponential approximation factor. It will be interesting to see whether such choices survive future improvements to cryptanalysis.

The next steps towards the security of lattice-based signature schemes were taken in '08, when two independent works described schemes that are provably secure based on the hardness of standard lattice problems. The first is by Gentry, Peikert and Vaikuntanathan, which follows the hash-and-sign paradigm and includes a useful method to sample from the discrete Gaussian distribution, which is used in most modern lattice-based crypto schemes. As the word 'hash' implies, the security proof for this scheme is in the random oracle model. The second scheme is by Lyubashevsky and Micciancio in the standard model, but it only provides one-time secure signatures. These can be converted into fully secure signatures using a tree construction, which requires a logarithmic number (in the security parameter) of applications of the one-time scheme.

These two works inspired two different lines of research. One line focuses on getting the best efficiency in the random oracle model, whereas the other focuses on getting security in the standard model while maintaining a good asymptotic efficiency as well. The focus paper of the study group was in this second line of research: Improved Short Lattice Signatures in the Standard Model by Ducas and Micciancio from this year's Crypto. It combines the 'vanishing trapdoor' technique due to Boyen and the 'confined guessing' method due to Böhl et al. For their lattice trapdoors, they use the work of Micciancio and Peikert from Eurocrypt '12. This non-trivial combination leads to a scheme where the signatures are short (consisting of only one vector) at the cost of having keys consisting of a logarithmic number of vectors. They also propose a stateful scheme which reduces the key sizes to a log log number of vectors and also tightens the reduction, removing a factor introduced by the confined guessing stage as well as tightening the approximation factor of the underlying lattice problem. Interestingly, the schemes by Ducas and Micciancio require the additional algebraic structure of ideal lattices, whereas previous works only use this structure for the efficiency improvement.

In conclusion, the result is a new scheme that compares favourably to previous schemes by either allowing for smaller signatures or smaller keys. But things move fast in the lattice world, as there is already a new paper on the ePrint archive that reduces the keys to a constant number of vectors, at the cost of a bigger approximation factor in the underlying lattice problem. It is still possible to choose parameters such that this approximation is polynomial, but it is also possible to pick them less conservatively, resulting in a subexponential approximation factor. It will be interesting to see whether such choices survive future improvements to cryptanalysis.

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