Cryptonight variant 2

Contains two modifications to improve ASIC resistance: shuffle and integer math.

Shuffle makes use of the whole 64-byte cache line instead of 16 bytes only, making Cryptonight 4 times more demanding for memory bandwidth.

Integer math adds 64:32 bit integer division followed by 64 bit integer square root, adding large and unavoidable computational latency to the main loop.

More details and performance numbers: https://github.com/SChernykh/xmr-stak-cpu/blob/master/README.md

2 files changed, 361 insertions, 54 deletions

diff --git a/src/crypto/slow-hash.c b/src/crypto/slow-hash.c
index 9d4fc0dfa..a4d2b58de 100644
--- a/src/crypto/slow-hash.c
+++ b/src/crypto/slow-hash.c
@@ -38,6 +38,7 @@
 #include "common/int-util.h"
 #include "hash-ops.h"
 #include "oaes_lib.h"
+#include "variant2_int_sqrt.h"
 
 #define MEMORY         (1 << 21) // 2MB scratchpad
 #define ITER           (1 << 20)
@@ -50,7 +51,7 @@ extern int aesb_single_round(const uint8_t *in, uint8_t*out, const uint8_t *expa
 extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *expandedKey);
 
 #define VARIANT1_1(p) \
-  do if (variant > 0) \
+  do if (variant == 1) \
   { \
     const uint8_t tmp = ((const uint8_t*)(p))[11]; \
     static const uint32_t table = 0x75310; \
@@ -59,7 +60,7 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
   } while(0)
 
 #define VARIANT1_2(p) \
-  do if (variant > 0) \
+  do if (variant == 1) \
   { \
     xor64(p, tweak1_2); \
   } while(0)
@@ -67,7 +68,7 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
 #define VARIANT1_CHECK() \
   do if (length < 43) \
   { \
-    fprintf(stderr, "Cryptonight variants need at least 43 bytes of data"); \
+    fprintf(stderr, "Cryptonight variant 1 needs at least 43 bytes of data"); \
     _exit(1); \
   } while(0)
 
@@ -75,7 +76,7 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
 
 #define VARIANT1_PORTABLE_INIT() \
   uint8_t tweak1_2[8]; \
-  do if (variant > 0) \
+  do if (variant == 1) \
   { \
     VARIANT1_CHECK(); \
     memcpy(&tweak1_2, &state.hs.b[192], sizeof(tweak1_2)); \
@@ -83,11 +84,119 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
   } while(0)
 
