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Now, we analyze the effectiveness of the ideal decomposition attack in detail.
Although the one-wayness of ASC was broken by the ideal decomposition attack [12], our analysis (section 4) shows that our cryptosystem has resistance against some possible attacks including the ideal decomposition attack.
However, in [12] Faugére and Spaenlehauer proposed a new attack called the ideal decomposition attack and claimed that this attack can fully break ASC.
Although we have not been able to give a security proof, we give some discussions on how secure our cryptosystem is against known attacks including the ideal decomposition attack, which can break the one-wayness of ASC.
We need to have (Lambda _{X} = Lambda _{tilde {m}} = Lambda _{f} = Lambda _{s_{i}} = Lambda _{r_{i}}) and translate m into (tilde {m}) to avoid the ideal decomposition attack and other attacks (see section 4).
Note that, because the absolute value of coefficients of f are as large as those of (tilde {m}), the above argument implies that choosing N satisfying N d>2128 H X) may complicate finding f from the ideal J or I 1. Next, we analyze the effectiveness of the ideal decomposition attack of Level 3 (see [12], section 3.3).
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