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

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]), _mm_xor_si128(_b, _c)); \
-  _mm_store_si128(R128(&hp_state[j]), _b); \
   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], veorq_u8(_b, _c)); \
-  vst1q_u8(&hp_state[j], _b); \
   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);
+      VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
-      swap_blocks(b, p);
+      xor_blocks(p, b);
-      swap_blocks(a, b);
       VARIANT1_1(p);
 
       // Iteration 2
-      p = &long_state[state_index(a)];
+      j = state_index(c1);
+      p = &long_state[j];
+      copy_block(c, p);
 
-      mul(a, p, d);
+      VARIANT2_PORTABLE_INTEGER_MATH(c, c1);
-      sum_half_blocks(b, d);
+      mul(c1, c, d);
-      swap_blocks(b, p);
+      VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
-      xor_blocks(b, p);
+      sum_half_blocks(a, d);
-      swap_blocks(a, b);
+      swap_blocks(a, c);
-      VARIANT1_2(U64(p) + 1);
+      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 b[AES_BLOCK_SIZE * 2];
-  uint8_t c[AES_BLOCK_SIZE];
+  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++) {
+  for (i = 0; i < AES_BLOCK_SIZE; i++) {
-    a[i] = state.k[     i] ^ state.k[32 + i];
+    a[i] = state.k[     i] ^ state.k[AES_BLOCK_SIZE * 2 + i];
-    b[i] = state.k[16 + i] ^ state.k[48 + 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);
+    j = e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE;
-    copy_block(c, &long_state[j * AES_BLOCK_SIZE]);
+    copy_block(c1, &long_state[j]);
-    aesb_single_round(c, c, a);
+    aesb_single_round(c1, c1, a);
-    xor_blocks(b, c);
+    VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
-    swap_blocks(b, c);
+    copy_block(&long_state[j], c1);
-    copy_block(&long_state[j * AES_BLOCK_SIZE], c);
+    xor_blocks(&long_state[j], b);
-    assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE));
+    assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE);
-    swap_blocks(a, b);
+    VARIANT1_1(&long_state[j]);
-    VARIANT1_1(&long_state[j * AES_BLOCK_SIZE]);
     /* Iteration 2 */
-    j = e2i(a, MEMORY / AES_BLOCK_SIZE);
+    j = e2i(c1, MEMORY / AES_BLOCK_SIZE) * AES_BLOCK_SIZE;
-    copy_block(c, &long_state[j * AES_BLOCK_SIZE]);
+    copy_block(c2, &long_state[j]);
-    mul(a, c, d);
+    VARIANT2_PORTABLE_INTEGER_MATH(c2, c1);
-    sum_half_blocks(b, d);
+    mul(c1, c2, d);
-    swap_blocks(b, c);
+    VARIANT2_PORTABLE_SHUFFLE_ADD(long_state, j);
-    xor_blocks(b, c);
+    swap_blocks(a, c1);
-    VARIANT1_2(c + 8);
+    sum_half_blocks(c1, d);
-    copy_block(&long_state[j * AES_BLOCK_SIZE], c);
+    swap_blocks(c1, c2);
-    assert(j == e2i(a, MEMORY / AES_BLOCK_SIZE));
+    xor_blocks(c1, c2);
-    swap_blocks(a, b);
+    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);

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

tests/hash/CMakeLists.txt

@@ -43,8 +43,12 @@ set_property(TARGET hash-tests
   PROPERTY
     FOLDER "tests")
 
-foreach (hash IN ITEMS fast slow slow-1 tree extra-blake extra-groestl extra-jh extra-skein)
+foreach (hash IN ITEMS fast slow slow-1 slow-2 tree extra-blake extra-groestl extra-jh extra-skein)
   add_test(
     NAME    "hash-${hash}"
     COMMAND hash-tests "${hash}" "${CMAKE_CURRENT_SOURCE_DIR}/tests-${hash}.txt")
 endforeach ()
+
+add_test(
+  NAME    "hash-variant2-int-sqrt"
+  COMMAND hash-tests "variant2_int_sqrt")

tests/hash/main.cpp

@@ -33,9 +33,11 @@
 #include <iomanip>
 #include <ios>
 #include <string>
+#include <cfenv>
 