 #define VARIANT1_INIT64() \
-  if (variant > 0) \
+  if (variant == 1) \
   { \
     VARIANT1_CHECK(); \
   } \
-  const uint64_t tweak1_2 = variant > 0 ? (state.hs.w[24] ^ (*((const uint64_t*)NONCE_POINTER))) : 0
+  const uint64_t tweak1_2 = (variant == 1) ? (state.hs.w[24] ^ (*((const uint64_t*)NONCE_POINTER))) : 0
+
+#define VARIANT2_INIT64() \
+  uint64_t division_result = 0; \
+  uint64_t sqrt_result = 0; \
+  do if (variant >= 2) \
+  { \
+    U64(b)[2] = state.hs.w[8] ^ state.hs.w[10]; \
+    U64(b)[3] = state.hs.w[9] ^ state.hs.w[11]; \
+    division_result = state.hs.w[12]; \
+    sqrt_result = state.hs.w[13]; \
+  } while (0)
+
+#define VARIANT2_PORTABLE_INIT() \
+  uint64_t division_result = 0; \
+  uint64_t sqrt_result = 0; \
+  do if (variant >= 2) \
+  { \
+    memcpy(b + AES_BLOCK_SIZE, state.hs.b + 64, AES_BLOCK_SIZE); \
+    xor64(b + AES_BLOCK_SIZE, state.hs.b + 80); \
+    xor64(b + AES_BLOCK_SIZE + 8, state.hs.b + 88); \
+    division_result = state.hs.w[12]; \
+    sqrt_result = state.hs.w[13]; \
+  } while (0)
+
+#define VARIANT2_SHUFFLE_ADD_SSE2(base_ptr, offset) \
+  do if (variant >= 2) \
+  { \
+    const __m128i chunk1 = _mm_load_si128((__m128i *)((base_ptr) + ((offset) ^ 0x10))); \
+    const __m128i chunk2 = _mm_load_si128((__m128i *)((base_ptr) + ((offset) ^ 0x20))); \
+    const __m128i chunk3 = _mm_load_si128((__m128i *)((base_ptr) + ((offset) ^ 0x30))); \
+    _mm_store_si128((__m128i *)((base_ptr) + ((offset) ^ 0x10)), _mm_add_epi64(chunk3, _b1)); \
+    _mm_store_si128((__m128i *)((base_ptr) + ((offset) ^ 0x20)), _mm_add_epi64(chunk1, _b)); \
+    _mm_store_si128((__m128i *)((base_ptr) + ((offset) ^ 0x30)), _mm_add_epi64(chunk2, _a)); \
+  } while (0)
+
+#define VARIANT2_SHUFFLE_ADD_NEON(base_ptr, offset) \
+  do if (variant >= 2) \
+  { \
+    const uint64x2_t chunk1 = vld1q_u64(U64((base_ptr) + ((offset) ^ 0x10))); \
+    const uint64x2_t chunk2 = vld1q_u64(U64((base_ptr) + ((offset) ^ 0x20))); \
+    const uint64x2_t chunk3 = vld1q_u64(U64((base_ptr) + ((offset) ^ 0x30))); \
+    vst1q_u64(U64((base_ptr) + ((offset) ^ 0x10)), vaddq_u64(chunk3, vreinterpretq_u64_u8(_b1))); \
+    vst1q_u64(U64((base_ptr) + ((offset) ^ 0x20)), vaddq_u64(chunk1, vreinterpretq_u64_u8(_b))); \
+    vst1q_u64(U64((base_ptr) + ((offset) ^ 0x30)), vaddq_u64(chunk2, vreinterpretq_u64_u8(_a))); \
+  } while (0)
+
+#define VARIANT2_PORTABLE_SHUFFLE_ADD(base_ptr, offset) \
+  do if (variant >= 2) \
+  { \
+    uint64_t* chunk1 = U64((base_ptr) + ((offset) ^ 0x10)); \
+    uint64_t* chunk2 = U64((base_ptr) + ((offset) ^ 0x20)); \
+    uint64_t* chunk3 = U64((base_ptr) + ((offset) ^ 0x30)); \
+    \
+    const uint64_t chunk1_old[2] = { chunk1[0], chunk1[1] }; \
+    \
+    uint64_t b1[2]; \
+    memcpy(b1, b + 16, 16); \
+    chunk1[0] = chunk3[0] + b1[0]; \
+    chunk1[1] = chunk3[1] + b1[1]; \
+    \
+    uint64_t a0[2]; \
+    memcpy(a0, a, 16); \
+    chunk3[0] = chunk2[0] + a0[0]; \
+    chunk3[1] = chunk2[1] + a0[1]; \
+    \
+    uint64_t b0[2]; \
+    memcpy(b0, b, 16); \
+    chunk2[0] = chunk1_old[0] + b0[0]; \
+    chunk2[1] = chunk1_old[1] + b0[1]; \
+  } while (0)
+
+#define VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr) \
+  ((uint64_t*)(b))[0] ^= division_result ^ (sqrt_result << 32); \
+  { \
+    const uint64_t dividend = ((uint64_t*)(ptr))[1]; \
+    const uint32_t divisor = (((uint64_t*)(ptr))[0] + (uint32_t)(sqrt_result << 1)) | 0x80000001UL; \
+    division_result = ((uint32_t)(dividend / divisor)) + \
+                     (((uint64_t)(dividend % divisor)) << 32); \
+  } \
+  const uint64_t sqrt_input = ((uint64_t*)(ptr))[0] + division_result
+
+#define VARIANT2_INTEGER_MATH_SSE2(b, ptr) \
+  do if (variant >= 2) \
+  { \
+    VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr); \
+    VARIANT2_INTEGER_MATH_SQRT_STEP_SSE2(); \
+    VARIANT2_INTEGER_MATH_SQRT_FIXUP(sqrt_result); \
+  } while(0)
+
+#if defined DBL_MANT_DIG && (DBL_MANT_DIG >= 50)
+  // double precision floating point type has enough bits of precision on current platform
+  #define VARIANT2_PORTABLE_INTEGER_MATH(b, ptr) \
+    do if (variant >= 2) \
+    { \
+      VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr); \
+      VARIANT2_INTEGER_MATH_SQRT_STEP_FP64(); \
+      VARIANT2_INTEGER_MATH_SQRT_FIXUP(sqrt_result); \
+    } while (0)
+#else
+  // double precision floating point type is not good enough on current platform
+  // fall back to the reference code (integer only)
+  #define VARIANT2_PORTABLE_INTEGER_MATH(b, ptr) \
+    do if (variant >= 2) \
+    { \
+      VARIANT2_INTEGER_MATH_DIVISION_STEP(b, ptr); \
+      VARIANT2_INTEGER_MATH_SQRT_STEP_REF(); \
+    } while (0)
+#endif
 
