C：用固定时间代码替换AES FIPS-197 SubBytes表

在FIPS-197（高级加密标准，称为AES）中，大量使用SubBytes，可以将其实现为

unsigned char SubBytes(unsigned char x) {
static const unsigned char t[256] = {
  0x63,0x7C,0x77,0x7B,0xF2,0x6B,0x6F,0xC5,0x30,0x01,0x67,0x2B,0xFE,0xD7,0xAB,0x76,
  0xCA,0x82,0xC9,0x7D,0xFA,0x59,0x47,0xF0,0xAD,0xD4,0xA2,0xAF,0x9C,0xA4,0x72,0xC0,
  0xB7,0xFD,0x93,0x26,0x36,0x3F,0xF7,0xCC,0x34,0xA5,0xE5,0xF1,0x71,0xD8,0x31,0x15,
  0x04,0xC7,0x23,0xC3,0x18,0x96,0x05,0x9A,0x07,0x12,0x80,0xE2,0xEB,0x27,0xB2,0x75,
  0x09,0x83,0x2C,0x1A,0x1B,0x6E,0x5A,0xA0,0x52,0x3B,0xD6,0xB3,0x29,0xE3,0x2F,0x84,
  0x53,0xD1,0x00,0xED,0x20,0xFC,0xB1,0x5B,0x6A,0xCB,0xBE,0x39,0x4A,0x4C,0x58,0xCF,
  0xD0,0xEF,0xAA,0xFB,0x43,0x4D,0x33,0x85,0x45,0xF9,0x02,0x7F,0x50,0x3C,0x9F,0xA8,
  0x51,0xA3,0x40,0x8F,0x92,0x9D,0x38,0xF5,0xBC,0xB6,0xDA,0x21,0x10,0xFF,0xF3,0xD2,
  0xCD,0x0C,0x13,0xEC,0x5F,0x97,0x44,0x17,0xC4,0xA7,0x7E,0x3D,0x64,0x5D,0x19,0x73,
  0x60,0x81,0x4F,0xDC,0x22,0x2A,0x90,0x88,0x46,0xEE,0xB8,0x14,0xDE,0x5E,0x0B,0xDB,
  0xE0,0x32,0x3A,0x0A,0x49,0x06,0x24,0x5C,0xC2,0xD3,0xAC,0x62,0x91,0x95,0xE4,0x79,
  0xE7,0xC8,0x37,0x6D,0x8D,0xD5,0x4E,0xA9,0x6C,0x56,0xF4,0xEA,0x65,0x7A,0xAE,0x08,
  0xBA,0x78,0x25,0x2E,0x1C,0xA6,0xB4,0xC6,0xE8,0xDD,0x74,0x1F,0x4B,0xBD,0x8B,0x8A,
  0x70,0x3E,0xB5,0x66,0x48,0x03,0xF6,0x0E,0x61,0x35,0x57,0xB9,0x86,0xC1,0x1D,0x9E,
  0xE1,0xF8,0x98,0x11,0x69,0xD9,0x8E,0x94,0x9B,0x1E,0x87,0xE9,0xCE,0x55,0x28,0xDF,
  0x8C,0xA1,0x89,0x0D,0xBF,0xE6,0x42,0x68,0x41,0x99,0x2D,0x0F,0xB0,0x54,0xBB,0x16};
return t[x];}

此函数不是任意的；它是可逆的映射，包括在Galois场中进行反演然后进行仿射变换。所有详细信息均在FIPS-197第5.1.1节或此处的第4.2.1节（名称略有不同）中。

以表的形式实现的一个问题是它容易受到所谓的缓存定时攻击。

因此，您的任务是为显示上述SubBytes()功能的确切功能设计一个精确的替代品；我们假设情况是这样的：不使用任何依赖于输入x的SubBytes内容：

• 作为数组索引，
• 作为控制的操作数if，while，for，case，或操作者?:;
• 由于运营商的任何操作数&&，||，!，==，!=，<，>，<=，>=，*，/，%;
• 作为运营商的右操作数>>，<<，*=，/=，%=，<<=，>>=。

中奖条目将是具有最低成本，从输入相关的数据路径执行操作符的数量而获得，具有重量的5为一元运算符-和~以及为<<1，>>1，+1，-1; 所有其他运算符的权重为7，与其他计数一起移位或其他常数的加/减（类型转换和提升是免费的）。原则上，通过展开循环（如果有），该成本不会改变，并且与输入无关x。作为决胜局，删除空格和注释后具有最短代码的答案将获胜。

