Lair Of The Multimedia Guru

Correlations based on the lower half of the LCG

Lets try picking some correlated combinations with (using my very simple pcg-dxsm implementation)

Multiplier	15750249268501108917	15750249268501108917	15750249268501108917
Increment	123456789	15750249268624565705	31500498537125674621
Seed	31415	31414	31413
Output 1	0x563B0DB76F45EEC2	0xDEE2CC4990CDBE1E	0xDEE2CC4990CDBE1E
Output 2	0x064C612B5E0A4853	0x064C612B5E0A4853	0x704228ECB09FC1D5
Output 3	0x4727428C93C1A285	0x5A7E3EA39E71C29F	0x5A7E3EA39E71C29F
Output 4	0x2745482D3BDBC1C0	0x2745482D3BDBC1C0	0x15D15980AE293340
Output 5	0x6055DF5FCE178AB2	0xC5DC9DC897A78F1E	0xC5DC9DC897A78F1E
Output 6	0x05DDF19C395C342F	0x05DDF19C395C342F	0x51705B299006C1C1
Output 7	0xD7AFDA4C1654626B	0x89A204161A53CA99	0x89A204161A53CA99
Output 8	0x2C1FCD6649DB8016	0x2C1FCD6649DB8016	0x338710EA7C027382
Output 9	0xEDE28080A228A6DC	0xED272B75717128A4	0xED272B75717128A4
Output 10	0xB05EFD2651E59BC7	0xB05EFD2651E59BC7	0x20911E0B5D6AE551
Output 11	0xD6045D380A566F2D	0xE6584BDE2C470387	0xE6584BDE2C470387
Output 12	0x1BF91549D870370F	0x1BF91549D870370F	0xF434399E25ED24B5
Output 13	0x1D9EB55CB071B60A	0x1DA3A160F1C93F66	0x1DA3A160F1C93F66
Output 14	0x892A16096BF94286	0x892A16096BF94286	0x8F5678078367E87A
Output 15	0xA7292EEE7AF4777F	0x281913D2D3F2DFC5	0x281913D2D3F2DFC5
Output 16	0x08AB29210A0B6FF1	0x08AB29210A0B6FF1	0xC0DD5B88BF6885A3
Output 17	0x20E581A9EA8BF155	0xB17BBB3957418DBB	0xB17BBB3957418DBB
Output 18	0x768E7BD7B5223C36	0x768E7BD7B5223C36	0x086B7F8F81A5CEDA
Output 19	0xF67313E1E5454A96	0xD3026BC096FEDDE2	0xD3026BC096FEDDE2
Output 20	0xAB091F6C4C169476	0xAB091F6C4C169476	0xF29E461BBD7C6952
Output 21	0xB18D1BC34F78E4D1	0x205033BF4C77D237	0x205033BF4C77D237
Output 22	0x5BBCD31929B10ED3	0x5BBCD31929B10ED3	0xFDCB8D05F8C862FD
Output 23	0xC64AFD5DFC7654A7	0x54C875057AAA9DCD	0x54C875057AAA9DCD
Output 24	0x34866A3DF2054E13	0x34866A3DF2054E13	0x4C9D903AB9738D79
Output 25	0x29CA92D156E93DD4	0x1174B427B597C82C	0x1174B427B597C82C
Output 26	0x46B0D6BFFE3BD1C3	0x46B0D6BFFE3BD1C3	0x18A4BCCC15242755
Output 27	0x74DF62C8A26F65CD	0xF5CAC48CFC7B4D47	0xF5CAC48CFC7B4D47
Output 28	0xE737155A33FC916F	0xE737155A33FC916F	0x2A8E83BF16B3C6F5
Output 29	0x0E0E87DB2724235B	0x82EA039F9945EE0D	0x82EA039F9945EE0D
Output 30	0x2367AB4D5FA3ABBE	0x2367AB4D5FA3ABBE	0x274FC0D71A60FF82

Besides the exactly equal ones, the more careful reader will have noticed that each line either has even or odd entries, so the 3 seed/increment pairs generate equal results in the lowest bit.

