用低密度格码改进GGH公钥方案
Improving GGH Public Key Scheme Using Low Density Lattice Codes
摘要：
Goldreich-Goldwasser-Halevi（GGH）公钥密码体制是格密码体制的一个实例，其安全性是基于格问题的难解性。GGH密码体制实际上是McEliece提出的第一个基于码的密码体制的格型版本。然而，它也有一些缺点，例如：大的公钥长度和低的安全级别。另一方面，低密度格码（LDLC）是一类实用的格码，它可以在加性高斯白噪声（AWGN）信道上以低复杂度的译码算法获得容量。针对GGH密码体制的不足，提出了一种基于LDLC的公钥密码体制。为了减少密钥长度，我们采用Hermite范式（HNF）中使用的LDLC的生成矩阵作为公钥。同时，利用LDLC的线性译码复杂度，与GGH密码体制相比降低了解密复杂度。这些提高的效率使我们能够使用更大的安全参数值。此外，我们利用特殊的高斯向量，其方差是上界的Poltyrev极限作为扰动向量。这些技术可以抵抗所提出的方案对最有效的攻击GGH类密码系统。
英文摘要：
Goldreich-Goldwasser-Halevi (GGH) public key cryptosystem is an instance of lattice-based cryptosystems whose security is based on the hardness of lattice problems. In fact, GGH cryptosystem is the lattice version of the first code-based cryptosystem, proposed by McEliece. However, it has a number of drawbacks such as; large public key length and low security level. On the other hand, Low Density Lattice Codes (LDLCs) are the practical classes of lattice codes which can achieve capacity on the additive white Gaussian noise (AWGN) channel with low complexity decoding algorithm. This paper introduces a public key cryptosystem based on LDLCs to withdraw the drawbacks of GGH cryptosystem. To reduce the key length, we employ the generator matrix of the used LDLC in Hermite normal form (HNF) as the public key. Also, by exploiting the linear decoding complexity of the used LDLC, the decryption complexity is decreased compared with GGH cryptosystem. These increased efficiencies allow us to use the bigger values of security parameters. Moreover, we exploit the special Gaussian vector whose variance is upper bounded by the Poltyrev limit as the perturbation vector. These techniques can resist the proposed scheme against the most efficient attacks to the GGH-like cryptosystems.
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