我计划最早在2013年UTC时指定一个条目作为答案。我将考虑用我所了解的语言来回答，将其作为对C的直接翻译（未针对大小进行优化）进行排名。

_{对于最初被遗漏+1并受其-1青睐的运营商，免费演员和晋升以及规模排名的歉意。请注意，*一元和乘法时都禁止这样做。}

得分：940933926910，现场塔式进场

public class SBox2
{
    public static void main(String[] args)
    {
        for (int i = 0; i < 256; i++) {
            int s = SubBytes(i);
            System.out.format("%02x  ", s);
            if (i % 16 == 15) System.out.println();
        }
    }

    private static int SubBytes(int x) {
        int fwd;
        fwd  = 0x010001 & -(x & 1); x >>= 1; //   7+5+7+5+ | 24+
        fwd ^= 0x1d010f & -(x & 1); x >>= 1; // 7+7+5+7+5+ | 31+
        fwd ^= 0x4f020b & -(x & 1); x >>= 1; // 7+7+5+7+5+ | 31+
        fwd ^= 0x450201 & -(x & 1); x >>= 1; // 7+7+5+7+5+ | 31+
        fwd ^= 0xce080d & -(x & 1); x >>= 1; // 7+7+5+7+5+ | 31+
        fwd ^= 0xa20f0f & -(x & 1); x >>= 1; // 7+7+5+7+5+ | 31+
        fwd ^= 0xc60805 & -(x & 1); x >>= 1; // 7+7+5+7+5+ | 31+
        fwd ^= 0x60070e & -x;                // 7+7+5+     | 19+

        // Running total so far: 229

        int p1;
        {
            int ma = fwd;
            int mb = fwd >> 16;         // 7+         | 7+
            p1  = ma & -(mb&1); ma<<=1; //   7+5+7+5+ | 24+
            p1 ^= ma & -(mb&2); ma<<=1; // 7+7+5+7+5+ | 31+
            p1 ^= ma & -(mb&4); ma<<=1; // 7+7+5+7+5+ | 31+
            p1 ^= ma & -(mb&8);         // 7+7+5+7+   | 26+
            int t = p1 >> 3;            // 7+         | 7+
            p1 ^= (t >> 1) ^ (t & 0xe); // 7+5+7+7+   | 26+
        }

        // Running total so far: 229 + 152 = 381

        int y3, y2, y1, y0;
        {
            int Kinv = (fwd >> 20) ^ p1;     // 7+7+
            int w0 = Kinv & 1; Kinv >>= 1;   // 7+5+
            int w1 = Kinv & 1; Kinv >>= 1;   // 7+5+
            int w2 = Kinv & 1; Kinv >>= 1;   // 7+5+
            int w3 = Kinv & 1;               // 7+

            int t0 = w1 ^ w0 ^ (w2 & w3);      // 7+7+7+
            int t1 = w2 ^ (w0 | w3);           // 7+7+
            int t2 = t0 ^ t1;                  // 7+

            y3 = t2 ^ (t1 & (w1 & w3));        // 7+7+7+
            y2 = t0 ^ (w0 | t2);               // 7+7+
            y1 = w0 ^ w3 ^ (t1 & t0);          // 7+7+7+
            y0 = w3 ^ (t0 | (w1 ^ (w0 | w2))); // 7+7+7+7


        }

        // Running total so far: 381 + 24*7 + 3*5 = 564

        int p2;
        {
            int ma = fwd;
            p2  = ma & -y0; ma<<=1;       //   7+5+5+ | 17+
            p2 ^= ma & -y1; ma<<=1;       // 7+7+5+5+ | 24+
            p2 ^= ma & -y2; ma<<=1;       // 7+7+5+5+ | 24+
            p2 ^= ma & -y3;               // 7+7+5+   | 19+
            int t = p2 >> 3;              // 7+       | 7+
            p2 ^= (t >> 1) ^ (t & 0xe0e); // 7+5+7+7+ | 26
        }

        // Running total so far: 564 + 117 = 681

        int inv8;
        inv8  =  31 & -(p2 & 1);           //   7+5+7+   | 19+
        inv8 ^= 178 & -(p2 & 2); p2 >>= 2; // 7+7+5+7+7+ | 33+
        inv8 ^= 171 & -(p2 & 1);           // 7+7+5+7+   | 26+
        inv8 ^=  54 & -(p2 & 2); p2 >>= 6; // 7+7+5+7+7+ | 33+
        inv8 ^= 188 & -(p2 & 1);           // 7+7+5+7+   | 26+
        inv8 ^=  76 & -(p2 & 2); p2 >>= 2; // 7+7+5+7+7+ | 33+
        inv8 ^= 127 & -(p2 & 1);           // 7+7+5+7+   | 26+
        inv8 ^= 222 & -(p2 & 2);           // 7+7+5+7    | 26+

        return inv8 ^ 0x63;                // 7+         | 7+

        // Grand total: 681 + 229 = 910
    }
}

结构基本上与Boyar和Peralta的实现相同-将GF（2 ^ 8）中的反转减少为GF（2 ^ 4）中的反转，将其分解为线性序言，非线性主体和线性尾声，然后分别将它们最小化。我为位提取付出了一些代价，但我通过能够并行执行操作（对的位进行了一些明智的填充fwd）来进行补偿。更详细地...