You might ask why does this all happen? Well, the identical numbers happen because the lowest bit is ored away in DXSM, that is forced to be 1. So any cases differing only by that bit at the LCG stage become identical in later stages. The other similarities happen because the mixing isnt as good as it needs to be to remove all differences from very similar input and LCGs can produce such very similar input when one picks the “wrong” parameters.

Did i hand pick the worst ? Is that the only case ?
lets see

Multiplier	15750249268501108917		15750249268501108917
Increment	123456789		1134925067785824341058664979018534165
Seed	31415		340282366920938463463302549837730314935
	0x56	3B0DB76F45EEC2	0x04	3B0DB76F45EEC2
	0x06	4C612B5E0A4853	0xC7	4C612B5E0A4853
	0x47	27428C93C1A285	0x54	27428C93C1A285
	0x27	45482D3BDBC1C0	0xE7	45482D3BDBC1C0
	0x60	55DF5FCE178AB2	0x96	55DF5FCE178AB2
	0x05	DDF19C395C342F	0xCE	DDF19C395C342F
	0xD7	AFDA4C1654626B	0xEE	AFDA4C1654626B
	0x2C	1FCD6649DB8016	0xE2	1FCD6649DB8016
	0xED	E28080A228A6DC	0xD1	E28080A228A6DC
	0xB0	5EFD2651E59BC7	0x75	5EFD2651E59BC7
	0xD6	045D380A566F2D	0x03	045D380A566F2D
	0x1B	F91549D870370F	0xEE	F91549D870370F
	0x1D	9EB55CB071B60A	0xCB	9EB55CB071B60A
	0x89	2A16096BF94286	0x83	2A16096BF94286
	0xA7	292EEE7AF4777F	0xCA	292EEE7AF4777F
	0x08	AB29210A0B6FF1	0xE1	AB29210A0B6FF1
	0x20	E581A9EA8BF155	0x53	E581A9EA8BF155
	0x76	8E7BD7B5223C36	0xC8	8E7BD7B5223C36
	0xF6	7313E1E5454A96	0x9C	7313E1E5454A96
	0xAB	091F6C4C169476	0x19	091F6C4C169476
	0xB1	8D1BC34F78E4D1	0x64	8D1BC34F78E4D1
	0x5B	BCD31929B10ED3	0x70	BCD31929B10ED3
	0xC6	4AFD5DFC7654A7	0x59	4AFD5DFC7654A7
	0x34	866A3DF2054E13	0xE7	866A3DF2054E13
	0x29	CA92D156E93DD4	0x55	CA92D156E93DD4
	0x46	B0D6BFFE3BD1C3	0x0F	B0D6BFFE3BD1C3
	0x74	DF62C8A26F65CD	0x31	DF62C8A26F65CD
	0xE7	37155A33FC916F	0xAA	37155A33FC916F
	0x0E	0E87DB2724235B	0x67	0E87DB2724235B
	0x23	67AB4D5FA3ABBE	0x05	67AB4D5FA3ABBE

Here 7 out of 8 bytes match apparently for each. Is that the worst ? no we could make part of the first byte match too, it just becomes harder to color the table to match that.

Correlations based on the upper half of the LCG

Update 2024-03-02, After thynson noticed that the above 2 classes of correlations, all have multiplier-1 as a coresidual of the increment. Ive looked more and found more classes of correlations.
The above 2 cases produce correlations because they introduce 2 specific classes of differences in the lower half of the underlying LCG, we can also introduce other specific differences in the upper half to create correlations. I first thought the mixing stage would make this difficult, but it doesnt seem that difficult.

