**Linearly Homomorphic Signatures over Binary Fields and New Tools for Lattice-Based Signatures**

*Dan Boneh and David Mandell Freeman*

**Abstract: **We propose a linearly homomorphic signature scheme that authenticates vector
subspaces of a given ambient space. Our system has several novel properties
not found in previous proposals:

- It is the first such scheme that authenticates vectors defined over *binary fields*; previous proposals could only authenticate vectors with large or growing coefficients.

- It is the first such scheme based on the problem of *finding short vectors in integer lattices*, and thus enjoys the worst-case security guarantees common to lattice-based cryptosystems.

Our scheme can be used to authenticate linear transformations of signed data, such as those arising when computing mean and Fourier transform or in networks that use network coding. Our construction gives an example of a cryptographic primitive --- homomorphic signatures over $\mathbb{F}_2$ --- that can be built using lattice methods, but cannot currently be built using bilinear maps or other traditional algebraic methods based on factoring or discrete-log type problems.

Security of our scheme (in the random oracle model) is based on a new hard problem on lattices, called k-SIS, that reduces to standard average-case and worst-case lattice problems. Our formulation of the k-SIS problem adds to the ``toolbox'' of lattice-based cryptography and may be useful in constructing other lattice-based cryptosystems.

As a second application of the new k-SIS tool, we construct an ordinary signature scheme and prove it k-time unforgeable in the standard model assuming the hardness of the k-SIS problem. Our construction can be viewed as ``removing the random oracle'' from the signatures of Gentry, Peikert, and Vaikuntanathan at the expense of only allowing a small number of signatures.

**Category / Keywords: **public-key cryptography / Lattice-based cryptography, homomorphic signatures, k-time signatures

**Publication Info: **An extended abstract of this paper will appear in PKC 2011.

**Date: **received 20 Aug 2010, last revised 9 May 2011

**Contact author: **dfreeman at cs stanford edu

**Available format(s): **PDF | BibTeX Citation

**Note: **New version with many updates.

**Version: **20110509:221952 (All versions of this report)

**Short URL: **ia.cr/2010/453

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