背景

如问题描述中所述，S盒包含在Galois字段GF（2 ^ 8）的特定实现中的求逆，然后进行仿射变换。如果您知道这两个都是什么意思，请跳过本节。

仿射（或线性）变换是一个函数f(x)，其方面f(x + y) = f(x) + f(y)和f(a*x) = a*f(x)。

字段是F具有两个特殊元素的元素集，我们将其称为0和1，并使用两个运算符将其称为+和*，它们尊重各种属性。在本节中，假设x，y和z是的元件F。

• 的元素F形式的阿贝尔群下+与0作为标识：即x + y是的元件F; x + 0 = 0 + x = x; 每个x都有相应的-x内容x + (-x) = (-x) + x = 0; x + (y + z) = (x + y) + z; 和x + y= y + x。
• 的元件F以外的其他0形式的阿贝尔群下*与1作为标识。
• 乘法分布于加法：x * (y + z) = (x * y) + (x * z)。

事实证明，有限域上存在一些相当严格的限制：

• 它们必须具有许多素数的元素。
• 它们与同尺寸的所有其他有限域是同构的（即，实际上只有一个给定尺寸的有限域，而其他域只是重新标记；称该域为GF（p ^ k），其中p素数k是幂） 。
• 乘法群F\{0}下*是环状的; 也就是说，至少有一个元素g使得每个元素都是的幂g。
• 对于大于1的幂，存在一个表示为k素数阶的阶的单变量多项式。例如，GF（2 ^ 8）具有多项式表示形式，这些多项式x超过GF（2）。实际上，通常有不止一种表示。考虑x^7 * x在GF（2 ^ 8）; 它必须等效于某些7阶多项式，但是哪个呢？有很多选择可以给出正确的结构；AES选择生成x^8 = x^4 + x^3 + x + 1（有效的字典上最小的多项式）。

那么我们如何计算GF（2 ^ 8）的特定表示形式的逆呢？这个问题过于庞大，无法直接解决，因此我们需要对其进行分解。

场塔：以GF（2 ^ 4）表示GF（2 ^ 8）

与其用GF（2）上的8个多项式表示GF（2 ^ 8），我们还可以用GF（2 ^ 4）上的2个多项式表示它。这次我们需要为选择线性多项式x^2。假设我们选择x^2 = px + q。然后(ax + b) * (cx + d) = (ad + bc + acp)x + (bd + acq)。

在这种表示形式中找到逆数是否更容易？如果(cx + d) = (ax + b)^-1我们得到联立方程

• ad + (b + ap)c = 0
• bd + (aq)c = 1

让D = [b(b+ap) + a^2 q]和设置c = a * D^-1; d = (b + ap) * D^-1。因此，对于转换为GF（2 ^ 4）的成本，我们可以在GF（2 ^ 8）中进行逆运算，在GF（2 ^ 4）中进行逆运算以及一些加法和乘法运算，然后再转换回去。即使我们通过表格进行逆运算，我们也将表格大小从256减少到了16。

实施细节

要构造GF（4）的表示形式，我们可以在三个多项式之间进行选择以减少x^4：

• x^4 = x + 1
• x^4 = x^3 + 1
• x^4 = x^3 + x^2 + x + 1

最重要的区别在于乘法的实现。对于这三个中的任何一个（对应于poly3、9，f），以下各项将起作用：

// 14x &, 7x unary -, 3x <<1, 3x >>1, 3x >>3, 6x ^ gives score 226
int mul(int a, int b) {
    // Call current values a = a0, b = b0
    int p = a & -(b & 1);
    a = ((a << 1) ^ (poly & -(a >> 3))) & 15;
    b >>= 1;
    // Now p = a0 * (b0 mod x); a = a0 x; b = b0 div x

    p ^= a & -(b & 1);
    a = ((a << 1) ^ (poly & -(a >> 3))) & 15;
    b >>= 1;
    // Now p = a0 * (b0 mod x^2); a = a0 x^2; b = b0 div x^2

    p ^= a & -(b & 1);
    a = ((a << 1) ^ (poly & -(a >> 3))) & 15;
    b >>= 1;
    // Now p = a0 * (b0 mod x^3); a = a0 x^3; b = b0 div x^3

    p ^= a & -(b & 1);
    // p = a0 * b0

    return p;
}

但是，如果我们选择，poly = 3我们可以更有效地处理溢出，因为它具有良好的结构：没有双重溢出，因为两个输入都是三次，而且x^6 = x^2 (x + 1)都是三次。另外，我们可以保存：的移位，b因为我们将溢出持续到最后，a0 x^2所以没有对应于x或1的任何设置位，因此我们可以使用-4而不是-1对其进行屏蔽。结果是