Multiplier	15750249268501108917		15750249268501108917		15750249268501108917		15750249268501108917
Increment	123456789		188175727042032768542098945617541451029		94087863521016384271049472808832453909		94087863521016384271049472808832453909
Seed	31415		170141183500083312988819472512656112311		85070591750041656494409736256328071863		255211775210510888226097039972212177591
Output1	0x563B0DB76F45E	EC2	0x4EAC9CA5AB826	EC2	0x5273D52E8D642	EC2	0x8BA757339BF3B	0C4
Output2	0x064C612B5E0A4	853	0x507FC7CA7B00C	853	0x7CF75FC4DBFEB	B3C	0x6132ADDBCF8F0	853
Output3	0x4727428C93C1A	285	0xE8681791D88ED	60B	0x821B65AAAE9D3	074	0x56EBE12BED519	60B
Output4	0x2745482D3BDBC	1C0	0x7D02EE9F8B954	1C0	0x7C6674F413FF0	1C0	0x064DB7AF79614	8E3
Output5	0x6055DF5FCE178	AB2	0x5284D85445650	AB2	0xCB9C54CE810BC	AB2	0x89BF5C086BDE9	447
Output6	0x05DDF19C395C3	42F	0xDA6CAC69AB735	98E	0x979601E7DBD3B	9D1	0x41C4B15A978E7	42F
Output7	0xD7AFDA4C16546	26B	0xD22DEF5B1E600	21E	0x81EE59290B934	21E	0xED6327A2A5747	2A1
Output8	0x2C1FCD6649DB8	016	0x3CD4A9AF3862F	39A	0x12F825A58C46C	679	0x798CE3702E873	39A
Output9	0xEDE28080A228A	6DC	0x8443929D63A67	C1C	0x02CE2330E2663	F1F	0xE422EEFF6C846	6DC
Output10	0xB05EFD2651E59	BC7	0x5BB46EBB794BC	378	0x9EC419F1B9850	378	0x3DD8F2BA415FE	2CE
Output11	0xD6045D380A566	F2D	0x909115D376D6E	F2D	0xB34AB985C096A	F2D	0xA49969ED9AE37	EBF
Output12	0x1BF91549D8703	70F	0x8B81A72468CE6	2BB	0x05043537DB072	5E2	0x20046693FD23A	2BB
Output13	0x1D9EB55CB071B	60A	0x264C5A17BA703	60A	0x458D7586D42BD	7B7	0x21F587BA3570F	60A
Output14	0x892A16096BF94	286	0x97EA766824D8C	286	0x908A4638C8690	286	0x8B8230900DC31	894
Output15	0xA7292EEE7AF47	77F	0x45B865A22102D	671	0xED910677E00EA	244	0xB9F1AF9C302C3	77F
Output16	0x08AB29210A0B6	FF1	0x66F131DB875BA	123	0xF5F9A395C5C1A	FF1	0x035C3B9F6264E	A8C
Output17	0x20E581A9EA8BF	155	0x2B7CBD378A1FB	44A	0x835FF11C3FF63	155	0xD6C0C0A58FD86	96A
Output18	0x768E7BD7B5223	C36	0x8282FAEF4390B	C36	0x7C88BB637C597	C36	0x4E30CF973257E	263
Output19	0xF67313E1E5454	A96	0xBCADB3276FBCA	D92	0x6A4F6314C50F9	11D	0x0DCC176A07468	A96
Output20	0xAB091F6C4C169	476	0x7C6C880401881	476	0x651E3C4FDC40D	476	0x87103D0FEAEBB	5A9
Output21	0xB18D1BC34F78E	4D1	0x182EF712B7CC9	6ED	0x632712BFD8439	536	0x7D1586E9295E2	4D1
Output22	0x5BBCD31929B10	ED3	0x177CCDEF531D3	167	0xC86CF8BD98EEF	167	0x6A7959BA3EC3B	1E5
Output23	0xC64AFD5DFC765	4A7	0xDBFE7B848484D	4A7	0x5124BC71407D9	4A7	0x7317F810CA105	929
Output24	0x34866A3DF2054	E13	0x2B7F37E87E70E	BA7	0x87E4C62AA01E0	E13	0xA9077B94F6670	644
Output25	0x29CA92D156E93	DD4	0x3AE2C3E7ABF63	199	0xBCF4EEF02BA57	199	0x190804CAE2166	D48
Output26	0x46B0D6BFFE3BD	1C3	0x714BB6B1E3FC4	96F	0x5E459BFFC60D8	F95	0xB9A7262288870	96F
Output27	0x74DF62C8A26F6	5CD	0x25B37CDE46C28	A09	0x4111E2836E4B0	090	0x01289C6FC845C	A09
Output28	0xE737155A33FC9	16F	0x8CB957BC0E0A1	16F	0xAEB26C1EB1942	EC4	0x1475F42946F5D	16F
Output29	0x0E0E87DB27242	35B	0xF94895647AA13	EF5	0xCA2ACE10F845E	35B	0x5234DF04DB0C2	7DE
Output30	0x2367AB4D5FA3A	BBE	0x56F8654413BC2	BBE	0xE69CD0B7EE3B4	D08	0x899F4E5205976	BBE