 #if !defined NO_AES && (defined(__x86_64__) || (defined(_MSC_VER) && defined(_WIN64)))
 // Optimised code below, uses x86-specific intrinsics, SSE2, AES-NI
@@ -164,19 +273,22 @@ extern int aesb_pseudo_round(const uint8_t *in, uint8_t *out, const uint8_t *exp
  * This code is based upon an optimized implementation by dga.
  */
 #define post_aes() \
+  VARIANT2_SHUFFLE_ADD_SSE2(hp_state, j); \
   _mm_store_si128(R128(c), _c); \
-  _b = _mm_xor_si128(_b, _c); \
-  _mm_store_si128(R128(&hp_state[j]), _b); \
+  _mm_store_si128(R128(&hp_state[j]), _mm_xor_si128(_b, _c)); \
   VARIANT1_1(&hp_state[j]); \
   j = state_index(c); \
   p = U64(&hp_state[j]); \
   b[0] = p[0]; b[1] = p[1]; \
+  VARIANT2_INTEGER_MATH_SSE2(b, c); \
   __mul(); \
+  VARIANT2_SHUFFLE_ADD_SSE2(hp_state, j); \
   a[0] += hi; a[1] += lo; \
   p = U64(&hp_state[j]); \
   p[0] = a[0];  p[1] = a[1]; \
   a[0] ^= b[0]; a[1] ^= b[1]; \
   VARIANT1_2(p + 1); \
+  _b1 = _b; \
   _b = _c; \
 
 #if defined(_MSC_VER)
@@ -570,10 +682,10 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
 
     uint8_t text[INIT_SIZE_BYTE];
     RDATA_ALIGN16 uint64_t a[2];
-    RDATA_ALIGN16 uint64_t b[2];
+    RDATA_ALIGN16 uint64_t b[4];
     RDATA_ALIGN16 uint64_t c[2];
     union cn_slow_hash_state state;
-    __m128i _a, _b, _c;
+    __m128i _a, _b, _b1, _c;
     uint64_t hi, lo;
 
     size_t i, j;
@@ -599,6 +711,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
     memcpy(text, state.init, INIT_SIZE_BYTE);
 
     VARIANT1_INIT64();
+    VARIANT2_INIT64();
 
     /* CryptoNight Step 2:  Iteratively encrypt the results from Keccak to fill
      * the 2MB large random access buffer.
@@ -637,6 +750,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
      */
 
     _b = _mm_load_si128(R128(b));
+    _b1 = _mm_load_si128(R128(b) + 1);
     // Two independent versions, one with AES, one without, to ensure that
     // the useAes test is only performed once, not every iteration.
     if(useAes)
@@ -761,19 +875,22 @@ union cn_slow_hash_state
   _a = vld1q_u8((const uint8_t *)a); \
 
