Hi,
as already threatened, I now annoy a little with more abstruse ideas to 340 
The deciding factor was Klaus Schmeh’s blog post to a magic square on a dog tag (doranchak linked this here: viewtopic.php?f=81&t=3196&start=1070#p63482). Jarlve had already posted ideas some time ago that certain magic squares have a kind of "relationship" to P19 in z340. That’s where I did some experiments that I would now like to continue.
In the above mentioned post it was stated that the magic square on the dog tag plays a role in numerology and astrology. I don’t think much of either and I don’t believe in either one. Nevertheless, a magic square remains a magic square and the mathematics behind it is clear. My thought: However, it is very interesting that there is a connection to the "Zodiac symbols". Maybe Zodiac picked up this connection somewhere and came up with the idea of transposition.
Sounds totally absurd? There’s even a coin showing a possible connection:
http://www.treasurenet.com/forums/token … bingo.html
Here you can find the magic squares, which are assigned to the individual planets:
https://www.thoughtco.com/planetary-mag … es-4123077
Well, it’s all quite mystical, I admit. However, I think there might be a very simple and logical connection. Zodiac has seen it somewhere, copied it and claimed it for himself (as he often did). Jarlve already had the idea that z340 could be a 2D-transposition based on a magic square. My idea: Why not from several magic squares? A simple hypothesis is that z340 consists of four single 9×9 magic squares (a 9×9 magic square is found in the previous link under "Moon"). Then z340 would originally be 18×18 and then transposed to 17×20. The remaining 16 symbols were then filled up as for z408. I have created such a cipher for testing purposes:
WT1=mwFnxAe3d5uy2 pK5keAqm+ssi=qJqu =M-9AcZ3J9G3rXqFg J1EYhoOw;W=o4j2dE uADF7vIxoAUEo3iCy J+Z51j0KRpYggGmqS bsIAXM=cLBZYO7HAN yl1dA3HCA;KDhVyr4 ibtCcQ+R05M:CXYLo wX74Zk7FR21qsA9p+ 7nBEJRNbXr=d=gYEn =F:KVYZ-MvHhtwbLb oF-Eqs5n7mINr1E2x KS=pLGYZlAZuw=YOV 3aN4A2y+-Re9SUdtO cSQ=YjCgux=BDuUgX K5oLq7O:7AQFAKv;I kHxXmcauNCWYU=q2M WJrYLJw=3yn7Q1;5Y 5AWRiA+cKZM0eGPsT
It has a spike at P19 and even a pivot (coincidence). 25% cycle randomization was chosen for the substitution. I tried to solve it with AZDecrypt’s transposition solver, but so far without success. BTW: With my new machine I get about 32 MIPS on 12 threads, yay! =)
What do you think? Too confused, too weird, too mystical? Could a 2D transposition based on multiple magic squares be a possibility? If not, what is the exclusion criterion?
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Nice ideas Largo. What is the pattern for the 9×9 magic square? Multiple inscription rectangles are usually stacked either horizontally or vertically, 1-dimensional. But yours looks like it is 2-dimensional which AZdecrypt does not support.
What do you think? Too confused, too weird, too mystical? Could a 2D transposition based on multiple magic squares be a possibility? If not, what is the exclusion criterion?
It is interesting. I would like to see another cipher with ioc similar to the 340.
BTW: With my new machine I get about 32 MIPS on 12 threads, yay!
Could you please try the benchmark, under options, with performance mode on?
You never know what might end up making a difference, and being close minded doesn’t help anything
I agree with that. However, I always try to favour the most simple explanations. Several magic squares may really be a possibility, but I’m still having a little trouble with it. The idea with four squares à 9×9 is still simple enough, so I will experiment with it a bit more.
By the way, thank you very much for the tip with the book! I looked at the description and find it really interesting. It is now on my "still-to-read"-list =)
What is the pattern for the 9×9 magic square?
37 78 29 70 21 62 13 54 05 06 38 79 30 71 22 63 14 46 47 07 39 80 31 72 23 55 15 16 48 08 40 81 32 64 24 56 57 17 49 09 41 73 33 65 25 26 58 18 50 01 42 74 34 66 67 27 59 10 51 02 43 75 35 36 68 19 60 11 52 03 44 76 77 28 69 20 61 12 53 04 45
Source: https://www.thoughtco.com/planetary-mag … es-4123077 (Magic Square of The Moon)
It is interesting. I would like to see another cipher with ioc similar to the 340.
I’ll try to create one. The current one is hand made with AZDecrypt and peek-a-boo. So I will write some code to generate such ciphers automatically (maybe I will extend my cipher factory).
Could you please try the benchmark, under options, with performance mode on?
