chore: add fast crc32c concat implementation

This new implementation provides a constant time computation regardless of the length of B, rather than the previous implementation which took linear time relative to the length of B.

#### Benchmark summary

CPU: AMD Ryzen Threadripper PRO 3945WX 12-Cores
OS: Debian Testing
```
# JMH version: 1.37
# VM version: JDK 11.0.17, OpenJDK 64-Bit Server VM, 11.0.17+8
# VM invoker: /home/benwhitehead/opt/java/jdk-11.0.17+8/bin/java
# VM options: -Xms1g -Xmx1g -XX:+AlwaysPreTouch
# Blackhole mode: full + dont-inline hint (auto-detected, use -Djmh.blackhole.autoDetect=false to disable)
# Warmup: 4 iterations, 30 s each
# Measurement: 8 iterations, 30 s each
# Timeout: 10 min per iteration
# Threads: 1 thread, will synchronize iterations
# Benchmark mode: Sampling time
```

```
Benchmark                                  (objectSize)    Mode      Cnt     Score    Error  Units
Crc32cBenchmark.concatCrc32c                     131072  sample  9243752     0.074 ±  0.001  us/op
Crc32cBenchmark.concatCrc32c                     262144  sample  9015106     0.078 ±  0.001  us/op
Crc32cBenchmark.concatCrc32c                     524288  sample  9010257     0.074 ±  0.001  us/op
Crc32cBenchmark.concatCrc32c                    1048576  sample  9165519     0.074 ±  0.001  us/op
Crc32cBenchmark.concatCrc32c                    2097152  sample  8993038     0.075 ±  0.001  us/op
Crc32cBenchmark.crc32cCombineGoogle              131072  sample  2070639   115.806 ±  0.018  us/op
Crc32cBenchmark.crc32cCombineGoogle              262144  sample  1029209   233.019 ±  0.045  us/op
Crc32cBenchmark.crc32cCombineGoogle              524288  sample   516517   464.360 ±  0.106  us/op
Crc32cBenchmark.crc32cCombineGoogle             1048576  sample   262118   915.067 ±  0.354  us/op
Crc32cBenchmark.crc32cCombineGoogle             2097152  sample   131385  1825.685 ±  0.834  us/op
```