Above attacks the first stage of the mixer, we can also attack the 2nd stage, it looks less impressive but none the less instead of 1 identical byte in 256 we see alot more

Multiplier	15750249268501108917		15750249268501108917		15750249268501108917
Increment	123456789		271162317658692578196538916254707862805		152861171144907760414978380921711611157
Seed	31415		39621334812049856571820243639		9905333703012464142955084471
Output1	0x563B0DB76F45EE	C2	0x0083130D590C82	C2	0x56399A48F66C29	C2
Output2	0x064C612B5E0A48	53	0xAFFAF7C625B7A0	55	0x426A227A69AC8B	55
Output3	0x4727428C93C1A2	85	0x3BFE265FC469DD	34	0xDF23B38901C67B	5C
Output4	0x2745482D3BDBC1	C0	0x28B8CF040CE59E	23	0xF2C12F43D3C5BC	98
Output5	0x6055DF5FCE178A	B2	0xEB4BAAA8C260D7	AD	0x52224A76A9AB37	DF
Output6	0x05DDF19C395C34	2F	0xD65E464613ADA6	A2	0xCF042916ADA3DB	F9
Output7	0xD7AFDA4C165462	6B	0x9FE6DE769FFF8E	1E	0xB816350D4D04B3	1E
Output8	0x2C1FCD6649DB80	16	0x754DC19986C65F	9A	0xCFC556D81AB400	D2
Output9	0xEDE28080A228A6	DC	0xD249BACC4FF71F	15	0xC03E7BF3E7423B	A8
Output10	0xB05EFD2651E59B	C7	0x794EA34DECBCC4	B4	0x9448CB72344E34	D5
Output11	0xD6045D380A566F	2D	0x2BB1BB3BA4DDCF	4A	0xC2956A532B866C	AB
Output12	0x1BF91549D87037	0F	0x8483638585B29A	C1	0xE085893BCA2906	D5
Output13	0x1D9EB55CB071B6	0A	0xB54C78D8826582	0A	0x11B567A953D0E2	2F
Output14	0x892A16096BF942	86	0x073360B2311E2E	86	0x0DD1F9F1A63F27	80
Output15	0xA7292EEE7AF477	7F	0x47C425E350061C	EF	0x06263999BE6442	AE
Output16	0x08AB29210A0B6F	F1	0xB6894B4E36C7FD	52	0xD38C235183235A	F9
Output17	0x20E581A9EA8BF1	55	0x08854CD5C6D98B	F9	0x0CD106F83B232A	55
Output18	0x768E7BD7B5223C	36	0xA1DDA8C4BE25F5	25	0xD0B32BB399C62F	36
Output19	0xF67313E1E5454A	96	0xCEFA2E1F1711E0	59	0x31AF5438CE9EA2	5F
Output20	0xAB091F6C4C1694	76	0x67110439B8E660	76	0x5FD39652AC17B6	A9
Output21	0xB18D1BC34F78E4	D1	0xBB66DD962205EA	F1	0x4BE1680D18C928	B1
Output22	0x5BBCD31929B10E	D3	0xFC0D37393C8893	EA	0xEB5BAA64A129E1	D3
Output23	0xC64AFD5DFC7654	A7	0x87EC987780AF59	FA	0xB31BA2E5123C18	CF
Output24	0x34866A3DF2054E	13	0xDB102ADD9EA7D7	0A	0x8885FA71C030D2	E7
Output25	0x29CA92D156E93D	D4	0xDCDD9AA636484C	7A	0xDD95314A5E5CA0	8D
Output26	0x46B0D6BFFE3BD1	C3	0x1669656599FB38	9E	0xF0C25ACE4573A6	35
Output27	0x74DF62C8A26F65	CD	0x72DFFFE5F48C19	5B	0xC494B0FF934BE3	46
Output28	0xE737155A33FC91	6F	0xBE560D45FF9E8B	C9	0x2D8821EB759B1F	50
Output29	0x0E0E87DB272423	5B	0xCB29008A08DC1A	DC	0xD712484F90D6E4	72
Output30	0x2367AB4D5FA3AB	BE	0x58E6BE59AE99D8	4B	0x7C289593C90335	F1