 #include "warnings.h"
 #include "crypto/hash.h"
+#include "crypto/variant2_int_sqrt.h"
 #include "../io.h"
 
 using namespace std;
@@ -57,6 +59,9 @@ extern "C" {
   static void cn_slow_hash_1(const void *data, size_t length, char *hash) {
     return cn_slow_hash(data, length, hash, 1/*variant*/, 0/*prehashed*/);
   }
+  static void cn_slow_hash_2(const void *data, size_t length, char *hash) {
+    return cn_slow_hash(data, length, hash, 2/*variant*/, 0/*prehashed*/);
+  }
 }
 POP_WARNINGS
 
@@ -67,7 +72,10 @@ struct hash_func {
 } hashes[] = {{"fast", cn_fast_hash}, {"slow", cn_slow_hash_0}, {"tree", hash_tree},
   {"extra-blake", hash_extra_blake}, {"extra-groestl", hash_extra_groestl},
   {"extra-jh", hash_extra_jh}, {"extra-skein", hash_extra_skein},
-  {"slow-1", cn_slow_hash_1}};
+  {"slow-1", cn_slow_hash_1}, {"slow-2", cn_slow_hash_2}};
+
+int test_variant2_int_sqrt();
+int test_variant2_int_sqrt_ref();
 
 int main(int argc, char *argv[]) {
   hash_f *f;
@@ -78,6 +86,36 @@ int main(int argc, char *argv[]) {
   size_t test = 0;
   bool error = false;
   if (argc != 3) {
+    if ((argc == 2) && (strcmp(argv[1], "variant2_int_sqrt") == 0)) {
+      if (test_variant2_int_sqrt_ref() != 0) {
+        return 1;
+      }
+      const int round_modes[3] = { FE_DOWNWARD, FE_TONEAREST, FE_UPWARD };
+      for (int i = 0; i < 3; ++i) {
+        std::fesetround(round_modes[i]);
+        const int result = test_variant2_int_sqrt();
+        if (result != 0) {
+          cerr << "FPU round mode was set to ";
+          switch (round_modes[i]) {
+            case FE_DOWNWARD:
+              cerr << "FE_DOWNWARD";
+              break;
+            case FE_TONEAREST:
+              cerr << "FE_TONEAREST";
+              break;
+            case FE_UPWARD:
+              cerr << "FE_UPWARD";
+              break;
+            default:
+              cerr << "unknown";
+              break;
+          }
+          cerr << endl;
+          return result;
+        }
+      }
+      return 0;
+    }
     cerr << "Wrong number of arguments" << endl;
     return 1;
   }
@@ -127,3 +165,165 @@ int main(int argc, char *argv[]) {
   }
   return error ? 1 : 0;
 }
+
+#if defined(__x86_64__) || (defined(_MSC_VER) && defined(_WIN64))
+
+#include <emmintrin.h>
+
+#if defined(_MSC_VER) || defined(__MINGW32__)
+  #include <intrin.h>
+#else
+  #include <wmmintrin.h>
+#endif
+
+#endif
+
+static inline bool test_variant2_int_sqrt_sse(const uint64_t sqrt_input, const uint64_t correct_result)
+{
+#if defined(__x86_64__) || (defined(_MSC_VER) && defined(_WIN64))
+  uint64_t sqrt_result;
+  VARIANT2_INTEGER_MATH_SQRT_STEP_SSE2();
+  VARIANT2_INTEGER_MATH_SQRT_FIXUP(sqrt_result);
+  if (sqrt_result != correct_result) {
+    cerr << "Integer sqrt (SSE2 version) returned incorrect result for N = " << sqrt_input << endl;
+    cerr << "Expected result: " << correct_result << endl;
+    cerr << "Returned result: " << sqrt_result << endl;
+    return false;
+  }
+#endif
+
+  return true;
+}
+
+static inline bool test_variant2_int_sqrt_fp64(const uint64_t sqrt_input, const uint64_t correct_result)
+{
+#if defined DBL_MANT_DIG && (DBL_MANT_DIG >= 50)
+  uint64_t sqrt_result;
+  VARIANT2_INTEGER_MATH_SQRT_STEP_FP64();
+  VARIANT2_INTEGER_MATH_SQRT_FIXUP(sqrt_result);
+  if (sqrt_result != correct_result) {