Here are the results:
6 Threads: 34,22937
8 Threads: 37,22656
12 Threads: 48,76792
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The sample cipher in the first post is based on 18×18 and the plaintext is from Jarlves plaintext library. There are four quadrants, each transposed with the matrix shown above:
HORSESARE TAUGHTTHE DANGEROUS VICEOFBAU LKINGORJI BBINGASIT ISCALLEDI NENGLANDB YIMPROPER MANAGEMEN TWHENAHOR SEJIBSINH ARNESSITI SGENERALL YFROMSOME MISMANAGE MENTEXCIT EMENTCONF USIONORFR OMNOTKNOW INGHOWTOP ULLBUTSEL DOMFROMAN YUNWILLIN GNESSTOPE RFORMALLT HATHEUNDE RSTANDSHI GHSPIRITE DFREEGOIN GHORSESAR ETHEMOSTS UBJECTTOB AULKINGAN DONLYSOBE CAUSEDRIV YXSOITGRE MCEAILEHH SICCMNIET TAONGNLOS WAMIAEGAR ETNNGEGSF OHRPTLYOR BJHAFLMAG NUEERMLFR EAIEGEAIS JESNHOEER INUBTEMNS OISDAOPND MTEVSAMED IMLSAHRET BABRIIUEN EISKINOSR TNNBACNGN HSNTMAHEN RDFGNTBNT OAOEOFROG KSROAWSUD HRTBSBRGW FNTIRNIET TSFHESUOH SROAVMALT OOPREDTBM HEEWNCAUL ARPIUUOOJ IELEODALL ENSIRSNNP KNMUGMSUL ECDENIIDL TIYOLONHS YGTOSGTOE ERNUSLIOI
Then the cipher was substituted with 25% randomness. The pivot was retained manually
Looking at the plaintext – this is great stuff Largo.
If you remember the special bigram peaks then your scheme is interesting in that regard: viewtopic.php?f=81&t=3591 Your plaintext has the period 2 row and column order peaks, though they are inverse to these of the 340. Such that in the 340, the second field, untransposed variation is higher and in your plaintext it is the first field, transposed variation which is higher. Also note the offset row and column order peak at 9 matches with your 9×9 rectangles. No doubt this is because of the 2-dimensional nature of your plaintext.
AZdecrypt, Stats, Plaintext direction: raw bigram ioc depth=2:
Largo’s plaintext:
Offset row order: (transposition) -------------------------------------------------------- Offset row order 0: 524 Offset row order 1: 514 Offset row order 2: 512 Offset row order 3: 498 Offset row order 4: 520 Offset row order 5: 486 Offset row order 6: 540 Offset row order 7: 520 Offset row order 8: 548 Offset row order 9: 552 <--- Offset row order 10: 516 Offset row order 11: 534 Offset row order 12: 512 Offset row order 13: 500 Offset row order 14: 526 Offset row order 15: 526 Offset row order 16: 532 Offset row order 17: 520 -------------------------------------------------------- Transposition average: 521.11 Offset column order: (transposition) -------------------------------------------------------- Offset column order 0: 524 Offset column order 1: 500 Offset column order 2: 518 Offset column order 3: 506 Offset column order 4: 486 Offset column order 5: 522 Offset column order 6: 512 Offset column order 7: 478 Offset column order 8: 496 Offset column order 9: 532 <--- Offset column order 10: 524 Offset column order 11: 498 Offset column order 12: 490 Offset column order 13: 512 Offset column order 14: 502 Offset column order 15: 508 Offset column order 16: 506 Offset column order 17: 496 -------------------------------------------------------- Transposition average: 506.11 Period row order: (transposition, untransposition) -------------------------------------------------------- Period row order 1: 524, 524 Period row order 2: 576, 466 <--- Period row order 3: 494, 478 Period row order 4: 492, 464 Period row order 5: 482, 452 Period row order 6: 478, 494 Period row order 7: 480, 472 Period row order 8: 460, 506 Period row order 9: 466, 576 <--- Period row order 10: 476, 558 Period row order 11: 468, 460 Period row order 12: 444, 458 Period row order 13: 448, 462 Period row order 14: 456, 468 Period row order 15: 450, 450 Period row order 16: 486, 482 Period row order 17: 502, 506 -------------------------------------------------------- Transposition average: 481.29 Untransposition average: 486.82 Period column order: (transposition, untransposition) -------------------------------------------------------- Period column order 1: 524, 524 Period column order 2: 546, 468 <--- Period column order 3: 508, 468 Period column order 4: 466, 456 Period column order 5: 478, 450 Period column order 6: 468, 508 Period column order 7: 506, 442 Period column order 8: 492, 494 Period column order 9: 468, 546 <--- Period column order 10: 442, 530 Period column order 11: 474, 482 Period column order 12: 450, 472 Period column order 13: 466, 450 Period column order 14: 470, 458 Period column order 15: 450, 464 Period column order 16: 474, 460 Period column order 17: 484, 462 -------------------------------------------------------- Transposition average: 480.35 Untransposition average: 478.47