We can also use this to generate more than 8bit matches, but they are spread out more so it doesnt show up nicely in the first 30 output values.
For example ./pcg-dxsm-64 15750249268501108917 164824157741578258584156320684186520853 3046549733826659956002159287 1000000 16 and ./pcg-dxsm-64 15750249268501108917 123456789 31415 1000000 16 produce a million values, we expect the low 20 bits to match in one case in a million on average but these have over a thousand such matches

Update 2024-03-03: Verifying that our PCG64-DXSM implementation is correct

./pcg-dxsm-64  15750249268501108917 123456789 31415 10 10
 6213575192384761538
  453844501346797651
 5127139872326394501
 2829747299987210688
 6941700003234024114
  422759593322230831
15541880859405869675
 3179485701156339734
17141404421690271452
12708873539510705095

python3
from randomgen import PCG64
g = PCG64(2**128+31415-123456789,123456788/2,variant="dxsm",mode="legacy"); 
for i in range(10): print(g.random_raw())
6213575192384761538
453844501346797651
5127139872326394501
2829747299987210688
6941700003234024114
422759593322230831
15541880859405869675
3179485701156339734
17141404421690271452
12708873539510705095

Update 2024-03-04: Add correlations with equal seeds

Correlations with equal seeds

For the case of equal seeds it must be first said that different pcg64-dxsm implementations mangle the seed in different ways. Thus whats equal in one is not equal in another. We show examples of equal seeds at the LCG level first, this is how the previous examples defined the seed.

Multiplier	15750249268501108917	15750249268501108917
Increment	340282366920937968669293837469431615317	340282366920937968685044086737932724233
Seed	31415	31415
Output1	0x0000000000000000	0x0000000000000000
Output2	0x0000000000000000	0x5542869E099E356E
Output3	0xCA1A90080FC41D1A	0x7CC2BF122332851C
Output4	0x7CC2BF122332851C	0xCB8D59E2E4C1218A
Output5	0x443C52CD5384C766	0x4E16E6CE9CA5478E
Output6	0x4E16E6CE9CA5478E	0x79339B1DB09444E9
Output7	0x6F763957F4353DDB	0x10EE4E400E4F428E
Output8	0x10EE4E400E4F428E	0x5F9CFCB33A3A60EF
Output9	0x4418DDB3F1FE88B1	0x2B907893DCD59873
Output10	0x2B907893DCD59873	0xC984F2B06253372B
Output11	0xA38FA95F35AB6739	0x3E6AE114E83A3619
Output12	0x3E6AE114E83A3619	0x52946D7362EEAA86
Output13	0x0E4B57A85E8670EA	0x04769CEADA5CC799
Output14	0x04769CEADA5CC799	0x2E71EE38F3683F6C
Output15	0xED4F32C1AF264384	0x54C64408D6FABE08
Output16	0x54C64408D6FABE08	0x6B65E69BB07E56F6
Output17	0xDAD0B79C5C6EFC2A	0xAB06E3184A632E04
Output18	0xAB06E3184A632E04	0x88E36EABEA546B63
Output19	0xE563E5FD7697D091	0x2709664DC434EDE3
Output20	0x2709664DC434EDE3	0x6277238F88A1D8D6
Output21	0x74B88A9EDBCF94FA	0xF4B4B8A667253240
Output22	0xF4B4B8A667253240	0x2B0DF01AF7F68886
Output23	0x23906462E660A212	0x9591B3C5A81F1E01
Output24	0x9591B3C5A81F1E01	0x5A67803048FA88EC
Output25	0x8C3FDB1C4F568714	0xEA48F0F9B2EE760B
Output26	0xEA48F0F9B2EE760B	0xA5785DE15372C0A0
Output27	0x6FB95CE223AFFEE0	0xE7F4EEB0FD473A91
Output28	0xE7F4EEB0FD473A91	0x83886CE76BF8FCDA
Output29	0xAE3D319DFBE64AF6	0xEBA429DA1AF5EE38
Output30	0xEBA429DA1AF5EE38	0x2E10BCE8F08A03B7