 #define post_aes() \
+  VARIANT2_SHUFFLE_ADD_NEON(hp_state, j); \
   vst1q_u8((uint8_t *)c, _c); \
-  _b = veorq_u8(_b, _c); \
-  vst1q_u8(&hp_state[j], _b); \
+  vst1q_u8(&hp_state[j], veorq_u8(_b, _c)); \
   VARIANT1_1(&hp_state[j]); \
   j = state_index(c); \
   p = U64(&hp_state[j]); \
   b[0] = p[0]; b[1] = p[1]; \
+  VARIANT2_PORTABLE_INTEGER_MATH(b, c); \
   __mul(); \
+  VARIANT2_SHUFFLE_ADD_NEON(hp_state, j); \
   a[0] += hi; a[1] += lo; \
   p = U64(&hp_state[j]); \
   p[0] = a[0];  p[1] = a[1]; \
   a[0] ^= b[0]; a[1] ^= b[1]; \
   VARIANT1_2(p + 1); \
+  _b1 = _b; \
   _b = _c; \
 
 
@@ -912,10 +1029,10 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
 
     uint8_t text[INIT_SIZE_BYTE];
     RDATA_ALIGN16 uint64_t a[2];
-    RDATA_ALIGN16 uint64_t b[2];
+    RDATA_ALIGN16 uint64_t b[4];
     RDATA_ALIGN16 uint64_t c[2];
     union cn_slow_hash_state state;
-    uint8x16_t _a, _b, _c, zero = {0};
+    uint8x16_t _a, _b, _b1, _c, zero = {0};
     uint64_t hi, lo;
 
     size_t i, j;
@@ -936,6 +1053,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
     memcpy(text, state.init, INIT_SIZE_BYTE);
 
     VARIANT1_INIT64();
+    VARIANT2_INIT64();
 
     /* CryptoNight Step 2:  Iteratively encrypt the results from Keccak to fill
      * the 2MB large random access buffer.
@@ -959,7 +1077,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
      */
 
     _b = vld1q_u8((const uint8_t *)b);
-
+    _b1 = vld1q_u8(((const uint8_t *)b) + AES_BLOCK_SIZE);
 
     for(i = 0; i < ITER / 2; i++)
     {
@@ -1075,6 +1193,11 @@ __asm__ __volatile__(
 #endif /* !aarch64 */
 #endif // NO_OPTIMIZED_MULTIPLY_ON_ARM
 
+STATIC INLINE void copy_block(uint8_t* dst, const uint8_t* src)
+{
+  memcpy(dst, src, AES_BLOCK_SIZE);
+}
+
 STATIC INLINE void sum_half_blocks(uint8_t* a, const uint8_t* b)
 {
   uint64_t a0, a1, b0, b1;
@@ -1109,7 +1232,9 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
 {
     uint8_t text[INIT_SIZE_BYTE];
     uint8_t a[AES_BLOCK_SIZE];
-    uint8_t b[AES_BLOCK_SIZE];
+    uint8_t b[AES_BLOCK_SIZE * 2];
+    uint8_t c[AES_BLOCK_SIZE];
+    uint8_t c1[AES_BLOCK_SIZE];
     uint8_t d[AES_BLOCK_SIZE];
     uint8_t aes_key[AES_KEY_SIZE];
     RDATA_ALIGN16 uint8_t expandedKey[256];
@@ -1138,11 +1263,12 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
     }
     memcpy(text, state.init, INIT_SIZE_BYTE);
 
-    VARIANT1_INIT64();
-
     aes_ctx = (oaes_ctx *) oaes_alloc();
     oaes_key_import_data(aes_ctx, state.hs.b, AES_KEY_SIZE);
 
+    VARIANT1_INIT64();
+    VARIANT2_INIT64();
+
     // use aligned data
     memcpy(expandedKey, aes_ctx->key->exp_data, aes_ctx->key->exp_data_len);
     for(i = 0; i < MEMORY / INIT_SIZE_BYTE; i++)
@@ -1163,23 +1289,33 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
       #define state_index(x) ((*(uint32_t *) x) & MASK)
 
       // Iteration 1
-      p = &long_state[state_index(a)];
+      j = state_index(a);
+      p = &long_state[j];
       aesb_single_round(p, p, a);
+      copy_block(c1, p);
 
