We propose a GSW-style fully homomorphic encryption scheme over the integers (FHE-OI) that is more efficient than the prior work by Benarroch et al. (PKC 2017). To reduce the expansion of ciphertexts, our scheme consists of two types of ciphertexts: integers and vectors. Moreover, the computational efficiency in the homomorphic evaluation can be improved by hybrid homomorphic operations between integers and vectors. In particular, when performing vector-integer multiplications, the evaluation has the computational complexity of and thus outperforms all prior FHE-OI schemes. To slow down the noise growth in homomorphic multiplications, we introduce a new noise management method called sequentialization; therefore, the noise in the resulting ciphertext increases by a factor of rather than in general multiplications, where is the number of multiplications. As a result, the circuit with larger multiplicative depth can be evaluated under the same parameter settings. Finally, to further reduce the size of ciphertexts, we apply ciphertext truncation and obtain the integer ciphertext of size , thus additionally reducing the size of the vector ciphertext in Benarroch’s scheme from to .

中文翻译：

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整数上的有效GSW风格全同态加密

我们提出了一种整数上的GSW样式完全同态加密方案（FHE-OI），该方案比Benarroch等人的现有工作更有效。（PKC 2017）。为了减少密文的扩展，我们的方案包括两种类型的密文：整数和向量。此外，可以通过整数和向量之间的混合同态运算来提高同态评估中的计算效率。特别地，当执行向量整数乘法时，评估具有计算复杂度，因此胜过所有先前的FHE-OI方案。为了减慢同态乘法中的噪声增长，我们引入了一种称为序列化的新噪声管理方法。因此，所得密文中的噪声增加了而不是一般的乘法，乘法的次数在哪里。结果，可以在相同的参数设置下评估具有较大乘法深度的电路。最后，为了进一步减小密文的大小，我们应用密文截断并获得size的整数密文，从而另外将Benarroch方案中的矢量密文的大小从减小为。