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.mdpull/88/head
SChernykh
parent
1f8b0ddc83
commit
40573dd79c
@ -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
|
||||
@ -0,0 +1,10 @@
|
||||
4cf1ff9ca46eb433b36cd9f70e02b14cc06bfd18ca77fa9ccaafd1fd96c674b0 5468697320697320612074657374205468697320697320612074657374205468697320697320612074657374
|
||||
7d292e43f4751714ec07dbcb0e4bbffe2a7afb6066420960684ff57d7474c871 4c6f72656d20697073756d20646f6c6f722073697420616d65742c20636f6e73656374657475722061646970697363696e67
|
||||
335563425256edebf1d92dc342369c2f4770ebb4112ba975659bd8a0f210abd0 656c69742c2073656420646f20656975736d6f642074656d706f7220696e6369646964756e74207574206c61626f7265
|
||||
47758e86d2f57210366cec36fff26f9464d89efd116fe6ef28b718b5da120801 657420646f6c6f7265206d61676e6120616c697175612e20557420656e696d206164206d696e696d2076656e69616d2c
|
||||
48787b48d5c68f0c1dd825c32580af741cc0ee314f08133135c1e86d87a24a95 71756973206e6f737472756420657865726369746174696f6e20756c6c616d636f206c61626f726973206e697369
|
||||
93bdf47495854f7cfaaca1af8c0f39ef4a3024c10eb0dea23726b0e06ef29e84 757420616c697175697020657820656120636f6d6d6f646f20636f6e7365717561742e20447569732061757465
|
||||
a375a71d0541057ccc96719150dfe10b6e6f486b19cf4a0835e19605413a8417 697275726520646f6c6f7220696e20726570726568656e646572697420696e20766f6c7570746174652076656c6974
|
||||
163478a76f8f1432533fbdd1284d65c89f37479e54f20841c6ce4eba56c73854 657373652063696c6c756d20646f6c6f726520657520667567696174206e756c6c612070617269617475722e
|
||||
356b0470c6eea75cad7a108179e232905b23bdaf03c2824c6e619d503ee93677 4578636570746575722073696e74206f6363616563617420637570696461746174206e6f6e2070726f6964656e742c
|
||||
a47e2b007dc25bb279e197a1b91f67ecebe2ddd8791cd32dd2cb76dd21ed943f 73756e7420696e2063756c706120717569206f666669636961206465736572756e74206d6f6c6c697420616e696d20696420657374206c61626f72756d2e
|
||||
Loading…
Reference in new issue