And another example:

Multiplier	15750249268501108917		15750249268501108917
Increment	18158801084572694679761040473092597769		340282366920937968670446758974038462293
Seed	31415		31415
Output1	0x2	8C57634CF3ADEA3	0x0	000000000000000
Output2	0x4	5E72764CBAC7D8A	0x1	8C57634CF3ADEA3
Output3	0x0	9F5D8F29B240248	0x0	9804DEEE6EB9D0E
Output4	0x4	8BD93FA680EE36C	0x8	9F5D8F29B240248
Output5	0x8	F2CF0B4671A5D87	0x4	2B52509EB19E6B4
Output6	0xE	57EE9B1AB50A947	0x7	F2CF0B4671A5D87
Output7	0x6	E92305B0D002F61	0xE	02E81E3F013B2B5
Output8	0x2	D513919164C9F11	0x5	E92305B0D002F61
Output9	0x2	FCAB132E105A9F2	0x1	C902225001A774F
Output10	0x6	E8A48BD7E37EE0C	0xC	FCAB132E105A9F2
Output11	0x9	A9240D1A1572F73	0x9	64DE7D8E859E804
Output12	0x6	EF0158EC1476E59	0x2	A9240D1A1572F73
Output13	0x9	34A4391489DB1D4	0x4	B66D2865E881D4F
Output14	0xB	634A42C2394888A	0xD	34A4391489DB1D4
Output15	0x8	E03CA0BB984E16B	0x5	5CC7B5271F792FE
Output16	0x5	5F6ACB2C66FDED7	0x5	E03CA0BB984E16B
Output17	0x5	CBEC3EC2C5C03DE	0x3	BF46F487533BDF9
Output18	0x7	C46F2BD428796EA	0xB	CBEC3EC2C5C03DE
Output19	0x0	39A4A7C2A084281	0xA	6A3E2D52E24926E
Output20	0x8	1A17AD6D8142476	0xB	39A4A7C2A084281
Output21	0x1	9223A7740BACC82	0x3	5A8AF9350D8645A
Output22	0x8	8DCB82922B55F8E	0x3	9223A7740BACC82
Output23	0x2	FB541FDDC8E5A86	0xE	DB016F19FE29D2A
Output24	0xC	61E8DBF6577A704	0xC	FB541FDDC8E5A86
Output25	0x8	2733F8F3CD82976	0x8	05A2C8C4F2E88FC
Output26	0x5	5A5329E50934786	0x6	2733F8F3CD82976
Output27	0xA	342EF7724025501	0x2	EBD1F90D6ADC342
Output28	0x1	A8635E212C32A2D	0xD	342EF7724025501
Output29	0x3	F87E671A19911B2	0x1	2B4DEB4213B90C8
Output30	0x7	C8BAC3F018FD61D	0x5	F87E671A19911B2

Now python randomgen + numpy

from randomgen import PCG64
g = PCG64(31415,(310562926860152171542224643668532814649>>1),variant="dxsm",mode="legacy"); g2 = PCG64(31415,(310562926860152171526474394400031705733>>1),variant="dxsm",mode="legacy");
for i in range(30): print("0x" + format(g.random_raw(), "016X") + " 0x" + format(g2.random_raw(), "016X"))

