Vandermonde meets Regev: public key encryption schemes based on partial Vandermonde problems

Boudgoust, Katharina; Sakzad, Amin; Steinfeld, Ron

doi:10.1007/s10623-022-01083-7

Cited by 3 publications

(3 citation statements)

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“…In 2015, Hoffstein and Silverman [HS15] designed a public key encryption scheme called PASS Encrypt whose mathematical building blocks resemble those of PASS Sign. Later, the scheme was slightly modified in order to provide a proof of security with respect to concretely defined hardness assumptions by Boudgoust et al [BSS22], accessible via one of the author's thesis manuscript [Bou21, Ch. 5+7].…”

Section: Implications To Cryptographymentioning

confidence: 99%

“…Partial Vandermonde Learning With Errors. Boudgoust et al [BSS22,Bou21] introduce a dual problem to PV-Knap that they call partial Vandermonde learning with errors (PV-LWE). The duality connection is similar as the one between the standard Knapsack and LWE problems.…”

Section: Appendix a Additional Preliminariesmentioning

confidence: 99%

“…The partial Vandermonde knapsack problem (PV-Knap) asks, given V Ω (g), to recover g(X). 3 As observed by Boudgoust et al [BSS22,Bou21], recovering a short polynomial while having access only to a partial list of its Vandermonde transform can be seen as a problem over an ideal lattice. More precisely, in the mathematical setting above, we know that the ideal generated by q in the mth cyclotomic ring O K = Z[X]/Φ m (X) completely splits into d prime ideals, where Φ m (X) denotes the m-th cyclotomic polynomial.…”

Section: Introductionmentioning

confidence: 99%

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Some Easy Instances of Ideal-SVP and Implications on the Partial Vandermonde Knapsack Problem

Boudgoust

Gachon

Pellet-Mary

2022

Lecture Notes in Computer Science

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In this article, we generalize the works of Pan et al. (Eurocrypt'21) and Porter et al. (ArXiv'21) and provide a simple condition under which an ideal lattice defines an easy instance of the shortest vector problem. Namely, we show that the more automorphisms stabilize the ideal, the easier it is to find a short vector in it. This observation was already made for prime ideals in Galois fields, and we generalize it to any ideal (whose prime factors are not ramified) of any number field. We then provide a cryptographic application of this result by showing that particular instances of the partial Vandermonde knapsack problem, also known as partial Fourier recovery problem, can be solved classically in polynomial time. As a proof of concept, we implemented our attack and managed to solve those particular instances for concrete parameter settings proposed in the literature. For random instances, we can halve the lattice dimension with non-negligible probability.

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Section: Implications To Cryptographymentioning

confidence: 99%

Section: Appendix a Additional Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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