+    cerr << "Integer sqrt (FP64 version) returned incorrect result for N = " << sqrt_input << endl;
+    cerr << "Expected result: " << correct_result << endl;
+    cerr << "Returned result: " << sqrt_result << endl;
+    return false;
+  }
+#endif
+
+  return true;
+}
+
+static inline bool test_variant2_int_sqrt_ref(const uint64_t sqrt_input, const uint64_t correct_result)
+{
+  uint64_t sqrt_result;
+  VARIANT2_INTEGER_MATH_SQRT_STEP_REF();
+  if (sqrt_result != correct_result) {
+    cerr << "Integer sqrt (reference version) returned incorrect result for N = " << sqrt_input << endl;
+    cerr << "Expected result: " << correct_result << endl;
+    cerr << "Returned result: " << sqrt_result << endl;
+    return false;
+  }
+
+  return true;
+}
+
+static inline bool test_variant2_int_sqrt(const uint64_t sqrt_input, const uint64_t correct_result)
+{
+  if (!test_variant2_int_sqrt_sse(sqrt_input, correct_result)) {
+    return false;
+  }
+  if (!test_variant2_int_sqrt_fp64(sqrt_input, correct_result)) {
+    return false;
+  }
+
+  return true;
+}
+
+int test_variant2_int_sqrt()
+{
+  if (!test_variant2_int_sqrt(0, 0)) {
+    return 1;
+  }
+  if (!test_variant2_int_sqrt(1ULL << 63, 1930543745UL)) {
+    return 1;
+  }
+  if (!test_variant2_int_sqrt(uint64_t(-1), 3558067407UL)) {
+    return 1;
+  }
+
+  for (uint64_t i = 1; i <= 3558067407UL; ++i) {
+    // "i" is integer part of "sqrt(2^64 + n) * 2 - 2^33"
+    // n = (i/2 + 2^32)^2 - 2^64
+
+    const uint64_t i0 = i >> 1;
+    uint64_t n1;
+    if ((i & 1) == 0) {
+      // n = (i/2 + 2^32)^2 - 2^64
+      // n = i^2/4 + 2*2^32*i/2 + 2^64 - 2^64
+      // n = i^2/4 + 2^32*i
+      // i is even, so i^2 is divisible by 4:
+      // n = (i^2 >> 2) + (i << 32)
+
+      // int_sqrt_v2(i^2/4 + 2^32*i - 1) must be equal to i - 1
+      // int_sqrt_v2(i^2/4 + 2^32*i) must be equal to i
+      n1 = i0 * i0 + (i << 32) - 1;
+    }
+    else {
+      // n = (i/2 + 2^32)^2 - 2^64
+      // n = i^2/4 + 2*2^32*i/2 + 2^64 - 2^64
+      // n = i^2/4 + 2^32*i
+      // i is odd, so i = i0*2+1 (i0 = i >> 1)
+      // n = (i0*2+1)^2/4 + 2^32*i
+      // n = (i0^2*4+i0*4+1)/4 + 2^32*i
+      // n = i0^2+i0+1/4 + 2^32*i
+      // i0^2+i0 + 2^32*i < n < i0^2+i0+1 + 2^32*i
+
+      // int_sqrt_v2(i0^2+i0 + 2^32*i) must be equal to i - 1
+      // int_sqrt_v2(i0^2+i0+1 + 2^32*i) must be equal to i
+      n1 = i0 * i0 + i0 + (i << 32);
+    }
+
+    if (!test_variant2_int_sqrt(n1, i - 1)) {
+      return 1;
+    }
+    if (!test_variant2_int_sqrt(n1 + 1, i)) {
+      return 1;
+    }
+  }
+
+  return 0;
+}
+
+int test_variant2_int_sqrt_ref()
+{
+  if (!test_variant2_int_sqrt_ref(0, 0)) {
+    return 1;
+  }
+  if (!test_variant2_int_sqrt_ref(1ULL << 63, 1930543745UL)) {
+    return 1;
+  }
+  if (!test_variant2_int_sqrt_ref(uint64_t(-1), 3558067407UL)) {
+    return 1;
+  }
+
+  // Reference version is slow, so we test only every 83th edge case
+  // "i += 83" because 1 + 83 * 42868282 = 3558067407
+  for (uint64_t i = 1; i <= 3558067407UL; i += 83) {
+    const uint64_t i0 = i >> 1;
+    uint64_t n1;
+    if ((i & 1) == 0) {
+      n1 = i0 * i0 + (i << 32) - 1;
+    }
+    else {
+      n1 = i0 * i0 + i0 + (i << 32);
+    }
+
+    if (!test_variant2_int_sqrt_ref(n1, i - 1)) {
+      return 1;
+    }
+    if (!test_variant2_int_sqrt_ref(n1 + 1, i)) {
+      return 1;
+    }
+  }
+
+  return 0;
+}

