is_prime for ideals uses factorization, can be VERY slow

[tornaria]

The implementation of is_prime() for ideals uses pari idealfactor(). This is a bad idea if the norm of the ideal is hard to factor.

Pari has ismaximalideal(), but it is broken in pari 2.13.3. As a workaround, since a prime ideal has prime power norm, we avoid factorization except on this (easy) case.

---

Was: VERY slow doctest in sage/rings/number_field/number_field.py

With sage 9.5:

$ time sage -t --long --warn-long 60.0 --random-seed=40834229707572602474454926153679777311 src/sage/rings/number_field/number_field.py 
Running doctests with ID 2022-02-16-15-58-06-58d0ee96.
Using --optional=build,pip,sage,sage_spkg,void
Features to be detected: 4ti2,benzene,bliss,buckygen,conway_polynomials,csdp,database_cremona_ellcurve,database_cremona_mini_ellcurve,database_jones_numfield,database_knotinfo,dvipng,graphviz,imagemagick,jupymake,kenzo,latte_int,lrslib,mcqd,meataxe,pandoc,pdf2svg,plantri,pynormaliz,rubiks,sage.combinat,sage.geometry.polyhedron,sage.graphs,sage.plot,sage.rings.number_field,sage.rings.real_double,sage.symbolic,sage_numerical_backends_coin,sagemath_doc_html,sphinx,tdlib
Doctesting 1 file.
sage -t --long --warn-long 60.0 --random-seed=40834229707572602474454926153679777311 src/sage/rings/number_field/number_field.py
**********************************************************************
File "src/sage/rings/number_field/number_field.py", line 2180, in sage.rings.number_field.number_field.NumberField_generic.?
Warning, slow doctest:
    while not K.random_element().is_prime():  # long time
        pass
Test ran for 1319.62 s
    [2265 tests, 1367.95 s]
----------------------------------------------------------------------
All tests passed!
----------------------------------------------------------------------
Total time for all tests: 1368.3 seconds
    cpu time: 1364.7 seconds
    cumulative wall time: 1367.9 seconds
Features detected for doctesting: 

real	22m51.641s
user	22m47.153s
sys	0m1.272s

CC: @yyyyx4 @fchapoton

Component: number fields

Author: Gonzalo Tornaría, Lorenz Panny

Reviewer: Lorenz Panny

Issue created by migration from https://trac.sagemath.org/ticket/33360

[tornaria]

comment:1

Tracked this down to

sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
sage: t = ((-2611940*c + 1925290/7653)*b - 1537130/7653*c + 10130950)*a + (1343014/7653*c - 8349770)*b + 6477058*c - 2801449990/4002519
sage: %time t.is_prime()
CPU times: user 27min 27s, sys: 1.74 s, total: 27min 28s
Wall time: 27min 32s
False

Behind the scenes this is handled by pari as follows:

sage: %time K.pari_nf().idealfactor(K.ideal(t).pari_hnf())
CPU times: user 27min 30s, sys: 2.26 s, total: 27min 32s
Wall time: 27min 36s
[[2, [1, 0, -1, 0, 0, 0, 0, 0]~, 4, 2, [1, 0, 1, 1, -1, 0, 3, 0; 0, 1, 0, 0, 1, 2, 0, -3; 1, 0, -1, 1, -1, 0, -1, 0; 1, 0, -1, 3, -1, 2, 1, -2; 0, 0, 0, 0, 2, 2, 0, -2; 0, 1, 0, 0, 1, 0, 0, 1; 1, 0, -1, 1, -1, 0, 1, 0; 0, -1, 0, 0, -1, 0, 0, 1]], 6; [3, [-1, 0, -1, 0, 0, 1, 1, -1]~, 2, 2, [0, 0, 3, -3, 3, 0, 3, -3; 0, 0, 0, 0, -3, 0, 3, -6; 1, 0, -2, 1, 0, -1, -3, 0; -1, -2, 0, 3, 1, 5, -3, 0; 0, -2, 1, 1, 1, 4, 0, 0; 0, 1, -2, 1, 1, -2, 0, 3; 1, 1, -1, -1, 1, 0, 0, 0; -1, -2, 0, 0, 1, 2, 0, 0]], -2; [5, [-2, 0, 1, 0, 0, -1, -2, 1]~, 2, 2, [0, 0, 1, 1, -3, -4, 6, -3; 0, 0, -2, -2, 1, -2, 3, -9; 1, -2, -4, 2, -1, -1, -1, -2; 1, -2, -1, 5, 0, 7, -3, 4; 0, 0, 1, 1, 2, 6, -4, 2; -2, -1, -2, 1, 2, -3, 2, -1; 2, 1, -1, 1, -3, 0, 0, 0; -1, -3, -2, 2, -1, 0, 0, 0]], 1; [5, [2, 0, 1, 0, 0, -1, -2, 1]~, 2, 2, [0, 0, 2, 2, -1, 2, 3, 6; 0, 0, 1, 1, 2, 6, -6, 3; 2, 1, -2, 1, -3, 2, -2, 1; 2, 1, -3, 0, -5, -4, 4, -7; 0, 0, -2, -2, 1, -2, 2, -6; 1, 3, 4, -2, 1, 1, -1, 3; 1, -2, -2, 2, -1, 0, 0, 0; 2, 1, 1, -1, -2, 0, 0, 0]], 1; [13, [-1, 3, 4, -5, 0, 4, 6, -4]~, 1, 2, [-1, -3, 11, -4, 0, 7, -9, -12; -3, -4, 11, -4, -4, 14, 12, -3; 6, 6, 5, -3, 7, -7, -11, 11; -4, -9, 10, -6, 8, 3, 3, -10; -5, -10, 7, 1, -4, 2, 7, -14; 6, 12, -11, 4, -3, -5, -11, 22; -3, 4, -6, 1, 0, -5, -1, 3; -4, -1, 6, -1, -1, 10, 3, -4]], 1; [523, [-226, 0, -4, 0, 0, 4, -2, -4]~, 1, 2, [-219, 0, 239, -91, -57, -296, -390, -780; 0, -219, -148, -148, -91, -148, 780, -390; 129, -148, 41, -130, 390, -260, -239, -148; 19, -258, 390, -219, 650, 520, -387, 296; 0, -110, 260, 260, -349, 260, -296, 148; -148, -19, -520, 260, -130, -349, 148, 91; -130, 260, -129, 19, -167, 0, -219, 0; -260, -130, -148, 148, -19, 110, 0, -219]], -1; [523, [-12, 0, -4, 0, 0, 4, -2, -4]~, 1, 2, [-240, 0, -286, 527, -768, -482, 105, 210; 0, -240, -241, -241, 527, -241, -210, 105; -15, -241, -310, 35, -105, 70, 286, -241; 256, 30, -105, -240, -175, -140, 45, 482; 0, 271, -70, -70, -205, -70, -482, 241; -241, -256, 140, -70, 35, -205, 241, -527; 35, -70, 15, 256, -497, 0, -240, 0; 70, 35, -241, 241, -256, -271, 0, -240]], -1; [523, [104, 0, -4, 0, 0, 4, -2, -4]~, 1, 2, [-27, 0, -445, 176, -430, -508, -306, -612; 0, -27, -254, -254, 176, -777, 612, -306; -238, -254, 177, -102, 306, -204, 445, -254; -31, -47, 306, -27, 510, 408, -332, 1031; 0, 207, 204, 204, -129, 204, -508, 777; -254, -492, -408, 204, -102, -129, 254, -699; -102, 204, 238, -31, -223, 0, -27, 0; -204, -102, -254, 254, 31, -207, 0, -27]], -1; [523, [134, 0, -4, 0, 0, 4, -2, -4]~, 1, 2, [142, 0, -422, 229, -36, 386, 657, -255; 0, 142, 193, 193, 229, 193, 255, -912; -205, 193, -296, 219, -134, -85, 422, 193; 12, 410, -134, 665, -49, 693, 615, -386; 0, 217, 85, 85, 361, 608, 386, -193; 193, -12, -170, 85, 219, -162, -193, -229; 219, 85, 205, 12, 181, 0, 142, 0; -85, -304, 193, -193, -12, -217, 0, 142]], -1; [2551, [-1124, -912, -460, -456, 0, 4, -458, -4]~, 1, 2, [-870, 539, 3724, -5054, 6203, -906, 1911, 2565; 539, -331, -2055, 1149, -5054, -2236, -2565, 654; 798, -987, -2144, 637, -1492, 1394, -3724, -2055; -1060, -1777, -2031, 88, -1808, -752, -5960, 1087; 1068, -1858, -1394, -316, -233, -436, -906, 2236; -987, -189, 2249, -855, 637, -113, 2055, 1669; 637, -855, -798, -2128, 2209, 1068, -870, -539; 855, 218, -987, -81, 2128, 1858, -539, -331]], -1; [2551, [64, 537, -1011, -1007, 0, 4, -1009, -4]~, 1, 2, [970, 1217, -1448, 1138, 274, 1189, -3648, 3141; 1217, 2187, -223, 1412, 1138, 879, -3141, 6789; -586, 322, 3402, -1216, 169, 2264, 1448, -223; 821, 2274, -1048, -1292, 339, -4356, 2327, -2291; 545, 1407, -2264, 170, -246, -4526, 1189, -879; 322, -264, 3311, -1047, -1216, 4450, 223, -1671; -1216, -1047, 586, 276, 591, 545, 970, -1217; 1047, 2263, 322, -867, -276, -1407, -1217, 2187]], -1; [2551, [515, 912, 452, 456, 0, 4, 454, -4]~, 1, 2, [-331, -539, -1197, 882, -2937, -906, 654, 2565; -539, -870, 1149, -2055, 882, -1221, -2565, 1911; -504, 81, -767, 218, -1073, 316, 1197, 1149; -879, -1362, -534, -1289, -2467, -2668, -24, 3276; -1068, -375, -316, -1394, -113, -1274, -906, 1221; 81, -423, 1171, -855, 218, -233, -1149, -48; 218, -855, 504, 189, -1176, -1068, -331, 539; 855, 637, 81, 987, -189, 375, 539, -870]], -1; [2551, [601, -537, 1003, 1007, 0, 4, 1005, -4]~, 1, 2, [-364, -1217, -2550, 1671, -1894, 1189, -864, 3141; -1217, -1581, 1412, -223, 1671, 310, -3141, 4005; -1143, 867, 212, -288, -759, -170, 2550, 1412; -281, 1184, 458, -3204, -3023, -4934, 2860, -87; -545, 862, 170, -2264, -652, -2670, 1189, -310; 867, -276, 877, -1047, -288, -246, -1412, -1138; -288, -1047, 1143, 264, 58, -545, -364, 1217; 1047, 1335, 867, -322, -264, -862, 1217, -1581]], -1; [769047171088666806556287061350418679, [368757955557913964915819035850256425, 2, 1, 1, 0, 0, 1, 0]~, 1, 1, [-360391442740426872271373983027096114, 339229659894556457632623858390572228, -913406273986809062698337363348986235, 761380669442646963502839806530380265, -1770246848665485453162356776077448450, -1052964222139293116898955978494394023, -150747332518885405159643602698614955, 584447833742338549861091003834805192; 339229659894556457632623858390572228, -21161782845870414638750124636523886, -44098042916454627239439008947325838, -1008866179222838489659516969547068185, 761380669442646963502839806530380265, -1204989826683455216094453535312999993, -584447833742338549861091003834805192, 735195166261223955020734606533420147; 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It is possible to use pari to test for primality much faster without factoring as follows:

