## Algorithms for the generalized NTRU equations and their storage analysis.(English)Zbl 07350027

Summary: 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 $$\ell\geq 3$$. For $$\ell=2$$, 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 $$\ell\geq 3$$, 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 $$\ell\geq 3$$. 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 $$O(\ell^2n^2\log B)$$ and that of the algorithm using a field norm is $$O(\ell^2n\log B)$$, 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 $$\ell=3$$ and the average storage requirements for $$\ell=3,4$$.

### MSC:

 68-XX Computer science

### Keywords:

NTRU; LATTE; hierarchical identity-based encryption

### Software:

NTRU; NTRUSign; Falcon
Full Text:

### References:

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