// 10x &, 4x unary -, 3x <<1, 1x >>1, 1x >>3, 5x ^ gives score 152
int mul(int a, int b) {
    int p;
    p  = a & -(b & 1); a <<= 1;
    p ^= a & -(b & 2); a <<= 1;
    p ^= a & -(b & 4); a <<= 1;
    p ^= a & -(b & 8);
    // Here p = a0 * b0 but with overflow, which we need to bring back down.

    int t = p >> 3;
    p ^= (t >> 1) ^ (t & 0xe);
    return p & 15;
}

我们仍然需要选择GF（2 ^ 8）的值p和qGF（2 ^ 4）的表示形式，但是对它们的约束并不多。重要的是，我们可以从原始表示的位到工作表示的位得到线性函数。这很重要，原因有两个：首先，进行线性转换很容易，而非线性转换则需要优化，其难度仅在于优化整个S-box；其次，因为我们可以获得一些附带利益。回顾一下结构：

GF256 SubBytes(GF256 x) {
    GF16 a, b, t, D, Dinv, c, d;

    (a, b) = f(x); // f is linear

    t = b + a * p;
    D = b * t + a * a * q;
    Dinv = inverse_GF16(D);
    c = a * Dinv;
    d = t * Dinv;

    return finv(c, d); // finv is also linear
}

如果的位为x，x7 x6 ... x0则由于变换是线性的，我们得到a = f({x7}0000000 + 0{x6}000000 + ... + 0000000{x0}) = f({x7}0000000) + f(0{x6}000000) + ... + f(0000000{x0})。将其平方，然后得出a^2 = f({x7}0000000)^2 + f(0{x6}000000)^2 + ... + f(0000000{x0})^2交叉项抵消的位置（由于GF（2），1 + 1 = 0）。因此a^2也可以计算为的线性函数x。我们可以扩大前向线性变换以获得：

GF256 SubBytes(GF256 x) {
    GF16 a, b, t, a2q, D, Dinv, c, d;

    (a, b, t, a2q) = faug(x);

    D = b * t + a2q;
    Dinv = inverse_GF16(D);
    c = a * Dinv;
    d = t * Dinv;

    return finv(c, d);
}

我们归结为三个乘法和一个加法。我们还可以扩展乘法代码以Dinv并行执行两个乘法。因此，我们的总成本是正向线性变换，加法，两个乘法，GF（2 ^ 4）的逆和反向线性变换。我们可以将S-box进行逆线性变换，然后从本质上免费获得它。

线性变换的系数的计算不是很有趣，微优化也没有在此处保存掩模并在此处进行移位的微优化。剩下的有趣部分是优化inverse_GF16。从4位到4位有2 ^ 64种不同的功能，因此直接优化需要大量的内存和时间。我所做的是考虑从4位到1位的4个函数，对任何一个函数允许的总开销设置上限（每个函数的最大开销为63，我可以在一分钟内枚举所有合适的表达式），并且对每个功能元组执行通用的子表达式消除。经过25分钟的运算后，我发现该上限可能的最佳逆运算的总成本为133（平均每位输出比特为33.25，考虑到任何单个比特的最便宜表达为35，这也不错） 。

我仍在尝试使用其他方法来最小化GF（2 ^ 4）中的反转，并且建立自下而上而不是自上而下的方法已从133改进到126。

得分：980 = 7 * 5 + 115 * 7 + 7 * 5 + 15 * 7，博亚尔和佩拉尔塔的极小化

我发现了Joan Boyar和RenéPeralta提出的一种新的组合逻辑最小化技术及其在密码学上的应用，（满足C形式主义要求）可以完成所需的工作。用于推导其方程的技术已获得美国的专利。我对他们的方程作了C语言的直接翻译，请链接到这里。

