/// <summary>
/// <paramref name="x"/>y≡1(mod <paramref name="m"/>) なる y のうち、0≤y<<paramref name="m"/> を満たすものを返します。
/// </summary>
/// <remarks>
/// <para>制約: gcd(<paramref name="x"/>,<paramref name="m"/>)=1, 1≤<paramref name="m"/></para>
/// <para>計算量: O(log<paramref name="m"/>)</para>
/// </remarks>
public static long InvMod(long x, int m)
{
Debug.Assert(1 <= m);
var(g, res) = InternalMath.InvGCD(x, m);
Debug.Assert(g == 1);
return(res);
}
public static long InvMod(long x, long m)
{
Contract.Assert(1 <= m, reason: $"1 <= {nameof(m)}");
var(g, res) = InternalMath.InvGCD(x, m);
Contract.Assert(g == 1, reason: $"gcd({nameof(x)}, {nameof(m)}) must be 1.");
return(res);
}
/// <summary>
/// 同じ長さ n の配列 <paramref name="r"/>, <paramref name="m"/> について、x≡<paramref name="r"/>[i] (mod <paramref name="m"/>[i]),∀i∈{0,1,⋯,n−1} を解きます。
/// </summary>
/// <remarks>
/// <para>制約: |<paramref name="r"/>|=|<paramref name="m"/>|, 1≤<paramref name="m"/>[i], lcm(m[i]) が ll に収まる</para>
/// <para>計算量: O(nloglcm(<paramref name="m"/>))</para>
/// </remarks>
/// <returns>答えは(存在するならば) y,z(0≤y<z=lcm(<paramref name="m"/>[i])) を用いて x≡y(mod z) の形で書ける。答えがない場合は(0,0)、n=0 の時は(0,1)、それ以外の場合は(y,z)。</returns>
public static (long y, long m) CRT(long[] r, long[] m)
{
Contract.Assert(r.Length == m.Length, reason: $"Length of {nameof(r)} and {nameof(m)} must be same.");
long r0 = 0, m0 = 1;
for (int i = 0; i < m.Length; i++)
{
Contract.Assert(1 <= m[i], reason: $"All of {nameof(m)} must be greater or equal 1.");
long r1 = InternalMath.SafeMod(r[i], m[i]);
long m1 = m[i];
if (m0 < m1)
{
(r0, r1) = (r1, r0);
(m0, m1) = (m1, m0);
}
if (m0 % m1 == 0)
{
if (r0 % m1 != r1)
{
return(0, 0);
}
continue;
}
var(g, im) = InternalMath.InvGCD(m0, m1);
long u1 = (m1 / g);
if ((r1 - r0) % g != 0)
{
return(0, 0);
}
long x = (r1 - r0) / g % u1 * im % u1;
r0 += x * m0;
m0 *= u1;
if (r0 < 0)
{
r0 += m0;
}
}
return(r0, m0);
}
/// <summary>
/// 同じ長さ n の配列 <paramref name="r"/>, <paramref name="m"/> について、x≡<paramref name="r"/>[i] (mod <paramref name="m"/>[i]),∀i∈{0,1,⋯,n−1} を解きます。
/// </summary>
/// <remarks>
/// <para>制約: |<paramref name="r"/>|=|<paramref name="m"/>|, 1≤<paramref name="m"/>[i], lcm(m[i]) が ll に収まる</para>
/// <para>計算量: O(nloglcm(<paramref name="m"/>))</para>
/// </remarks>
/// <returns>答えは(存在するならば) y,z(0≤y<z=lcm(<paramref name="m"/>[i])) を用いて x≡y(mod z) の形で書ける。答えがない場合は(0,0)、n=0 の時は(0,1)、それ以外の場合は(y,z)。</returns>
public static (long, long) CRT(long[] r, long[] m)
{
Debug.Assert(r.Length == m.Length);
long r0 = 0, m0 = 1;
for (int i = 0; i < m.Length; i++)
{
Debug.Assert(1 <= m[i]);
long r1 = InternalMath.SafeMod(r[i], m[i]);
long m1 = m[i];
if (m0 < m1)
{
(r0, r1) = (r1, r0);
(m0, m1) = (m1, m0);
}
if (m0 % m1 == 0)
{
if (r0 % m1 != r1)
{
return(0, 0);
}
continue;
}
var(g, im) = InternalMath.InvGCD(m0, m1);
long u1 = (m1 / g);
if ((r1 - r0) % g != 0)
{
return(0, 0);
}
long x = (r1 - r0) / g % u1 * im % u1;
r0 += x * m0;
m0 *= u1;
if (r0 < 0)
{
r0 += m0;
}
}
return(r0, m0);
}
public static long[] ConvolutionLong(ReadOnlySpan <long> a, ReadOnlySpan <long> b)
{
unchecked
{
var n = a.Length;
var m = b.Length;
if (n == 0 || m == 0)
{
return(Array.Empty <long>());
}
const ulong Mod1 = 754974721; // 2^24
const ulong Mod2 = 167772161; // 2^25
const ulong Mod3 = 469762049; // 2^26
const ulong M2M3 = Mod2 * Mod3;
const ulong M1M3 = Mod1 * Mod3;
const ulong M1M2 = Mod1 * Mod2;
// (m1 * m2 * m3) % 2^64
const ulong M1M2M3 = Mod1 * Mod2 * Mod3;
const ulong i1 = 190329765;
const ulong i2 = 58587104;
const ulong i3 = 187290749;
Debug.Assert(default(FFTMod1).Mod == Mod1);
Debug.Assert(default(FFTMod2).Mod == Mod2);
Debug.Assert(default(FFTMod3).Mod == Mod3);
Debug.Assert(i1 == (ulong)InternalMath.InvGCD((long)M2M3, (long)Mod1).Item2);
Debug.Assert(i2 == (ulong)InternalMath.InvGCD((long)M1M3, (long)Mod2).Item2);
Debug.Assert(i3 == (ulong)InternalMath.InvGCD((long)M1M2, (long)Mod3).Item2);
var c1 = Convolution <FFTMod1>(a, b);
var c2 = Convolution <FFTMod2>(a, b);
var c3 = Convolution <FFTMod3>(a, b);
var c = new long[n + m - 1];
//ReadOnlySpan<ulong> offset = stackalloc ulong[] { 0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3 };
for (int i = 0; i < c.Length; i++)
{
ulong x = 0;
x += ((ulong)c1[i] * i1) % Mod1 * M2M3;
x += ((ulong)c2[i] * i2) % Mod2 * M1M3;
x += ((ulong)c3[i] * i3) % Mod3 * M1M2;
long diff = c1[i] - InternalMath.SafeMod((long)x, (long)Mod1);
if (diff < 0)
{
diff += (long)Mod1;
}
// 真値を r, 得られた値を x, M1M2M3 % 2^64 = M', B = 2^63 として、
// r = x,
// x - M' + (0 or 2B),
// x - 2M' + (0 or 2B or 4B),
// x - 3M' + (0 or 2B or 4B or 6B)
// のいずれかが成り立つ、らしい
// -> see atcoder/convolution.hpp
switch (diff % 5)
{
case 2:
x -= M1M2M3;
break;
case 3:
x -= 2 * M1M2M3;
break;
case 4:
x -= 3 * M1M2M3;
break;
}
c[i] = (long)x;
}
return(c);
}
}