Algorithms for the Generalized NTRU Equations and their Storage Analysis

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(ℓ2n2 logB) and that of the algorithm using a field norm is O(ℓ2n 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.