340:
Offset row order: (transposition) -------------------------------------------------------- Offset row order 0: 90 Offset row order 1: 90 Offset row order 2: 88 Offset row order 3: 88 Offset row order 4: 90 Offset row order 5: 88 Offset row order 6: 90 Offset row order 7: 82 Offset row order 8: 84 Offset row order 9: 88 Offset row order 10: 96 Offset row order 11: 94 Offset row order 12: 80 Offset row order 13: 80 Offset row order 14: 90 Offset row order 15: 88 Offset row order 16: 98 Offset row order 17: 104 Offset row order 18: 106 <--- Offset row order 19: 90 -------------------------------------------------------- Transposition average: 90.2 Offset column order: (transposition) -------------------------------------------------------- Offset column order 0: 90 Offset column order 1: 114 <--- Offset column order 2: 96 Offset column order 3: 86 Offset column order 4: 82 Offset column order 5: 80 Offset column order 6: 82 Offset column order 7: 82 Offset column order 8: 82 Offset column order 9: 80 Offset column order 10: 80 Offset column order 11: 88 Offset column order 12: 90 Offset column order 13: 90 Offset column order 14: 86 Offset column order 15: 76 Offset column order 16: 76 -------------------------------------------------------- Transposition average: 85.88 Period row order: (transposition, untransposition) -------------------------------------------------------- Period row order 1: 90, 90 Period row order 2: 98, 76 <--- Period row order 3: 84, 84 Period row order 4: 78, 82 Period row order 5: 82, 78 Period row order 6: 96, 74 Period row order 7: 84, 84 Period row order 8: 80, 76 Period row order 9: 82, 80 Period row order 10: 76, 98 <--- Period row order 11: 78, 80 Period row order 12: 76, 78 Period row order 13: 96, 76 Period row order 14: 82, 70 Period row order 15: 80, 78 Period row order 16: 76, 72 Period row order 17: 84, 80 Period row order 18: 78, 80 Period row order 19: 90, 90 -------------------------------------------------------- Transposition average: 83.68 Untransposition average: 80.31 Period column order: (transposition, untransposition) -------------------------------------------------------- Period column order 1: 90, 90 Period column order 2: 86, 108 <--- Period column order 3: 84, 82 Period column order 4: 82, 70 Period column order 5: 78, 76 Period column order 6: 82, 84 Period column order 7: 78, 70 Period column order 8: 82, 74 Period column order 9: 108, 86 <--- Period column order 10: 90, 80 Period column order 11: 78, 80 Period column order 12: 84, 70 Period column order 13: 78, 74 Period column order 14: 76, 74 Period column order 15: 74, 82 Period column order 16: 72, 82 -------------------------------------------------------- Transposition average: 82.62 Untransposition average: 80.12
Jarlve: Thank you for the analysis. I need to take a closer look so that I can better understand your findings.
Here is another cipher, this time with a Raw-IOC of 2426. no pivots available. It has a P19 peak, but not a P15/Mirror peak. Unfortunately it behaves completely different in the "plaintext direction: raw bigram ioc depth=2" statistics (to me, at least).
I didn’t had time to automate cipher generation yet. However, this cipher clearly shows the effect of a badly choosen key. In this case, the plaintext very often contains ‘O’, but in the key ‘O’ is only substituted with ‘X’ and ‘=’. Hence there is not the smooth distribution of unigrams one expect in a homophonic substitution.
