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Algorithms for the Generalized NTRU Equations and their Storage Analysis

Cho Gook Hwa, Lim Seongan, Lee Hyang-Sook

Fundamenta Informaticae

2020

Vol. 177, nr 2

115--139

In LATTE, a lattice based hierarchical identity-based encryption (HIBE) scheme, each hierarchical level user delegates a trapdoor basis to the next level by solving a generalized NTRU equation of level ℓ ≥ 3. For ℓ = 2, Howgrave-Graham, Pipher, Silverman, and Whyte presented an algorithm using resultant and Pornin and Prest presented an algorithm using a field norm with complexity analysis. Even though their ideas of solving NTRU equations can be conceptually extended for ℓ ≥ 3, no explicit algorithmic extensions with the storage analysis are known so far. In this paper, we interpret the generalized NTRU equation as the determinant of a matrix. By using the mathematical properties of the determinant, we show that how to construct algorithms for solving the generalized NTRU equation either using resultant or a field norm for any ℓ ≥ 3. We also obtain an upper bound of the size of solutions by using the properties of the determinant. From our analysis, the storage requirement of the algorithm using resultant is O (ℓ2 n 2 logB ) and that of the algorithm using a field norm is O (ℓ2 n logB ), where B is an upper bound of the coefficients of the input polynomials of the generalized NTRU equations. We present examples of our algorithms for ℓ = 3 and the average storage requirements for ℓ = 3, 4.

Cryptanalysis of Two Efficient HIBE Schemes in the Standard Model

Wang X.A., Yang Y., Zhang M.

Fundamenta Informaticae

2011

Vol. 109, nr 2

189-200

In Informatica 32 (2008), Ren and Gu proposed an anonymous hierarchical identity based encryption scheme based on the q-ABDHE problem with full security in the standard model. Later in Indocrypt’08, they proposed another secure hierarchical identity based encryption scheme based on the q-TBDHE problem with full security in the standard model. They claimed that their schemes have short parameters, high efficiency and tight reduction. However, in this paper we give attacks to show their schemes are not secure at all. Concretely, from any first level private key, the adversary can easily derive a “private key” which can decrypt any ciphertexts for the target identity. That is to say, a query on any first level identity is enough to decrypt any ciphertext in the system.