-      xor_blocks(b, p);
-      swap_blocks(b, p);
-      swap_blocks(a, b);
+      VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+      xor_blocks(p, b);
       VARIANT1_1(p);
 
       // Iteration 2
-      p = &long_state[state_index(a)];
-
-      mul(a, p, d);
-      sum_half_blocks(b, d);
-      swap_blocks(b, p);
-      xor_blocks(b, p);
-      swap_blocks(a, b);
-      VARIANT1_2(U64(p) + 1);
+      j = state_index(c1);
+      p = &long_state[j];
+      copy_block(c, p);
+
+      VARIANT2_PORTABLE_INTEGER_MATH(c, c1);
+      mul(c1, c, d);
+      VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+      sum_half_blocks(a, d);
+      swap_blocks(a, c);
+      xor_blocks(a, c);
+      VARIANT1_2(U64(c) + 1);
+      copy_block(p, c);
+
+      if (variant >= 2) {
+        copy_block(b + AES_BLOCK_SIZE, b);
+      }
+      copy_block(b, c1);
     }
 
     memcpy(text, state.init, INIT_SIZE_BYTE);
@@ -1298,8 +1434,9 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
   union cn_slow_hash_state state;
   uint8_t text[INIT_SIZE_BYTE];
   uint8_t a[AES_BLOCK_SIZE];
-  uint8_t b[AES_BLOCK_SIZE];
-  uint8_t c[AES_BLOCK_SIZE];
+  uint8_t b[AES_BLOCK_SIZE * 2];
+  uint8_t c1[AES_BLOCK_SIZE];
+  uint8_t c2[AES_BLOCK_SIZE];
   uint8_t d[AES_BLOCK_SIZE];
   size_t i, j;
   uint8_t aes_key[AES_KEY_SIZE];
@@ -1315,6 +1452,7 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
   aes_ctx = (oaes_ctx *) oaes_alloc();
 
   VARIANT1_PORTABLE_INIT();
+  VARIANT2_PORTABLE_INIT();
 
   oaes_key_import_data(aes_ctx, aes_key, AES_KEY_SIZE);
   for (i = 0; i < MEMORY / INIT_SIZE_BYTE; i++) {
@@ -1324,9 +1462,9 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
     memcpy(&long_state[i * INIT_SIZE_BYTE], text, INIT_SIZE_BYTE);
   }
 
-  for (i = 0; i < 16; i++) {
-    a[i] = state.k[     i] ^ state.k[32 + i];
-    b[i] = state.k[16 + i] ^ state.k[48 + i];
+  for (i = 0; i < AES_BLOCK_SIZE; i++) {
+    a[i] = state.k[     i] ^ state.k[AES_BLOCK_SIZE * 2 + i];
+    b[i] = state.k[AES_BLOCK_SIZE + i] ^ state.k[AES_BLOCK_SIZE * 3 + i];
   }
 
   for (i = 0; i < ITER / 2; i++) {
@@ -1335,26 +1473,32 @@ void cn_slow_hash(const void *data, size_t length, char *hash, int variant, int
      * next address  <-+
      */
     /* Iteration 1 */
-    j = e2i(a, MEMORY / AES_BLOCK_SIZE);
-    copy_block(c, &long_state[j * AES_BLOCK_SIZE]);
-    aesb_single_round(c, c, a);
-    xor_blocks(b, c);
-    swap_blocks(b, c);
-    copy_block(&long_state[j * AES_BLOCK_SIZE], c);
-    assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE));
-    swap_blocks(a, b);
-    VARIANT1_1(&long_state[j * AES_BLOCK_SIZE]);
+    j = e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE;
+    copy_block(c1, &long_state[j]);
+    aesb_single_round(c1, c1, a);
+    VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+    copy_block(&long_state[j], c1);
+    xor_blocks(&long_state[j], b);
+    assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE);
+    VARIANT1_1(&long_state[j]);
     /* Iteration 2 */
-    j = e2i(a, MEMORY / AES_BLOCK_SIZE);
-    copy_block(c, &long_state[j * AES_BLOCK_SIZE]);
-    mul(a, c, d);
-    sum_half_blocks(b, d);
-    swap_blocks(b, c);
-    xor_blocks(b, c);
-    VARIANT1_2(c + 8);
-    copy_block(&long_state[j * AES_BLOCK_SIZE], c);
-    assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE));
-    swap_blocks(a, b);
+    j = e2i(c1, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE;
+    copy_block(c2, &long_state[j]);
+    VARIANT2_PORTABLE_INTEGER_MATH(c2, c1);
+    mul(c1, c2, d);
+    VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
+    swap_blocks(a, c1);
+    sum_half_blocks(c1, d);
+    swap_blocks(c1, c2);
+    xor_blocks(c1, c2);
+    VARIANT1_2(c2 + 8);
+    copy_block(&long_state[j], c2);
+    assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE);
+    if (variant >= 2) {
+      copy_block(b + AES_BLOCK_SIZE, b);
+    }
+    copy_block(b, a);
+    copy_block(a, c1);
   }
 