unsigned char SubBytes_Boyar_Peralta(unsigned char x7){
  unsigned char 
  x6=x7>>1,x5=x6>>1,x4=x5>>1,x3=x4>>1,x2=x3>>1,x1=x2>>1,x0=x1>>1,
  y14=x3^x5,y13=x0^x6,y9=x0^x3,y8=x0^x5,t0=x1^x2,y1=t0^x7,y4=y1^x3,y12=y13^y14,y2=y1^x0,
  y5=y1^x6,y3=y5^y8,t1=x4^y12,y15=t1^x5,y20=t1^x1,y6=y15^x7,y10=y15^t0,y11=y20^y9,y7=x7^y11,
  y17=y10^y11,y19=y10^y8,y16=t0^y11,y21=y13^y16,y18=x0^y16,t2=y12&y15,t3=y3&y6,t4=t3^t2,
  t5=y4&x7,t6=t5^t2,t7=y13&y16,t8=y5&y1,t9=t8^t7,t10=y2&y7,t11=t10^t7,t12=y9&y11,
  t13=y14&y17,t14=t13^t12,t15=y8&y10,t16=t15^t12,t17=t4^t14,t18=t6^t16,t19=t9^t14,
  t20=t11^t16,t21=t17^y20,t22=t18^y19,t23=t19^y21,t24=t20^y18,t25=t21^t22,t26=t21&t23,
  t27=t24^t26,t28=t25&t27,t29=t28^t22,t30=t23^t24,t31=t22^t26,t32=t31&t30,t33=t32^t24,
  t34=t23^t33,t35=t27^t33,t36=t24&t35,t37=t36^t34,t38=t27^t36,t39=t29&t38,t40=t25^t39,
  t41=t40^t37,t42=t29^t33,t43=t29^t40,t44=t33^t37,t45=t42^t41,z0=t44&y15,z1=t37&y6,
  z2=t33&x7,z3=t43&y16,z4=t40&y1,z5=t29&y7,z6=t42&y11,z7=t45&y17,z8=t41&y10,z9=t44&y12,
  z10=t37&y3,z11=t33&y4,z12=t43&y13,z13=t40&y5,z14=t29&y2,z15=t42&y9,z16=t45&y14,z17=t41&y8,
  t46=z15^z16,t47=z10^z11,t48=z5^z13,t49=z9^z10,t50=z2^z12,t51=z2^z5,t52=z7^z8,t53=z0^z3,
  t54=z6^z7,t55=z16^z17,t56=z12^t48,t57=t50^t53,t58=z4^t46,t59=z3^t54,t60=t46^t57,
  t61=z14^t57,t62=t52^t58,t63=t49^t58,t64=z4^t59,t65=t61^t62,t66=z1^t63,s0=t59^t63,
  s6=t56^t62,s7=t48^t60,t67=t64^t65,s3=t53^t66,s4=t51^t66,s5=t47^t65,s1=t64^s3,s2=t55^t67;
  return (((((((s0<<1|s1&1)<<1|s2&1)<<1|s3&1)<<1|s4&1)<<1|s5&1)<<1|s6&1)<<1|s7&1)^0x63;}

得分：10965

这使用展开数组查找的相同原理。可能需要额外的演员。

感谢ugoren指出了如何改进is_zero。

// Cost: 255 * (5+7+24+7) = 10965
unsigned char SubBytes(unsigned char x) {
    unsigned char r = 0x63;
    char c = (char)x;
    c--; r ^= is_zero(c) & (0x63^0x7c); // 5+7+24+7 inlining the final xor
    c--; r ^= is_zero(c) & (0x63^0x77); // 5+7+24+7
    // ...
    c--; r ^= is_zero(c) & (0x63^0x16); // 5+7+24+7
    return r;
}

// Cost: 24
// Returns (unsigned char)-1 when input is 0 and 0 otherwise
unsigned char is_zero(char c) {
    // Shifting a signed number right is unspecified, so use unsigned
    unsigned char u;
    c |= -c;               // 7+5+
    u = (unsigned char)c;
    u >>= (CHAR_BITS - 1); // 7+
    c = (char)u;
    // c is 0 if we want -1 and 1 otherwise.
    c--;                   // 5
    return (unsigned char)c;
}

得分：9109，采用代数方法

万一有人可以大幅度改进它，我将保留查找方法，但事实证明，一种好的代数方法是可行的。此实现找到乘法逆使用Euclid算法。我已经用Java编写了它，但是原则上它可以移植到C-我评论了如果您只想使用8位类型对其进行修改的部分。

感谢ugoren is_nonzero在我的其他答案的评论中指出了如何缩短支票。

public class SBox
{
    public static void main(String[] args)
    {
        for (int i = 0; i < 256; i++) {
            int s = SubBytes(i);
            System.out.format("%02x  ", s);
            if (i % 16 == 15) System.out.println();
        }
    }

    // Total cost: 9109
    public static int SubBytes(int x)
    {
        x = inv_euclid(x); // 9041
        x = affine(x);     // 68
        return x;
    }

    // Total cost: 68
    private static int affine(int s0) {
        int s = s0;
        s ^= (s0 << 1) ^ (s0 >> 7); // 5 + 7
        s ^= (s0 << 2) ^ (s0 >> 6); // 7 + 7
        s ^= (s0 << 3) ^ (s0 >> 5); // 7 + 7
        s ^= (s0 << 4) ^ (s0 >> 4); // 7 + 7
        return (s ^ 0x63) & 0xff;   // 7 + 7
    }

    // Does the inverse in the Galois field for a total cost of 24 + 9010 + 7 = 9041
    private static int inv_euclid(int s) {
        // The first part of handling the special case: cost of 24
        int zeromask = is_nonzero(s);

        // NB the special value of r would complicate the unrolling slightly with unsigned bytes
        int r = 0x11b, a = 0, b = 1;

        // Total cost of loop: 7*(29+233+566+503+28) - 503 = 9010
        for (int depth = 0; depth < 7; depth++) { // 7*(
            // Calculate mask to fake out when we're looping further than necessary: cost 29
            int mask = is_nonzero(s >> 1);

