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The correctness in decrypting a ciphertext after some operations in the DGVH scheme depends heavily on the dimension of the secret key. In this paper we compute two bounds on the size of the secret key for the DGHV scheme to decrypt correctly a ciphertext after a fixed number of additions and a fixed number of multiplication. Moreover we improve the original bound on the dimension of the secret key for a general circuit. [ABSTRACT FROM AUTHOR]

In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve H over the finite field F q 2 . We focus on those with distance d ≥ q 2 − q and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing μ q + λ (q + 1) of the distance d with μ , λ non negative integers, and μ ≤ q , and consider all the curves X of the affine plane A F q 2 2 of degree μ + λ defined by polynomials with x μ y λ as leading monomial with respect to the DegRevLex term ordering (with y > x). We prove that a zero-dimensional subscheme Z of A F q 2 2 is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple F q 2 -points and there is a curve X such that Z coincides with the scheme theoretic intersection H ∩ X (namely, as a cycle, Z = H ⋅ X). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords. [ABSTRACT FROM AUTHOR]