Trapdoor One-way Functions From Tensors

Weyman and Zelevinsky generalised Vandermonde matrices to higher dimensions, which we call Vandermonde-Weyman-Zelevinsky tensors [35]. We generalise Lagrange interpolation to higher dimensions by devising a nearly linear time algorithm that given a Vandermonde-Weyman-Zelevinsky tensor and a sparse target vector, finds a tuple of vectors that hit the target under tensor evaluation. Tensor evaluation to us means evaluating the usual multilinear form associated with the tensor in all but one chosen dimension. Yet, this interpolation problem phrased with respect to a random tensor appears to be a hard multilinear system. Leveraging this dichotomy, we propose preimage sampleable trapdoor one-way functions in the spirit of Gentry-Peikert-Vaikuntanathan (GPV) lattice trapdoors [16]. We design and analyse “Hash-and-Sign” digital signatures from such trapdoor one-way functions, yielding short signatures whose lengths scale nearly linearly in the security parameter. We also describe an encryption scheme.

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Our trapdoor is a random Vandermonde-Weyman-Zelevinsky tensor over a finite field and a random basis change. We hide the Vandermonde-Weyman-Zelevinsky tensor under the basis change and publish the resulting pseudorandom tensor. The one way function is the tensor evaluation derived from the public tensor, restricted so as to only map to sparse vectors. We then design the domain sampler and preimage sampler demanded by the GPV framework. The former samples inputs that map to uniform images under the one-way function. The latter samples preimages given supplementary knowledge of the trapdoor. Preimage sampling is a randomised version of interpolation and knowing the basis change allows efficient translation between interpolation corresponding to the public and trapdoor tensors. An adversary seeking a preimage must solve a pseudorandom multilinear system, which seems cryptographically hard.