            // Cost: 233
            int ord = polynomial_order(s);

            // This next block does div/rem at a total cost of 6*(24+49) + 69 + 59 = 566
            int quot = 0, rem = r;
            for (int i = 7; i > 1; i--) {                   // 6*(
                int divmask = is_nonzero(ord & (rem >> i)); // 24+7+7
                quot ^= (1 << i) & divmask;                 // 7+0+7+ since 1<<i is inlined on unrolling
                rem ^= (s << i) & divmask;                  // 7+7+7) +
            }
            int divmask1 = is_nonzero(ord & (rem >> 1));    // 24+7+5
            quot ^= 2 & divmask1;                           // 7+7+
            rem ^= (s << 1) & divmask1;                     // 7+5+7+
            int divmask0 = is_nonzero(ord & rem);           // 24+7
            quot ^= 1 & divmask0;                           // 7+7+
            rem ^= s & divmask0;                            // 7+7

            // This next block does the rest for the cost of a mul (503) plus 28
            // When depth = 0, b = 1 so we can skip the mul on unrolling
            r = s;
            s = rem;
            quot = mul(quot, b) ^ a;
            a = b;
            b ^= (quot ^ b) & mask;
        }

        // The rest of handling the special case: cost 7
        return b & zeromask;
    }

    // Gets the highest set bit in the input. Assumes that it's always at least 1<<1
    // Cost: 233
    private static int polynomial_order(int s) {
        int ord = 2;
        ord ^= 6 & -((s >> 2) & 1);           // 7+7+5+7+7 = 33 +
        ord ^= (ord ^ 8) & -((s >> 3) & 1);   // 7+7+7+5+7+7 = 40 +
        ord ^= (ord ^ 16) & -((s >> 4) & 1);  // 40 +
        ord ^= (ord ^ 32) & -((s >> 5) & 1);  // 40 +
        ord ^= (ord ^ 64) & -((s >> 6) & 1);  // 40 +
        ord ^= (ord ^ 128) & -((s >> 7) & 1); // 40
        return ord;
    }

    // Returns 0 if c is 0 and -1 otherwise
    // Cost: 24
    private static int is_nonzero(int c) {
        c |= -c;   // 7+5+
        c >>>= 31; // 7+ (simulating a cast to unsigned and right shift by CHAR_BIT)
        c = -c;    // 5+ (could be saved assuming a particular implementation of signed right shift)
        return c;
    }

    // Performs a multiplication in the Rijndael finite field
    // Cost: 503 (496 if working with unsigned bytes)
    private static int mul(int a, int b) {
        int p = 0;
        for (int counter = 0; counter < 8; counter++) { // 8*(_+_
            p ^= a & -(b & 1);                          // +7+7+5+7
            a = (a << 1) ^ (0x1b & -(a >> 7));          // +5+7+7+5+7
            b >>= 1;                                    // +5)
        }
        p &= 0xff;                                      // +7 avoidable with unsigned bytes
        return p;
    }
}

得分：256 *（7+（8 *（7 + 7 + 7）-（2 + 2））+ 5 + 7 + 7）= 48640（假设循环展开）

unsigned char SubBytes(unsigned char x) {
static const unsigned char t[256] = {
  0x63,0x7C,0x77,0x7B,0xF2,0x6B,0x6F,0xC5,0x30,0x01,0x67,0x2B,0xFE,0xD7,0xAB,0x76,
  0xCA,0x82,0xC9,0x7D,0xFA,0x59,0x47,0xF0,0xAD,0xD4,0xA2,0xAF,0x9C,0xA4,0x72,0xC0,
  0xB7,0xFD,0x93,0x26,0x36,0x3F,0xF7,0xCC,0x34,0xA5,0xE5,0xF1,0x71,0xD8,0x31,0x15,
  0x04,0xC7,0x23,0xC3,0x18,0x96,0x05,0x9A,0x07,0x12,0x80,0xE2,0xEB,0x27,0xB2,0x75,
  0x09,0x83,0x2C,0x1A,0x1B,0x6E,0x5A,0xA0,0x52,0x3B,0xD6,0xB3,0x29,0xE3,0x2F,0x84,
  0x53,0xD1,0x00,0xED,0x20,0xFC,0xB1,0x5B,0x6A,0xCB,0xBE,0x39,0x4A,0x4C,0x58,0xCF,
  0xD0,0xEF,0xAA,0xFB,0x43,0x4D,0x33,0x85,0x45,0xF9,0x02,0x7F,0x50,0x3C,0x9F,0xA8,
  0x51,0xA3,0x40,0x8F,0x92,0x9D,0x38,0xF5,0xBC,0xB6,0xDA,0x21,0x10,0xFF,0xF3,0xD2,
  0xCD,0x0C,0x13,0xEC,0x5F,0x97,0x44,0x17,0xC4,0xA7,0x7E,0x3D,0x64,0x5D,0x19,0x73,
  0x60,0x81,0x4F,0xDC,0x22,0x2A,0x90,0x88,0x46,0xEE,0xB8,0x14,0xDE,0x5E,0x0B,0xDB,
  0xE0,0x32,0x3A,0x0A,0x49,0x06,0x24,0x5C,0xC2,0xD3,0xAC,0x62,0x91,0x95,0xE4,0x79,
  0xE7,0xC8,0x37,0x6D,0x8D,0xD5,0x4E,0xA9,0x6C,0x56,0xF4,0xEA,0x65,0x7A,0xAE,0x08,
  0xBA,0x78,0x25,0x2E,0x1C,0xA6,0xB4,0xC6,0xE8,0xDD,0x74,0x1F,0x4B,0xBD,0x8B,0x8A,
  0x70,0x3E,0xB5,0x66,0x48,0x03,0xF6,0x0E,0x61,0x35,0x57,0xB9,0x86,0xC1,0x1D,0x9E,
  0xE1,0xF8,0x98,0x11,0x69,0xD9,0x8E,0x94,0x9B,0x1E,0x87,0xE9,0xCE,0x55,0x28,0xDF,
  0x8C,0xA1,0x89,0x0D,0xBF,0xE6,0x42,0x68,0x41,0x99,0x2D,0x0F,0xB0,0x54,0xBB,0x16};