sage: %time K.pari_nf().idealismaximal(K.ideal(t).pari_hnf())
CPU times: user 5.18 ms, sys: 0 ns, total: 5.18 ms
Wall time: 7.81 ms
0

I'll try to cook up a patch for is_prime() to use idealismaximal() instead of idealfactor().

Another issue: during the call to idealfactor(), the one that takes 27 minutes, hitting control C doesn't work for me, but I'm using all system packages (in particular cypari, cysignals, python 3.10, etc) can someone confirm that Ctrl-C is broken on a standard build of sage?

[DaveWitteMorris]

comment:2

Ctrl-C works for me (I instantaneously get KeyboardInterrupt) with 9.6b1 on MacOS 11.5.2.

[tornaria]

comment:3

For the record:

• I have a working patch for is_prime() that I'll polish and submit later
• About the issue with Ctrl-C, it turns out this is caused by a recent upgrade of prompt_toolkit (a system package) from 3.0.24 to 3.0.26. I didn't try with 3.0.25 (the bundled version in sage is 3.0.22 and that one works ok)

[tornaria]

comment:5

Cf: prompt-toolkit/python-prompt-toolkit#1576

[tornaria]

Branch: u/tornaria/33360-ideal.is_prime

[tornaria]

comment:7

To fix the issue, we only factor the ideal if the norm is a prime power.

Note that a simpler way would be to use pari ismaximalideal() as in:

    K = self.number_field().pari_nf()
    I = self.pari_hnf()
    self._pari_prime = K.idealismaximal(I) or None
    return self._pari_prime is not None

However it turns out that ismaximalideal() is broken in pari 2.13.3. For a quick example in gp:

? K = nfinit(y^2 + 1); P = idealhnf(K, 3); idealismaximal(K, P)
%1 = 0

This is fixed in current development version of pari by this commit.

---

New commits:

6330cee

trac 33360: avoid factoring in is_prime()

[tornaria]

Author: Gonzalo Tornaría

[tornaria]

Commit: 6330cee

[yyyyx4]

comment:9

The patch looks good as far as I can tell.

Just one question: How much faster is idealismaximal() when using a non-broken version of PARI? Could it be worthwhile to detect the PARI version at runtime to switch between idealismaximal() and the workaround?

[yyyyx4]

Reviewer: Lorenz Panny

[tornaria]

comment:10

Replying to @yyyyx4:

The patch looks good as far as I can tell.

Just one question: How much faster is idealismaximal() when using a non-broken version of PARI? Could it be worthwhile to detect the PARI version at runtime to switch between idealismaximal() and the workaround?