Author: BenWhitehead

google-cloud-storage/src/main/java/com/google/cloud/storage/Crc32cUtility.java

@@ -19,288 +19,70 @@
 class Crc32cUtility {
   private Crc32cUtility() {}
 
+  // Castagnoli polynomial and its degree.
+  private static final long CASTAGNOLI_POLY = 4812730177L;
+  private static final int DEGREE = 32;
+
+  // Table storing values of x^(2^k) mod CASTANOLI_POLY for all k < 31. This is sufficient since
+  // x^(2^31) = x.
+  private static final long[] X_POW_2K_TABLE = {
+    2L, 4L, 16L, 256L, 65536L, 517762881L, 984302966L,
+    408362264L, 1503875210L, 2862076957L, 3884826397L, 1324787473L, 621200174L, 1758783527L,
+    1416537776L, 1180494764L, 648569364L, 2521473789L, 994858823L, 1728245375L, 3498467999L,
+    4059169852L, 3345064394L, 2828422810L, 2429203150L, 3336788029L, 860151998L, 2102628683L,
+    1033187991L, 4243778976L, 1123580069L
+  };
+
+  // Multiplies two polynomials together modulo CASTAGNOLI_POLY.
+  private static int multiply(int p, int q) {
+    long q64 = q;
+    int result = 0;
+    long topBit = (1L << DEGREE);
+    for (int i = 0; i < DEGREE; i++) {
+      if ((p & 1) != 0) {
+        result ^= (int) q64;
+      }
+      q64 <<= 1; // Multiply by x.
+
+      // If multiplying by x gave q64 a non-zero 32nd coefficient, it no longer encodes the desired
+      // representative of that polynomial modulo CASTAGNOLI_POLY, so we subtract the generator.
+      if ((q64 & topBit) != 0) {
+        q64 ^= CASTAGNOLI_POLY;
+      }
+      p >>= 1;
+    }
+    return result;
+  }
+
+  // Given crc representing polynomial P(x), compute P(x)*x^numBits.
+  private static int extendByZeros(int crc, long numBits) {
+    // Begin by reversing the bits to most-significant coefficient first for comprehensibility.
+    crc = Integer.reverse(crc);
+    int i = 0;
+    // Iterate over the binary representation of numBits, multiplying by x^(2^k) for numBits_k = 1.
+    while (numBits != 0) {
+      if ((numBits & 1) != 0) {
+        crc = multiply(crc, (int) X_POW_2K_TABLE[i % X_POW_2K_TABLE.length]);
+      }
+      i += 1;
+      numBits >>= 1;
+    }
+    crc = Integer.reverse(crc); // Return to the standard bit-order.
+    return crc;
+  }
+
   /**
-   * Straight forward implementation to combine two crc32c values
+   * Efficiently computes CRC32C for concat(A, B) given crc(A), crc(B) and len(B).
    *
-   * <p>There is a more efficient implementation in https://github.com/google/crc32c, however for
-   * brevity we are following the byte by byte computation, implemented in:
-   * https://github.com/google/crc32c/blob/main/src/crc32c_portable.cc#L252-L257
-   *
-   * @param crc1 crc32c int of an object to combine crc2 with
-   * @param crc2 crc32c int of the second object
-   * @param crc2ObjectSize byte length of object which generated crc2
-   * @return
-   * @see <a target="_blank"
-   *     href="https://github.com/google/crc32c/blob/21fc8ef30415a635e7351ffa0e5d5367943d4a94/src/crc32c_portable.cc#L252-L257">https://github.com/google/crc32c/blob/21fc8ef30415a635e7351ffa0e5d5367943d4a94/src/crc32c_portable.cc#L252-L257</a>
+   * @param crcA A 32-bit integer representing crc(A) with least-significant coefficient first.
+   * @param crcB Same as crcA for B.
+   * @param numBytesInB Length of B in bytes.
+   * @return CRC32C for concat(A, B) PiperOrigin-RevId: 158626905
    */
-  public static int crc32cCombineGoogle(int crc1, int crc2, long crc2ObjectSize) {
-    int l = crc1;
-    // run number of 0's based on size
-    for (long i = 0; i < crc2ObjectSize; i++) {
-      l = (l >>> 8) ^ kByteExtensionTable[l & 0xff];
+  public static int concatCrc32c(int crcA, int crcB, long numBytesInB) {
+    if (numBytesInB == 0) {
+      return crcA;
     }
-    return l ^ crc2;
+    return extendByZeros(crcA, 8 * numBytesInB) ^ crcB;
   }
-
-  // Ported from:
-  // https://github.com/google/crc32c/blob/21fc8ef30415a635e7351ffa0e5d5367943d4a94/src/crc32c_portable.cc#L16-L59
-  private static final int[] kByteExtensionTable =
-      new int[] {
-        0x00000000,
-        0xf26b8303,
-        0xe13b70f7,
-        0x1350f3f4,
-        0xc79a971f,
-        0x35f1141c,
-        0x26a1e7e8,
-        0xd4ca64eb,
-        0x8ad958cf,
-        0x78b2dbcc,
-        0x6be22838,
-        0x9989ab3b,
-        0x4d43cfd0,
-        0xbf284cd3,
-        0xac78bf27,
-        0x5e133c24,
-        0x105ec76f,
-        0xe235446c,
-        0xf165b798,
-        0x030e349b,
-        0xd7c45070,
-        0x25afd373,
-        0x36ff2087,
-        0xc494a384,
-        0x9a879fa0,
-        0x68ec1ca3,
-        0x7bbcef57,
-        0x89d76c54,
-        0x5d1d08bf,
-        0xaf768bbc,
-        0xbc267848,
-        0x4e4dfb4b,
