Harper

seal库BGV同态加密解析

seal库参数设置

先看seal库源码

const map<size_t, vector<Modulus>> &GetDefaultCoeffModulus128()
            {
                static const map<size_t, vector<Modulus>> default_coeff_modulus_128{
                    /*
                    Polynomial modulus: 1x^1024 + 1
                    Modulus count: 1
                    Total bit count: 27
                    */
                    { 1024, { 0x7e00001 } },

                    /*
                    Polynomial modulus: 1x^2048 + 1
                    Modulus count: 1
                    Total bit count: 54
                    */
                    { 2048, { 0x3fffffff000001 } },

                    /*
                    Polynomial modulus: 1x^4096 + 1
                    Modulus count: 3
                    Total bit count: 109 = 2 * 36 + 37
                    */
                    { 4096, { 0xffffee001, 0xffffc4001, 0x1ffffe0001 } },

                    /*
                    Polynomial modulus: 1x^8192 + 1
                    Modulus count: 5
                    Total bit count: 218 = 2 * 43 + 3 * 44
                    */
                    { 8192, { 0x7fffffd8001, 0x7fffffc8001, 0xfffffffc001,                               0xffffff6c001, 0xfffffebc001 } },

                    /*
                    Polynomial modulus: 1x^16384 + 1
                    Modulus count: 9
                    Total bit count: 438 = 3 * 48 + 6 * 49
                    */
                    { 16384,
                      { 0xfffffffd8001, 0xfffffffa0001, 0xfffffff00001, 0x1fffffff68001, 0x1fffffff50001,
                        0x1ffffffee8001, 0x1ffffffea0001, 0x1ffffffe88001, 0x1ffffffe48001 } },

                    /*
                    Polynomial modulus: 1x^32768 + 1
                    Modulus count: 16
                    Total bit count: 881 = 15 * 55 + 56
                    */
                    { 32768,
                      { 0x7fffffffe90001, 0x7fffffffbf0001, 0x7fffffffbd0001, 0x7fffffffba0001, 0x7fffffffaa0001,
                        0x7fffffffa50001, 0x7fffffff9f0001, 0x7fffffff7e0001, 0x7fffffff770001, 0x7fffffff380001,
                        0x7fffffff330001, 0x7fffffff2d0001, 0x7fffffff170001, 0x7fffffff150001, 0x7ffffffef00001,
                        0xfffffffff70001 } }
                };

                return default_coeff_modulus_128;
            }

这是BGV的128bit安全性参数设置，随着模多项式的次数\(n\)不同，对应的密文模数的大小\(log(q)\)也有调整，具体的公式可以按照下面的方法计算： \[ \lambda \approx -log(\cfrac{A*log(q)}{n})\cfrac{Bn}{logq} + C\sqrt{\cfrac{logq}{n}}log(\cfrac{n}{log(q)}) \]

BGV噪声变化