unsigned char ret_val = 0;
int i,j;
for(i=0;i<256;i++) {
  unsigned char is_index = (x ^ ((unsigned char) i));
  for(j=0;j<8;j++) {
   is_index |= (is_index << (1 << j)) | (is_index >> (1 << j));
  }
  is_index = ~is_index;
  ret_val |= is_index & t[i];
}

return ret_val;}

说明：

本质上，使用位运算符重新实现数组查找，并始终处理整个数组。我们遍历数组，然后x对每个索引进行异或运算，然后使用按位运算符对结果进行逻辑求反，因此当x=i，否则为0。我们将其与数组值进行按位和运算，以便所选值保持不变，而其他值变为0。然后我们对该数组进行按位或运算，从而仅提取所选值。

1 << j作为展开循环的一部分，这两个操作将消失，以1到128的2的幂代替它们。

使用查找表对1968 1692进行评分

注意：由于，此解决方案未通过标准w >> b。

使用查找表，但一次读取8个字节。
3 * 7 + 32 *（6 * 7 + 2 * 5）+ 7 = 692

unsigned char SubBytes(unsigned char x){

static const unsigned char t[256] = {
  0x63,0x7C,0x77,0x7B,0xF2,0x6B,0x6F,0xC5,0x30,0x01,0x67,0x2B,0xFE,0xD7,0xAB,0x76,
  0xCA,0x82,0xC9,0x7D,0xFA,0x59,0x47,0xF0,0xAD,0xD4,0xA2,0xAF,0x9C,0xA4,0x72,0xC0,
  0xB7,0xFD,0x93,0x26,0x36,0x3F,0xF7,0xCC,0x34,0xA5,0xE5,0xF1,0x71,0xD8,0x31,0x15,
  0x04,0xC7,0x23,0xC3,0x18,0x96,0x05,0x9A,0x07,0x12,0x80,0xE2,0xEB,0x27,0xB2,0x75,
  0x09,0x83,0x2C,0x1A,0x1B,0x6E,0x5A,0xA0,0x52,0x3B,0xD6,0xB3,0x29,0xE3,0x2F,0x84,
  0x53,0xD1,0x00,0xED,0x20,0xFC,0xB1,0x5B,0x6A,0xCB,0xBE,0x39,0x4A,0x4C,0x58,0xCF,
  0xD0,0xEF,0xAA,0xFB,0x43,0x4D,0x33,0x85,0x45,0xF9,0x02,0x7F,0x50,0x3C,0x9F,0xA8,
  0x51,0xA3,0x40,0x8F,0x92,0x9D,0x38,0xF5,0xBC,0xB6,0xDA,0x21,0x10,0xFF,0xF3,0xD2,
  0xCD,0x0C,0x13,0xEC,0x5F,0x97,0x44,0x17,0xC4,0xA7,0x7E,0x3D,0x64,0x5D,0x19,0x73,
  0x60,0x81,0x4F,0xDC,0x22,0x2A,0x90,0x88,0x46,0xEE,0xB8,0x14,0xDE,0x5E,0x0B,0xDB,
  0xE0,0x32,0x3A,0x0A,0x49,0x06,0x24,0x5C,0xC2,0xD3,0xAC,0x62,0x91,0x95,0xE4,0x79,
  0xE7,0xC8,0x37,0x6D,0x8D,0xD5,0x4E,0xA9,0x6C,0x56,0xF4,0xEA,0x65,0x7A,0xAE,0x08,
  0xBA,0x78,0x25,0x2E,0x1C,0xA6,0xB4,0xC6,0xE8,0xDD,0x74,0x1F,0x4B,0xBD,0x8B,0x8A,
  0x70,0x3E,0xB5,0x66,0x48,0x03,0xF6,0x0E,0x61,0x35,0x57,0xB9,0x86,0xC1,0x1D,0x9E,
  0xE1,0xF8,0x98,0x11,0x69,0xD9,0x8E,0x94,0x9B,0x1E,0x87,0xE9,0xCE,0x55,0x28,0xDF,
  0x8C,0xA1,0x89,0x0D,0xBF,0xE6,0x42,0x68,0x41,0x99,0x2D,0x0F,0xB0,0x54,0xBB,0x16};