   memcpy(text, state.init, INIT_SIZE_BYTE);
diff --git a/src/crypto/variant2_int_sqrt.h b/src/crypto/variant2_int_sqrt.h
new file mode 100644
index 000000000..b405bb798
--- /dev/null
+++ b/src/crypto/variant2_int_sqrt.h
@@ -0,0 +1,163 @@
+#ifndef VARIANT2_INT_SQRT_H
+#define VARIANT2_INT_SQRT_H
+
+#include <math.h>
+#include <float.h>
+
+#define VARIANT2_INTEGER_MATH_SQRT_STEP_SSE2() \
+  do { \
+    const __m128i exp_double_bias = _mm_set_epi64x(0, 1023ULL << 52); \
+    __m128d x = _mm_castsi128_pd(_mm_add_epi64(_mm_cvtsi64_si128(sqrt_input >> 12), exp_double_bias)); \
+    x = _mm_sqrt_sd(_mm_setzero_pd(), x); \
+    sqrt_result = (uint64_t)(_mm_cvtsi128_si64(_mm_sub_epi64(_mm_castpd_si128(x), exp_double_bias))) >> 19; \
+  } while(0)
+
+#define VARIANT2_INTEGER_MATH_SQRT_STEP_FP64() \
+  do { \
+    sqrt_result = sqrt(sqrt_input + 18446744073709551616.0) * 2.0 - 8589934592.0; \
+  } while(0)
+
+#define VARIANT2_INTEGER_MATH_SQRT_STEP_REF() \
+  sqrt_result = integer_square_root_v2(sqrt_input)
+
+// Reference implementation of the integer square root for Cryptonight variant 2
+// Computes integer part of "sqrt(2^64 + n) * 2 - 2^33"
+//
+// In other words, given 64-bit unsigned integer n:
+// 1) Write it as x = 1.NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN000... in binary (1 <= x < 2, all 64 bits of n are used)
+// 2) Calculate sqrt(x) = 1.0RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR... (1 <= sqrt(x) < sqrt(2), so it will always start with "1.0" in binary)
+// 3) Take 32 bits that come after "1.0" and return them as a 32-bit unsigned integer, discard all remaining bits
+//
+// Some sample inputs and outputs:
+//
+// Input            | Output     | Exact value of "sqrt(2^64 + n) * 2 - 2^33"
+// -----------------|------------|-------------------------------------------
+// 0                | 0          | 0
+// 2^32             | 0          | 0.99999999994179233909330885695244...
+// 2^32 + 1         | 1          | 1.0000000001746229827200734316305...
+// 2^50             | 262140     | 262140.00012206565608606978175873...
+// 2^55 + 20963331  | 8384515    | 8384515.9999999997673963974959744...
+// 2^55 + 20963332  | 8384516    | 8384516
+// 2^62 + 26599786  | 1013904242 | 1013904242.9999999999479374853545...
+// 2^62 + 26599787  | 1013904243 | 1013904243.0000000001561875439364...
+// 2^64 - 1         | 3558067407 | 3558067407.9041987696409179931096...
+
+// The reference implementation as it is now uses only unsigned int64 arithmetic, so it can't have undefined behavior
+// It was tested once for all edge cases and confirmed correct
+static inline uint32_t integer_square_root_v2(uint64_t n)
+{
+  uint64_t r = 1ULL << 63;
+
+  for (uint64_t bit = 1ULL << 60; bit; bit >>= 2)
