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implement fast ExpTrunc() and CharPolyMod() #23

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implement fast ExpTrunc() and CharPolyMod() #23

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7 changes: 7 additions & 0 deletions include/NTL/ZZ_pX.h
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Original file line number Diff line number Diff line change
Expand Up @@ -548,6 +548,13 @@ inline ZZ_pX power(const ZZ_pX& a, long e)
{ ZZ_pX x; power(x, a, e); NTL_OPT_RETURN(ZZ_pX, x); }


void ExpTrunc(ZZ_pX& x, const ZZ_pX& a, long n);
inline ZZ_pX ExpTrunc(const ZZ_pX& a, long n)
{ ZZ_pX x; ExpTrunc(x, a, n); NTL_OPT_RETURN(ZZ_pX, x); }




// The following data structures and routines allow one
// to hand-craft various algorithms, using the FFT convolution
// algorithms directly.
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3 changes: 3 additions & 0 deletions include/NTL/lzz_pX.h
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Expand Up @@ -553,6 +553,9 @@ inline zz_pX power(const zz_pX& a, long e)
{ zz_pX x; power(x, a, e); NTL_OPT_RETURN(zz_pX, x); }


void ExpTrunc(zz_pX& x, const zz_pX& a, long n);
inline zz_pX ExpTrunc(const zz_pX& a, long n)
{ zz_pX x; ExpTrunc(x, a, n); NTL_OPT_RETURN(zz_pX, x); }



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72 changes: 72 additions & 0 deletions src/ZZ_pX1.cpp
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Expand Up @@ -1753,6 +1753,78 @@ void SqrTrunc(ZZ_pX& x, const ZZ_pX& a, long n)
}


// slight variant of https://homepages.loria.fr/PZimmermann/papers/fastnewton.ps.gz §5
static void ExpTruncFast(ZZ_pX& f, ZZ_pX& g, const ZZ_pX& h, long prec)
{
switch (prec) {
case 0:
clear(f);
clear(g);
return;
case 1:
f = 1;
clear(g);
return;
case 2:
f = 1 + trunc(h, 2);
g = 1;
return;
case 3:
f = 1 + trunc(h, 3) + MulTrunc(h, h, 3) / 2;
g = 1 - trunc(h, 2);
return;
}

{
ZZ_pX h2 = MulTrunc(h, h, 4) / 2;
ZZ_pX h3 = MulTrunc(h, h2, 4) / 3;
f = 1 + trunc(h, 4) + h2 + h3;
g = 1 - trunc(h, 2);
}

// invariant: f = exp(h) + O(x^n), g = exp(-h) + O(x^(n/2))
for (long n = 4; n < prec; n *= 2) {

// step 2
ZZ_pX m1 = MulTrunc(f, sqr(g), n);
g.SetLength(n);
for (long i = n/2; i < n; ++i)
g[i] = -coeff(m1, i);
g.normalize();

// step 3
ZZ_pX q = diff(trunc(h, n));

// step 4
ZZ_pX s = MulTrunc(g, f*q >> (n-1), n);
q.SetLength(2*n-1);
for (long i = 0; i <= deg(s); ++i)
q[n-1+i] = -s[i];
q.normalize();

// step 5
unsigned long length = NTL::min(deg(q)+2, prec);
ZZ_pX intq;
intq.SetLength(length);
intq[0] = 0;
for (long i = 1; i < length; ++i)
intq[i] = q[i-1] / i;
intq.normalize();
f = MulTrunc(f, 1 + h - intq, NTL::min(2*n, prec));
}
}

void ExpTrunc(ZZ_pX& x, const ZZ_pX& a, long n)
{
if (n < 1)
LogicError("ExpTrunc: bad args");
if (ConstTerm(a) != 0)
LogicError("ExpTrunc: nonzero constant term");
ZZ_pX y;
ExpTruncFast(x, y, a, n);
}


void FastTraceVec(vec_ZZ_p& S, const ZZ_pX& f)
{
long n = deg(f);
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50 changes: 28 additions & 22 deletions src/ZZ_pXCharPoly.cpp
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Expand Up @@ -29,48 +29,54 @@ void HessCharPoly(ZZ_pX& g, const ZZ_pX& a, const ZZ_pX& f)
CharPoly(g, M);
}

