  Modern information communications use cryptography to keep the contents of
communications confidential. RSA (Rivest-Shamir-Adleman) cryptography and
elliptic curve cryptography, which are public-key cryptosystems, are widely
used cryptographic schemes. However, it is known that these cryptographic
schemes can be deciphered in a very short time by Shor's algorithm when a
quantum computer is put into practical use. Therefore, several methods have
been proposed for quantum computer-resistant cryptosystems that cannot be
cracked even by a quantum computer. A simple implementation of LWE-based
lattice cryptography based on the LWE (Learning With Errors) problem requires a
key length of $O(n^2)$ to ensure the same level of security as existing
public-key cryptography schemes such as RSA and elliptic curve cryptography. In
this paper, we attacked the Ring-LWE (RLWE) scheme, which can be implemented
with a short key length, with a modified LLL (Lenstra-Lenstra-Lov\'asz) basis
reduction algorithm and investigated the trend in the degree of field extension
required to generate a secure and small key. Results showed that the
lattice-based cryptography may be strengthened by employing Cullen or Mersenne
prime numbers as the degree of field extension.
