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Though the multivariable cryptosystems first suggested by Matsumoto and Imai was defeated by the linearization method of Patarin due to the special properties of the Matsumoto-Imai (MI) cryptosystem, many variants and extensions of the MI system were suggested mainly by Patarin and his collaborators. In this paper, we propose a new variant of the MI system, which was inspired by the idea of “perturbation”. This method uses a set of r (a small number) linearly independent linear functions z_i=∑_{j=1}^n α_{ij}x_j + β_i, i=1,..,r, over the variables x_i , which are variables of the MI system. The perturbation is performed by adding random quadratic function of z i to the MI systems. The difference between our idea and a very similar idea of the Hidden Field Equation and Oil-Vinegar system is that our perturbation is internal, where we do not introduce any new variables, while the Hidden Field Equation and Oil-Vinegar system is an “external” perturbation of the HFE system, where a few extra (external) new variables are introduced to perform the perturbation. A practical implementation example of 136 bits, its security analysis and efficiency analysis are presented. The attack complexity of this perturbed Matsumoto-Imai cryptosystem is estimated.

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