// "well-known" algorithm according to https://arxiv.org/pdf/1109.4323.pdf p.11
void CharPolyMod(ZZ_pX& g, const ZZ_pX& a, const ZZ_pX& ff)
{
ZZ_pX f = ff;
MakeMonic(f);
long n = deg(f);

if (n <= 0 || deg(a) >= n)
LogicError("CharPoly: bad args");
if (n <= 0 || deg(a) >= n)
LogicError("CharPolyMod: bad args");

if (IsZero(a)) {
clear(g);
SetCoeff(g, n);
return;
}

if (n > 25) {
ZZ_pX h;
MinPolyMod(h, a, f);
if (deg(h) == n) {
g = h;
return;
}
}

if (ZZ_p::modulus() < n+1) {
HessCharPoly(g, a, f);
MinPolyMod(g, a, f);
if (deg(g) < n)
HessCharPoly(g, a, f);
return;
}

vec_ZZ_p u(INIT_SIZE, n+1), v(INIT_SIZE, n+1);
// compute tr(x^0 mod f), tr(x^1 mod f), ..., tr(x^(n-1) mod f) as rev(f')/rev(f)
// see for instance https://cr.yp.to/papers/newton.pdf
vec_ZZ_p T(INIT_SIZE, n);
{
ZZ_pX rev_f = reverse(f);
ZZ_pX rev_df = reverse(diff(f));
ZZ_pX inv_rev_f = InvTrunc(rev_f, n);
ZZ_pX quot = MulTrunc(rev_df, inv_rev_f, n);
VectorCopy(T, quot, n);
}

ZZ_pX h, h1;
negate(h, a);
long i;
// compute tr(a^0 mod f), tr(a^1 mod f), ..., tr(a^n mod f)
vec_ZZ_p tr = ProjectPowers(T, n+1, a, ff);

for (i = 0; i <= n; i++) {
u[i] = i;
add(h1, h, u[i]);
resultant(v[i], f, h1);
}
// recover characteristic polynomial using exp(∫g'/g) = g;
// "well-known" according to https://dl.acm.org/doi/pdf/10.1145/1145768.1145814 Prop. 9
ZZ_pX trpoly;
trpoly.SetLength(n+1);
trpoly[0] = 0;
for (long i = 1; i < n+1; ++i)
trpoly[i] = -tr[i] / i;
trpoly.normalize();

interpolate(g, u, v);
ExpTrunc(g, trpoly, n+1);
reverse(g, g, n);
}

NTL_END_IMPL
73 changes: 73 additions & 0 deletions src/lzz_pX1.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -1668,6 +1668,79 @@ void SqrTrunc(zz_pX& x, const zz_pX& a, long n)



// slight variant of https://homepages.loria.fr/PZimmermann/papers/fastnewton.ps.gz §5
static void ExpTruncFast(zz_pX& f, zz_pX& g, const zz_pX& h, long prec)
{
switch (prec) {
case 0:
clear(f);
clear(g);
return;
case 1:
f = 1;
clear(g);
return;
case 2:
f = 1 + trunc(h, 2);
g = 1;
return;
case 3:
f = 1 + trunc(h, 3) + MulTrunc(h, h, 3) / 2;
g = 1 - trunc(h, 2);
return;
}

{
zz_pX h2 = MulTrunc(h, h, 4) / 2;
zz_pX h3 = MulTrunc(h, h2, 4) / 3;
f = 1 + trunc(h, 4) + h2 + h3;
g = 1 - trunc(h, 2);
}

// invariant: f = exp(h) + O(x^n), g = exp(-h) + O(x^(n/2))
for (long n = 4; n < prec; n *= 2) {