  unsigned long long *t2 = (unsigned long long *)t;
  int a = x>>3, b=(x&7)<<3;                       // 7+7+7
  int i;
  int ret = 0;
  for (i=0;i<256/8;i++) {                         // 32 *
      unsigned long long w = t2[i];
      int badi = ((unsigned int)(a|-a))>>31;      // 7+7+5
      w &= (badi-1);                              // +7+7
      a--;                                        // +5
      ret |= w >> b;                              // +7+7
  }
  return ret & 0xff;                              // +7
}

查表和掩码，分数= 256 *（5 * 7 + 1 * 5）= 10240

使用带掩码的表查找仅挑选我们想要的结果。使用j|-j负数（当i！= x时）或零（当i == x时）的事实。移位生成一个全一或全零的掩码，用于仅选择我们想要的条目。

static const unsigned char t[256] = {
  0x63,0x7C,0x77,0x7B,0xF2,0x6B,0x6F,0xC5,0x30,0x01,0x67,0x2B,0xFE,0xD7,0xAB,0x76,
  0xCA,0x82,0xC9,0x7D,0xFA,0x59,0x47,0xF0,0xAD,0xD4,0xA2,0xAF,0x9C,0xA4,0x72,0xC0,
  0xB7,0xFD,0x93,0x26,0x36,0x3F,0xF7,0xCC,0x34,0xA5,0xE5,0xF1,0x71,0xD8,0x31,0x15,
  0x04,0xC7,0x23,0xC3,0x18,0x96,0x05,0x9A,0x07,0x12,0x80,0xE2,0xEB,0x27,0xB2,0x75,
  0x09,0x83,0x2C,0x1A,0x1B,0x6E,0x5A,0xA0,0x52,0x3B,0xD6,0xB3,0x29,0xE3,0x2F,0x84,
  0x53,0xD1,0x00,0xED,0x20,0xFC,0xB1,0x5B,0x6A,0xCB,0xBE,0x39,0x4A,0x4C,0x58,0xCF,
  0xD0,0xEF,0xAA,0xFB,0x43,0x4D,0x33,0x85,0x45,0xF9,0x02,0x7F,0x50,0x3C,0x9F,0xA8,
  0x51,0xA3,0x40,0x8F,0x92,0x9D,0x38,0xF5,0xBC,0xB6,0xDA,0x21,0x10,0xFF,0xF3,0xD2,
  0xCD,0x0C,0x13,0xEC,0x5F,0x97,0x44,0x17,0xC4,0xA7,0x7E,0x3D,0x64,0x5D,0x19,0x73,
  0x60,0x81,0x4F,0xDC,0x22,0x2A,0x90,0x88,0x46,0xEE,0xB8,0x14,0xDE,0x5E,0x0B,0xDB,
  0xE0,0x32,0x3A,0x0A,0x49,0x06,0x24,0x5C,0xC2,0xD3,0xAC,0x62,0x91,0x95,0xE4,0x79,
  0xE7,0xC8,0x37,0x6D,0x8D,0xD5,0x4E,0xA9,0x6C,0x56,0xF4,0xEA,0x65,0x7A,0xAE,0x08,
  0xBA,0x78,0x25,0x2E,0x1C,0xA6,0xB4,0xC6,0xE8,0xDD,0x74,0x1F,0x4B,0xBD,0x8B,0x8A,
  0x70,0x3E,0xB5,0x66,0x48,0x03,0xF6,0x0E,0x61,0x35,0x57,0xB9,0x86,0xC1,0x1D,0x9E,
  0xE1,0xF8,0x98,0x11,0x69,0xD9,0x8E,0x94,0x9B,0x1E,0x87,0xE9,0xCE,0x55,0x28,0xDF,
  0x8C,0xA1,0x89,0x0D,0xBF,0xE6,0x42,0x68,0x41,0x99,0x2D,0x0F,0xB0,0x54,0xBB,0x16};

unsigned char SubBytes(unsigned char x) {
  unsigned char r = 255;
  for (int i = 0; i < 256; i++) {
    int j = i - x;
    r &= t[i] | ((j | -j) >> 31);
  }
  return r;
}