+  {
+    const bool b = (n < r + bit);
+    const uint64_t n_next = n - (r + bit);
+    const uint64_t r_next = r + bit * 2;
+    n = b ? n : n_next;
+    r = b ? r : r_next;
+    r >>= 1;
+  }
+
+  return r * 2 + ((n > r) ? 1 : 0);
+}
+
+/*
+VARIANT2_INTEGER_MATH_SQRT_FIXUP checks that "r" is an integer part of "sqrt(2^64 + sqrt_input) * 2 - 2^33" and adds or subtracts 1 if needed
+It's hard to understand how it works, so here is a full calculation of formulas used in VARIANT2_INTEGER_MATH_SQRT_FIXUP
+
+The following inequalities must hold for r if it's an integer part of "sqrt(2^64 + sqrt_input) * 2 - 2^33":
+1) r <= sqrt(2^64 + sqrt_input) * 2 - 2^33
+2) r + 1 > sqrt(2^64 + sqrt_input) * 2 - 2^33
+
+We need to check them using only unsigned integer arithmetic to avoid rounding errors and undefined behavior
+
+First inequality: r <= sqrt(2^64 + sqrt_input) * 2 - 2^33
+-----------------------------------------------------------------------------------
+r <= sqrt(2^64 + sqrt_input) * 2 - 2^33
+r + 2^33 <= sqrt(2^64 + sqrt_input) * 2
+r/2 + 2^32 <= sqrt(2^64 + sqrt_input)
+(r/2 + 2^32)^2 <= 2^64 + sqrt_input
+
+Rewrite r as r = s * 2 + b (s = trunc(r/2), b is 0 or 1)
+
+((s*2+b)/2 + 2^32)^2 <= 2^64 + sqrt_input
+(s*2+b)^2/4 + 2*2^32*(s*2+b)/2 + 2^64 <= 2^64 + sqrt_input
+(s*2+b)^2/4 + 2*2^32*(s*2+b)/2 <= sqrt_input
+(s*2+b)^2/4 + 2^32*r <= sqrt_input
+(s^2*4+2*s*2*b+b^2)/4 + 2^32*r <= sqrt_input
+s^2+s*b+b^2/4 + 2^32*r <= sqrt_input
+s*(s+b) + b^2/4 + 2^32*r <= sqrt_input
+
+Let r2 = s*(s+b) + r*2^32
+r2 + b^2/4 <= sqrt_input
+
+If this inequality doesn't hold, then we must decrement r: IF "r2 + b^2/4 > sqrt_input" THEN r = r - 1
+
+b can be 0 or 1
+If b is 0 then we need to compare "r2 > sqrt_input"
+If b is 1 then b^2/4 = 0.25, so we need to compare "r2 + 0.25 > sqrt_input"
+Since both r2 and sqrt_input are integers, we can safely replace it with "r2 + 1 > sqrt_input"
+-----------------------------------------------------------------------------------
+Both cases can be merged to a single expression "r2 + b > sqrt_input"
+-----------------------------------------------------------------------------------
+There will be no overflow when calculating "r2 + b", so it's safe to compare with sqrt_input:
+r2 + b = s*(s+b) + r*2^32 + b
+The largest value s, b and r can have is s = 1779033703, b = 1, r = 3558067407 when sqrt_input = 2^64 - 1
+r2 + b <= 1779033703*1779033704 + 3558067407*2^32 + 1 = 18446744068217447385 < 2^64
+
+Second inequality: r + 1 > sqrt(2^64 + sqrt_input) * 2 - 2^33
+-----------------------------------------------------------------------------------
+r + 1 > sqrt(2^64 + sqrt_input) * 2 - 2^33
+r + 1 + 2^33 > sqrt(2^64 + sqrt_input) * 2
+((r+1)/2 + 2^32)^2 > 2^64 + sqrt_input