-        0x20bd8ede,
-        0xd2d60ddd,
-        0xc186fe29,
-        0x33ed7d2a,
-        0xe72719c1,
-        0x154c9ac2,
-        0x061c6936,
-        0xf477ea35,
-        0xaa64d611,
-        0x580f5512,
-        0x4b5fa6e6,
-        0xb93425e5,
-        0x6dfe410e,
-        0x9f95c20d,
-        0x8cc531f9,
-        0x7eaeb2fa,
-        0x30e349b1,
-        0xc288cab2,
-        0xd1d83946,
-        0x23b3ba45,
-        0xf779deae,
-        0x05125dad,
-        0x1642ae59,
-        0xe4292d5a,
-        0xba3a117e,
-        0x4851927d,
-        0x5b016189,
-        0xa96ae28a,
-        0x7da08661,
-        0x8fcb0562,
-        0x9c9bf696,
-        0x6ef07595,
-        0x417b1dbc,
-        0xb3109ebf,
-        0xa0406d4b,
-        0x522bee48,
-        0x86e18aa3,
-        0x748a09a0,
-        0x67dafa54,
-        0x95b17957,
-        0xcba24573,
-        0x39c9c670,
-        0x2a993584,
-        0xd8f2b687,
-        0x0c38d26c,
-        0xfe53516f,
-        0xed03a29b,
-        0x1f682198,
-        0x5125dad3,
-        0xa34e59d0,
-        0xb01eaa24,
-        0x42752927,
-        0x96bf4dcc,
-        0x64d4cecf,
-        0x77843d3b,
-        0x85efbe38,
-        0xdbfc821c,
-        0x2997011f,
-        0x3ac7f2eb,
-        0xc8ac71e8,
-        0x1c661503,
-        0xee0d9600,
-        0xfd5d65f4,
-        0x0f36e6f7,
-        0x61c69362,
-        0x93ad1061,
-        0x80fde395,
-        0x72966096,
-        0xa65c047d,
-        0x5437877e,
-        0x4767748a,
-        0xb50cf789,
-        0xeb1fcbad,
-        0x197448ae,
-        0x0a24bb5a,
-        0xf84f3859,
-        0x2c855cb2,
-        0xdeeedfb1,
-        0xcdbe2c45,
-        0x3fd5af46,
-        0x7198540d,
-        0x83f3d70e,
-        0x90a324fa,
-        0x62c8a7f9,
-        0xb602c312,
-        0x44694011,
-        0x5739b3e5,
-        0xa55230e6,
-        0xfb410cc2,
-        0x092a8fc1,
-        0x1a7a7c35,
-        0xe811ff36,
-        0x3cdb9bdd,
-        0xceb018de,
-        0xdde0eb2a,
-        0x2f8b6829,
-        0x82f63b78,
-        0x709db87b,
-        0x63cd4b8f,
-        0x91a6c88c,
-        0x456cac67,
-        0xb7072f64,
-        0xa457dc90,
-        0x563c5f93,
-        0x082f63b7,
-        0xfa44e0b4,
-        0xe9141340,
-        0x1b7f9043,
-        0xcfb5f4a8,
-        0x3dde77ab,
-        0x2e8e845f,
-        0xdce5075c,
-        0x92a8fc17,
-        0x60c37f14,
-        0x73938ce0,
-        0x81f80fe3,
-        0x55326b08,
-        0xa759e80b,
-        0xb4091bff,
-        0x466298fc,
-        0x1871a4d8,
-        0xea1a27db,
-        0xf94ad42f,
-        0x0b21572c,
-        0xdfeb33c7,
-        0x2d80b0c4,
-        0x3ed04330,
-        0xccbbc033,
-        0xa24bb5a6,
-        0x502036a5,
-        0x4370c551,
-        0xb11b4652,
-        0x65d122b9,
-        0x97baa1ba,
-        0x84ea524e,
-        0x7681d14d,
-        0x2892ed69,
-        0xdaf96e6a,
-        0xc9a99d9e,
-        0x3bc21e9d,
-        0xef087a76,
-        0x1d63f975,
-        0x0e330a81,
-        0xfc588982,
-        0xb21572c9,
-        0x407ef1ca,
-        0x532e023e,
-        0xa145813d,
-        0x758fe5d6,
-        0x87e466d5,
-        0x94b49521,
-        0x66df1622,
-        0x38cc2a06,
-        0xcaa7a905,
-        0xd9f75af1,
-        0x2b9cd9f2,
-        0xff56bd19,
-        0x0d3d3e1a,
-        0x1e6dcdee,
-        0xec064eed,
-        0xc38d26c4,
-        0x31e6a5c7,
-        0x22b65633,
-        0xd0ddd530,
-        0x0417b1db,
-        0xf67c32d8,
-        0xe52cc12c,
-        0x1747422f,
-        0x49547e0b,
-        0xbb3ffd08,
-        0xa86f0efc,
-        0x5a048dff,
-        0x8ecee914,
-        0x7ca56a17,
-        0x6ff599e3,
-        0x9d9e1ae0,
-        0xd3d3e1ab,
-        0x21b862a8,
-        0x32e8915c,
-        0xc083125f,
-        0x144976b4,
-        0xe622f5b7,
-        0xf5720643,
-        0x07198540,
-        0x590ab964,
-        0xab613a67,
-        0xb831c993,
-        0x4a5a4a90,
-        0x9e902e7b,
-        0x6cfbad78,
-        0x7fab5e8c,
-        0x8dc0dd8f,
-        0xe330a81a,
-        0x115b2b19,
-        0x020bd8ed,
-        0xf0605bee,
-        0x24aa3f05,
-        0xd6c1bc06,
-        0xc5914ff2,
-        0x37faccf1,
-        0x69e9f0d5,
-        0x9b8273d6,
-        0x88d28022,
-        0x7ab90321,
-        0xae7367ca,
-        0x5c18e4c9,
-        0x4f48173d,
-        0xbd23943e,
-        0xf36e6f75,
-        0x0105ec76,
-        0x12551f82,
-        0xe03e9c81,
-        0x34f4f86a,
-        0xc69f7b69,
-        0xd5cf889d,
-        0x27a40b9e,
-        0x79b737ba,
-        0x8bdcb4b9,
-        0x988c474d,
-        0x6ae7c44e,
-        0xbe2da0a5,
-        0x4c4623a6,
-        0x5f16d052,
-        0xad7d5351
-      };
 }