// step 2
zz_pX m1 = MulTrunc(f, sqr(g), n);
g.SetLength(n);
for (long i = n/2; i < n; ++i)
g[i] = -coeff(m1, i);
g.normalize();

// step 3
zz_pX q = diff(trunc(h, n));

// step 4
zz_pX s = MulTrunc(g, f*q >> (n-1), n);
q.SetLength(2*n-1);
for (long i = 0; i <= deg(s); ++i)
q[n-1+i] = -s[i];
q.normalize();

// step 5
unsigned long length = NTL::min(deg(q)+2, prec);
zz_pX intq;
intq.SetLength(length);
intq[0] = 0;
for (long i = 1; i < length; ++i)
intq[i] = q[i-1] / i;
intq.normalize();
f = MulTrunc(f, 1 + h - intq, NTL::min(2*n, prec));
}
}

void ExpTrunc(zz_pX& x, const zz_pX& a, long n)
{
if (n < 1)
LogicError("ExpTrunc: bad args");
if (ConstTerm(a) != 0)
LogicError("ExpTrunc: nonzero constant term");
zz_pX y;
ExpTruncFast(x, y, a, n);
}



void FastTraceVec(vec_zz_p& S, const zz_pX& f)
{
long n = deg(f);
Expand Down
50 changes: 28 additions & 22 deletions src/lzz_pXCharPoly.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -29,48 +29,54 @@ void HessCharPoly(zz_pX& g, const zz_pX& a, const zz_pX& f)
CharPoly(g, M);
}

// "well-known" algorithm according to https://arxiv.org/pdf/1109.4323.pdf p.11
void CharPolyMod(zz_pX& g, const zz_pX& a, const zz_pX& ff)
{
zz_pX f = ff;
MakeMonic(f);
long n = deg(f);

if (n <= 0 || deg(a) >= n)
LogicError("CharPoly: bad args");
if (n <= 0 || deg(a) >= n)
LogicError("CharPolyMod: bad args");

if (IsZero(a)) {
clear(g);
SetCoeff(g, n);
return;
}

if (n > 90 || (zz_p::PrimeCnt() <= 1 && n > 45)) {
zz_pX h;
MinPolyMod(h, a, f);
if (deg(h) == n) {
g = h;
return;
}
}

if (zz_p::modulus() < n+1) {
HessCharPoly(g, a, f);
MinPolyMod(g, a, f);
if (deg(g) < n)
HessCharPoly(g, a, f);
return;
}

vec_zz_p u(INIT_SIZE, n+1), v(INIT_SIZE, n+1);
// compute tr(x^0 mod f), tr(x^1 mod f), ..., tr(x^(n-1) mod f) as rev(f')/rev(f)
// see for instance https://cr.yp.to/papers/newton.pdf
vec_zz_p T(INIT_SIZE, n);
{
zz_pX rev_f = reverse(f);
zz_pX rev_df = reverse(diff(f));
zz_pX inv_rev_f = InvTrunc(rev_f, n);
zz_pX quot = MulTrunc(rev_df, inv_rev_f, n);
VectorCopy(T, quot, n);
}

zz_pX h, h1;
negate(h, a);
long i;
// compute tr(a^0 mod f), tr(a^1 mod f), ..., tr(a^n mod f)
vec_zz_p tr = ProjectPowers(T, n+1, a, ff);

for (i = 0; i <= n; i++) {
u[i] = i;
add(h1, h, u[i]);
resultant(v[i], f, h1);
}
// recover characteristic polynomial using exp(∫g'/g) = g;
// "well-known" according to https://dl.acm.org/doi/pdf/10.1145/1145768.1145814 Prop. 9
zz_pX trpoly;
trpoly.SetLength(n+1);
trpoly[0] = 0;
for (long i = 1; i < n+1; ++i)
trpoly[i] = -tr[i] / i;
trpoly.normalize();

interpolate(g, u, v);
ExpTrunc(g, trpoly, n+1);
reverse(g, g, n);
}

NTL_END_IMPL