Multiplier	15750249268501108917	15750249268501108917
Increment	310562926860152171526474394400031705733>>1	310562926860152171542224643668532814649>>1
Seed	31415	31415
Output1	0xEE846300E83A27DC	0x7256D1D22864FB31
Output2	0x62B36CDF71DD1743	0x1C2239BDA2F793C6
Output3	0x1C2239BDA2F793C6	0xA25FB4606E9F9955
Output4	0x2C21FF9FA58D879B	0xA90BC860E5BC5E65
Output5	0xA90BC860E5BC5E65	0xD9A217425A7117AF
Output6	0x9B802E5C51C811D5	0x31A30A053121F011
Output7	0x31A30A053121F011	0xDF91893796BE498C
Output8	0xB6F6FCF3CC23CF14	0x27FFF0C3DFDAE842
Output9	0x27FFF0C3DFDAE842	0x5653E5F3D78910B6
Output10	0x043FFA9868AB2B22	0xC77C03211C0201CE
Output11	0xC77C03211C0201CE	0x1A2DC56C688DBDE8
Output12	0x1F3C5871B8AA1618	0x0C12CBDC34DFABB0
Output13	0x0C12CBDC34DFABB0	0xDFD2D074C5127C15
Output14	0xC7FEC3AED543EDE7	0x127120DC1B75A20E
Output15	0x127120DC1B75A20E	0xAA580CD3C44B571D
Output16	0x800C8F5DBF2D5C4B	0xD0D231D10F85BB11
Output17	0xD0D231D10F85BB11	0xB433A15370448948
Output18	0xB678DA426A582A58	0x4E5DC360D27E4857
Output19	0x4E5DC360D27E4857	0x52B4BF17EE5FB665
Output20	0x91A4127B348A8BAB	0xAD7A55AF7A1E62B2
Output21	0xAD7A55AF7A1E62B2	0xAAE73D4A877AEB9E
Output22	0xE6F6BA90233BFF2A	0x537BA3170D10A98F
Output23	0x537BA3170D10A98F	0x71778DBD0A1F1858
Output24	0x22D07B79FDA625E8	0xE03DFB31EDF18C5A
Output25	0xE03DFB31EDF18C5A	0x610809577FF8326B
Output26	0x82BA957BB887F8A1	0x5C430AB7C74B61D9
Output27	0x5C430AB7C74B61D9	0x2C63932EC021D830
Output28	0x0FD2B2B66AF18DD0	0x779C96BF20FAAC65
Output29	0x779C96BF20FAAC65	0x93C6F9F9FA2026BD
Output30	0xFCB0833058A7183F	0x5397DA646711A886

g = PCG64(31415,(328721727944725361000316204835460899641>>1),variant="dxsm",mode="legacy"); g2 = PCG64(31415,(310562926860152171527627315904638552709>>1),variant="dxsm",mode="legacy");
for i in range(30): print("0x" + format(g.random_raw(), "016X") + " 0x" + format(g2.random_raw(), "016X"))

Multiplier	15750249268501108917		15750249268501108917
Increment	328721727944725361000316204835460899641>>1		310562926860152171527627315904638552709>>1
Seed	31415		31415
Output1	0x0	70E695F12FAC872	0x3	847B59A89C311E2
Output2	0x6	C673BB08B64385F	0xB	565D0C976C04FB6
Output3	0x8	D353EED117D7BC2	0xF	C673BB08B64385F
Output4	0x9	EA14B4ED2206371	0x0	06B1E786214329E
Output5	0xB	FE0FB0D66377C3C	0xE	EA14B4ED2206371
Output6	0x8	F48FEB9E21CF90C	0xB	C9512DDD19B53D4
Output7	0x9	FE6F7ED18A90B3E	0xC	F48FEB9E21CF90C
Output8	0x8	554B31E0632BF9F	0x2	82E54E6060CFF52
Output9	0xB	D2444268D86C2E0	0xF	554B31E0632BF9F
Output10	0x6	A77908E23FC25DB	0xE	BE330DEB6411006
Output11	0x7	F5401944A241692	0xB	A77908E23FC25DB
Output12	0x3	78E7CFCA99985EC	0x6	1E2C802A1208A6E
Output13	0xE	B3C49C3FC1B80BD	0xF	78E7CFCA99985EC
Output14	0x0	CE124E21F2F325C	0xA	F5923BC63DFE91F
Output15	0xF	2DAAFA5A3C0D13C	0x4	CE124E21F2F325C
Output16	0x1	4B999F605D4C93B	0xD	00E469E490FE6A4
Output17	0x3	4CAE614E3FBF077	0xC	4B999F605D4C93B
Output18	0x3	BA4BA78F570240F	0xE	5AFD46A52CC2195
Output19	0xD	935DBA5BB7508AA	0x6	84B89E8D66E1609
Output20	0xE	B3D1F1A9262C709	0x7	8C2397180142176
Output21	0xB	7AEF7982643BADC	0xB	B3D1F1A9262C709
Output22	0x1	C258C8417669CD1	0x2	542033E81E98834
Output23	0x4	81A95DC0F79E1A6	0xC	C258C8417669CD1
Output24	0x1	227C217600F6ECC	0xF	3E4B6B862CC276A
Output25	0x6	F0860C9B8E5E021	0xD	227C217600F6ECC
Output26	0xC	72BDAFE8A09D355	0x5	3D268E5FE4D6083
Output27	0x2	69759DA86213109	0x7	72BDAFE8A09D355
Output28	0x8	46C4CD6EC69A47D	0xB	D9032549CBE8217
Output29	0xE	3FC3F0A98C99CF1	0x1	46C4CD6EC69A47D
Output30	0xC	7C7EC7A33D0292F	0x3	4D9727EA1FBECFB