google-cloud-storage/src/main/java/com/google/cloud/storage/Crc32cValue.java

@@ -81,7 +81,7 @@ public int getValue() {
 
     @Override
     public Crc32cLengthUnknown concat(Crc32cLengthKnown other) {
-      int combined = Crc32cUtility.crc32cCombineGoogle(value, other.value, other.length);
+      int combined = Crc32cUtility.concatCrc32c(value, other.value, other.length);
       return new Crc32cLengthUnknown(combined);
     }
 
@@ -137,7 +137,7 @@ public long getLength() {
 
     @Override
     public Crc32cLengthKnown concat(Crc32cLengthKnown other) {
-      int combined = Crc32cUtility.crc32cCombineGoogle(value, other.value, other.length);
+      int combined = Crc32cUtility.concatCrc32c(value, other.value, other.length);
       return new Crc32cLengthKnown(combined, length + other.length);
     }
 

google-cloud-storage/src/test/java/com/google/cloud/storage/Crc32cUtilityPropertyTest.java

@@ -29,8 +29,7 @@ public void testCrc32cCombinePropertyTest(
     int secondPartHash = Hashing.crc32c().hashBytes(secondObject).asInt();
     int expected =
         Hashing.crc32c().newHasher().putBytes(firstObject).putBytes(secondObject).hash().asInt();
-    int actual =
-        Crc32cUtility.crc32cCombineGoogle(firstPartHash, secondPartHash, secondObject.length);
+    int actual = Crc32cUtility.concatCrc32c(firstPartHash, secondPartHash, secondObject.length);
     assertThat(actual).isEqualTo(expected);
   }
 }

google-cloud-storage/src/test/java/com/google/cloud/storage/Crc32cUtilityTest.java

@@ -30,7 +30,7 @@ public void testCrc32cCombine() {
     int object2_hash = 0x31AA814E;
     // length("world") -> 5
     int object2_size = 5;
-    int combined = Crc32cUtility.crc32cCombineGoogle(object1_hash, object2_hash, object2_size);
+    int combined = Crc32cUtility.concatCrc32c(object1_hash, object2_hash, object2_size);
     Assert.assertEquals(expected, combined);
   }
 
@@ -42,7 +42,7 @@ public void testCrc32cCombineGuavaValues() {
     int expected = Hashing.crc32c().hashBytes(helloWorld.getBytes()).asInt();
     int object1Hash = Hashing.crc32c().hashBytes(hello.getBytes()).asInt();
     int object2Hash = Hashing.crc32c().hashBytes(world.getBytes()).asInt();
-    int combined = Crc32cUtility.crc32cCombineGoogle(object1Hash, object2Hash, world.length());
+    int combined = Crc32cUtility.concatCrc32c(object1Hash, object2Hash, world.length());
     Assert.assertEquals(expected, combined);
   }
 }