I doubt the difference is significative. For the example in the doctest:

sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
sage: t = (((-2611940*c + 1925290/7653)*b - 1537130/7653*c
....:       + 10130950)*a + (1343014/7653*c - 8349770)*b
....:       + 6477058*c - 2801449990/4002519)
sage: I = K.ideal(t).pari_hnf()
sage: 
sage: 
sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
sage: t = (((-2611940*c + 1925290/7653)*b - 1537130/7653*c
....:       + 10130950)*a + (1343014/7653*c - 8349770)*b
....:       + 6477058*c - 2801449990/4002519)
sage: I = K.ideal(t)
sage: Kp = K.pari_nf()
sage: Ip = I.pari_hnf()
sage: %timeit Kp.idealismaximal(Ip)
14.4 µs ± 125 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: %timeit Kp.idealnorm(Ip)
2.26 µs ± 11.1 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)

which probably just means pari is optimizing in a different way but that will depend on the ideal. In the long run, I think using idealismaximal() is better since pari is in a better position to optimize (and any effort to optimize is better spent in pari), but not worth complicating things.

Using idealfactor() on that ideal would take 20-30 minutes so a microsecond here and there is just insignificant in contrast.

For a prime ideal, I guess the time would be dominated by isprime() anyway...

[yyyyx4]

comment:11

The idealnorm() is indeed negligible, but idealfactor() is much slower than idealismaximal() even in the prime-power case.

Example:

R.<x> = QQ[]
K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
Kpari = K.pari_nf()

for bits in (50, 100, 200, 300, 500, 800):
    p = random_prime(2^bits, proof=False)

    I = choice([f for f,_ in K.ideal(p).factor()])  # a prime above p
    Ipari = I.pari_hnf()

    print(f'{bits} bits')
    %timeit Kpari.idealismaximal(Ipari)
    %timeit assert Kpari.idealnorm(Ipari).isprimepower(); Kpari.idealfactor(Ipari)

Result:

50 bits
790 µs ± 2.97 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
799 µs ± 5 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
100 bits
3.35 ms ± 69.5 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
17.4 ms ± 617 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
200 bits
7.52 ms ± 394 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
52.9 ms ± 733 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
300 bits
11.9 ms ± 389 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
178 ms ± 6.06 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
500 bits
64 ms ± 1.3 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
973 ms ± 11 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
800 bits
61.4 ms ± 5 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
4.45 s ± 131 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

[yyyyx4]

comment:12

We have to be careful with idealismaximal() though because (according to the PARI documentation) it uses a probabilistic primality test. So if Sage's flag for strict primality proving is set, we should still run isprimepower() prior to calling idealismaximal(), or ensure primality of the relevant entry of the structure returned by idealismaximal() afterwards.

Also, a small optimization: Calling self.norm() instead of idealnorm() will cache the norm instead of computing it from scratch every time.

[yyyyx4]

Changed author from Gonzalo Tornaría to Gonzalo Tornaría, Lorenz Panny

[yyyyx4]

comment:15

I implemented my suggestions from comment:9 and comment:12. Someone else will have to review the new changes.

---

New commits:

6330cee	trac 33360: avoid factoring in is_prime()
7f19cea	Merge branch 'u/tornaria/33360-ideal.is_prime' into public/33360
b938f7b	opportunistically use idealismaximal() when PARI version is recent enough

[yyyyx4]

Changed branch from u/tornaria/33360-ideal.is_prime to public/33360

[yyyyx4]

Changed commit from 6330cee to b938f7b

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

647bd45	Merge tag '9.8.beta1' into public/33360
e3af4e3	#34537 requires PARI >= 2.15

[sagetrac-git]

Changed commit from b938f7b to e3af4e3

[yyyyx4]

Dependencies: #34537

[sagetrac-git]

Changed commit from e3af4e3 to 58511ea

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

58511ea

remove unused import

[sagetrac-git]

Changed commit from 58511ea to 7eee1c4

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

a9051b1	Merge branch 'u/tornaria/33360-ideal.is_prime' into public/33360
7eee1c4	opportunistically use idealismaximal() when PARI version is recent enough

[yyyyx4]

comment:20

Reverted earlier changes from comment:16 as #34537 appears to be stuck. Can we get this in?

[yyyyx4]

Changed dependencies from #34537 to none

[sagetrac-git]

Changed commit from 7eee1c4 to 689eb88

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

689eb88

fix docstring markup

[sagetrac-git]

Changed commit from 689eb88 to 4ac2c9b

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

4ac2c9b

fix docstring markup some more

[sagetrac-git]

Changed commit from 4ac2c9b to b31f59b

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

100b39c	Merge tag '9.8.beta6' into public/33360
9a2bddb	#34537 requires PARI >= 2.15
b31f59b	Merge tag '9.8.rc0' into public/33360

[mkoeppe]

Removed branch from issue description; replaced by PR #34980