+
+Rewrite r as r = s * 2 + b (s = trunc(r/2), b is 0 or 1)
+
+((s*2+b+1)/2 + 2^32)^2 > 2^64 + sqrt_input
+(s*2+b+1)^2/4 + 2*(s*2+b+1)/2*2^32 + 2^64 > 2^64 + sqrt_input
+(s*2+b+1)^2/4 + (s*2+b+1)*2^32 > sqrt_input
+(s*2+b+1)^2/4 + (r+1)*2^32 > sqrt_input
+(s*2+(b+1))^2/4 + r*2^32 + 2^32 > sqrt_input
+(s^2*4+2*s*2*(b+1)+(b+1)^2)/4 + r*2^32 + 2^32 > sqrt_input
+s^2+s*(b+1)+(b+1)^2/4 + r*2^32 + 2^32 > sqrt_input
+s*(s+b) + s + (b+1)^2/4 + r*2^32 + 2^32 > sqrt_input
+
+Let r2 = s*(s+b) + r*2^32
+
+r2 + s + (b+1)^2/4 + 2^32 > sqrt_input
+r2 + 2^32 + (b+1)^2/4 > sqrt_input - s
+
+If this inequality doesn't hold, then we must decrement r: IF "r2 + 2^32 + (b+1)^2/4 <= sqrt_input - s" THEN r = r - 1
+b can be 0 or 1
+If b is 0 then we need to compare "r2 + 2^32 + 1/4 <= sqrt_input - s" which is equal to "r2 + 2^32 < sqrt_input - s" because all numbers here are integers
+If b is 1 then (b+1)^2/4 = 1, so we need to compare "r2 + 2^32 + 1 <= sqrt_input - s" which is also equal to "r2 + 2^32 < sqrt_input - s"
+-----------------------------------------------------------------------------------
+Both cases can be merged to a single expression "r2 + 2^32 < sqrt_input - s"
+-----------------------------------------------------------------------------------
+There will be no overflow when calculating "r2 + 2^32":
+r2 + 2^32 = s*(s+b) + r*2^32 + 2^32 = s*(s+b) + (r+1)*2^32
+The largest value s, b and r can have is s = 1779033703, b = 1, r = 3558067407 when sqrt_input = 2^64 - 1
+r2 + b <= 1779033703*1779033704 + 3558067408*2^32 = 18446744072512414680 < 2^64
+
+There will be no integer overflow when calculating "sqrt_input - s", i.e. "sqrt_input >= s" at all times:
+s = trunc(r/2) = trunc(sqrt(2^64 + sqrt_input) - 2^32) < sqrt(2^64 + sqrt_input) - 2^32 + 1
+sqrt_input > sqrt(2^64 + sqrt_input) - 2^32 + 1
+sqrt_input + 2^32 - 1 > sqrt(2^64 + sqrt_input)
+(sqrt_input + 2^32 - 1)^2 > sqrt_input + 2^64
+sqrt_input^2 + 2*sqrt_input*(2^32 - 1) + (2^32-1)^2 > sqrt_input + 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 2) + (2^32-1)^2 > sqrt_input + 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 3) + (2^32-1)^2 > 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 3) + 2^64-2^33+1 > 2^64
+sqrt_input^2 + sqrt_input*(2^33 - 3) - 2^33 + 1 > 0
+This inequality is true if sqrt_input > 1 and it's easy to check that s = 0 if sqrt_input is 0 or 1, so there will be no integer overflow
+*/
+
+#define VARIANT2_INTEGER_MATH_SQRT_FIXUP(r) \
+  do { \
+    const uint64_t s = r >> 1; \
+    const uint64_t b = r & 1; \
+    const uint64_t r2 = (uint64_t)(s) * (s + b) + (r << 32); \
+    r += ((r2 + b > sqrt_input) ? -1 : 0) + ((r2 + (1ULL << 32) < sqrt_input - s) ? 1 : 0); \
+  } while(0)
+
+#endif