An example of the upper part of the LCG in PCG64-DXSM being modified with identical seed in numpy/randomgen

g = PCG64(31415,(292528383318202715988945170563770901305>>1),variant="dxsm",mode="legacy"); g2 = PCG64(31415,(140421743439297021051919259480919575173>>1),variant="dxsm",mode="legacy");
for i in range(30): print("0x" + format(g.random_raw(), "016X") + " 0x" + format(g2.random_raw(), "016X"))

Multiplier	15750249268501108917		15750249268501108917
Increment	292528383318202715988945170563770901305>>1		140421743439297021051919259480919575173>>1
Seed	31415		31415
Output1	0x97F9738805252	FBF	0x3A48F167D2DF3	6B7
Output2	0x964B11CABAA4E	501	0x88DECF43C2B52	E8D
Output3	0xB6AC157505409	594	0x73D73CEDB2296	501
Output4	0xCEC2A042E362F	2A4	0xE36B6340C1CDC	FC3
Output5	0xD0A2096F73ED9	41A	0x28A909ED3D7C7	2A4
Output6	0x7127C22885CC5	4A3	0x2F3751079E58B	CB4
Output7	0x90A13F2459C9B	6CE	0x5EB33E911942C	F94
Output8	0x496FE1695C009	C55	0xE05801B1AA7F2	AF6
Output9	0x80126568D4AA5	FDF	0x8204715BB013E	169
Output10	0x85BC9BD98E5DE	FA4	0x53456ECD3C705	21D
Output11	0xA7380BD92D338	178	0x901391E7BC83E	69A
Output12	0x3E324902274F1	0C2	0xC1698DE771445	A88
Output13	0xA5584406CB702	00A	0x54EC4E180FAC9	0C2
Output14	0x736CBD359BEE5	8B5	0xF43324BCA3F47	26E
Output15	0x7F72C7D6A6C6B	778	0x48C873AED7047	0E2
Output16	0x7CAD149EE868F	197	0x6F3A19369FE36	048
Output17	0x30B0B3F452E1F	475	0x31B0038A96D97	197
Output18	0x36514492F155F	FD8	0x2586DFDD55D21	2EF
Output19	0x0CE19FA5F24DA	C42	0x88058C6DBD897	FD8
Output20	0xE5C9DECAE4A12	C0D	0x5040EC24F3386	95E
Output21	0xE43F8183F92C1	450	0x064D651C1466D	7C3
Output22	0x4E84BB152C42A	D6C	0x36D952B70BFA0	870
Output23	0xA4CD58BA22302	BB5	0x3C74E5804A742	D6C
Output24	0x360F2AB984C9C	065	0x33E5305BEC81D	936
Output25	0x50B420A99712E	267	0x0E3141A4260BF	B79
Output26	0xD5F0C6071AE99	AEB	0xA9AEDA617F487	C03
Output27	0x8354DDF7179B0	638	0x4027BB1478F1E	C11
Output28	0xCE21CF50C0F0E	D3F	0xAA188AB00050F	05E
Output29	0xD08A6974E1276	809	0x21370D1D12353	313
Output30	0x2D33EC6185398	A0C	0x5F439A9086378	B2C