Digital signatures are an important primitive for building secure systems and are used in most real world security protocols. However, almost all popular signature schemes are either based on the factoring as-sumption (RSA) or the hardness of the discrete logarithm problem (DSA/ECDSA). In the case of classical cryptanalytic advances or progress on the development of quantum computers the hardness of these closely related problems might be seriously weakened. A potential alternative approach is the construction of sig-nature schemes based on the hardness of certain lattices problems which are assumed to be intractable by quantum computers. Due to significant research advancements in recent years, lattice-based schemes have now become practical and appear to be a very viable alternative to number-theoretic cryptography. In this paper we focus on recent developments and the current state-of-the-art in lattice-based digital signatures and provide a comprehensive survey discussing signature schemes with respect to practicality. Additionally, we discuss future research areas that are essential for the continued development of lattice-based cryptography.

I would like to thank my parents for always believing in me. I would also like to thank my supervisors Richard Lindner and Johannes Buchmann for their useful comments and suggestions on how to improve the queality of the thesis. Not on last place I would like to thank Vadim Lyubashevsky and Luis Carlos Coronado Garcia for their kindness and readiness to answer my questions. Warranty I hereby warrant that the content of this thesis is the direct result of my own work and that any use made in it of published or unpublished material is fully and correctly referenced. I also warrant that the presented work has

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