tests/hash/tests-slow-2.txt

@@ -0,0 +1,10 @@
+4cf1ff9ca46eb433b36cd9f70e02b14cc06bfd18ca77fa9ccaafd1fd96c674b0 5468697320697320612074657374205468697320697320612074657374205468697320697320612074657374
+7d292e43f4751714ec07dbcb0e4bbffe2a7afb6066420960684ff57d7474c871 4c6f72656d20697073756d20646f6c6f722073697420616d65742c20636f6e73656374657475722061646970697363696e67
+335563425256edebf1d92dc342369c2f4770ebb4112ba975659bd8a0f210abd0 656c69742c2073656420646f20656975736d6f642074656d706f7220696e6369646964756e74207574206c61626f7265
+47758e86d2f57210366cec36fff26f9464d89efd116fe6ef28b718b5da120801 657420646f6c6f7265206d61676e6120616c697175612e20557420656e696d206164206d696e696d2076656e69616d2c
+48787b48d5c68f0c1dd825c32580af741cc0ee314f08133135c1e86d87a24a95 71756973206e6f737472756420657865726369746174696f6e20756c6c616d636f206c61626f726973206e697369
+93bdf47495854f7cfaaca1af8c0f39ef4a3024c10eb0dea23726b0e06ef29e84 757420616c697175697020657820656120636f6d6d6f646f20636f6e7365717561742e20447569732061757465
+a375a71d0541057ccc96719150dfe10b6e6f486b19cf4a0835e19605413a8417 697275726520646f6c6f7220696e20726570726568656e646572697420696e20766f6c7570746174652076656c6974
+163478a76f8f1432533fbdd1284d65c89f37479e54f20841c6ce4eba56c73854 657373652063696c6c756d20646f6c6f726520657520667567696174206e756c6c612070617269617475722e
+356b0470c6eea75cad7a108179e232905b23bdaf03c2824c6e619d503ee93677 4578636570746575722073696e74206f6363616563617420637570696461746174206e6f6e2070726f6964656e742c
+a47e2b007dc25bb279e197a1b91f67ecebe2ddd8791cd32dd2cb76dd21ed943f 73756e7420696e2063756c706120717569206f666669636961206465736572756e74206d6f6c6c697420616e696d20696420657374206c61626f72756d2e