Plaintext ("Dracula", Project Gutenberg), 18×18:
BOUTTOPERFORMWHATW ECALLTRANSFUSIONOF BLOODTOTRANSFERFRO MFULLVEINSOFONETOT HEEMPTYVEINSWHICHP INEFORHIMJOHNWASTO GIVEHISBLOODASHEIS THEMOREYOUNGANDSTR ONGTHANMEHEREARTHU RTOOKMYHANDANDWRUN GITHARDINSILENCEBU TNOWYOUAREHEREYOUA REMOREGOODTHANUSOL DORYOUNGWHOTOILMUC HINTHEWORLDOFTHOUG HTOURNERVESARENOTS OCALMANDOURBLOODNO TSOBRIGHTTHANYOURS
Transposed with 9×9 magic squares. Added 16 filler symbols:
HAFEOBLMTIROSSISO WOENUYOLLIHNTFTFS IJNPEMLODGTOASHOR EOOREEMELTTINHTWU NUROVAFRPOVHODONW HHENFTENOBTNEERAF WFIETGMRHEROYGIAO SSAOCRONOBICHUVTT NAHFSRHEHMRLSAITE ASDNEUTMPDIENORHR KHOTDETEGDMOGMDWV AHWOUHNRSNLIYRHOO YHRDRTRAOEECDNHYT ROOTEOUOSNUYSOITA OTECUABFNITUROAUN HRUSGAURUTNLHSBLR RGETNOOLAESHDMAOO MTNIWOGBLOENIOAUN RRANETOOWYDOHOLUN CQXZAIDJEMFNSKQLW
Final cipher:
FdIyX9:;Lhn=M1cKX l=xArTXuuC27wILIM Cm7Y3;:=kENXDop=G 0XXV+0;PuLwcRFslv AStXQiInY=Q2=H=ql pFa7INaRX4LAx3VZI lIcPwJ;V2+t=TECdX MKZ=enXqX95eprQsN 7iFIMG20p;VUoihL3 d1HRgvw;YkcaAXtFn j2=LBxN3JH;=E;klQ Dpl=r2qGo7:CTV2X= TptBVLGiXy+bHRFTw V=Xs0=vXMAvTo=5NZ XLPeSd4Iqhsrt=Dv7 FnSMJirGvLRupK9Ut tEgNAX=:iaMFk;iX= ;Nqcl=E4u=37CXDSR ntiA3w=XlTB=2XUrq eO-WD5Hmy;I7ojOul
As soon as I have a generator for such ciphers, I will post more material that hopefully will have more in common with z340.
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I did some tests with the idea shown above. To do this, I automatically generated ciphers from a large amount of plain text and only sorted out those that have certain matches with z340. For this test I have chosen a 100% cyclic homophonic substitution. Here is the result (from a total of 296189 ciphers):
Number of ciphers with IOC >= 2200 : 768 (=0.259%) Number of ciphers with bigram peak at P19 : 37468 (= 12.65%) Number of ciphers with IOC >= 2200 + bigram peak at P19 : 157 (= 0,053%)
Ciphers consisting of four 9×9 magic squares therefore have a certain tendency to generate P19. This is not surprising when you look at the overall picture "geometrically" (e.g. with peek-a-boo).
I’m not sure anything can be determined by such tests. There are simply too many factors involved. An extremely important aspect is the key used and the resulting IOC. I would be interested to know how much weight should be given to the IOC in homophonic encryption. If the key was generated based on the letter distribution of the English language, but the text does not correspond to these statistics, then one does not get a "smooth" distribution, that is obvious.
The ciphers generated in the test usually only had a high IOC if they contained a large number of "O’s" for which there were only two symbols in the key. In short: The worse the key, the higher the IOC.
A similar assumption has already been made for z340 and the + symbol. The + symbol may only map to a single plain text letter.
Substituting the + symbol in z340 with four symbols (for example +;;:#+;:#+;:#+;:#) results in an IOC of only 1804, so low IOC scores are not an exclusion criterion for certain theories, or am I missing something?
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Higher ioc should increase bigram repeats. I suspect that your test ciphers with ioc > 2200 may be unnatural. Allow some randomness in your key and try more than 157 samples.
Allow some randomness in your key
Okay, I’ll try that.
I tested a total of 296189 ciphers, not just 157. Each of the ciphers was transposed with four 9×9 squares. Of these ciphers only 157 had an IOC >= 2200 and at the same time a peak at P19. The plaintexts come from different books of project Gutenberg and Wortschatz (news 2005) and seem very natural to me. The ciphers have 16 fillers at the end in addition to the plain text. Here are two samples:
ONONTHEPROPERTYINT HESMANYITFIRMSOFFE RFLEXIBLEWORKINGHO URSANDTHEOPTIONTOW ORKFROMHOMEBUTSOME GOBEYONDTHISOFFERI NGFREEFOODEXTRADAY SOFFFREETRAININGOR EVENFREEBEERTOLURE ANDKEEPEMPLOYEESMA TTRICIANIPREFERSTO PRAISETHEEFFORTSOF THEHOUSETOPASSADDI TIONALTORTREFORMSD URINGTHISYEARSREGU LARLEGISLATIVESESS IONAARONNARVAALSOW ASCHARGEDWITHVIOLA OSTmGPJ01JHQvUbJA 2xVnWopOcw!q43PF! XyOrFsd09KLBCfSDT AjP!gQufMYBvuZN5t OUKmuCnVSokPg9uO1 2pLewIT3!PO70uz1q uPWr2DustxOMJmyNP UuuQOVPnuSopKOzL8 A8bBqrJ3u6TuuXlMP OPNOjUVSs0TxPcYCZ t1DQWuXu4mnAvoKBG h5pUOdPOYq!VQPryL ZHCsWtIX8SeTbDmMk OAncYdJueZb23UNKJ cd6P0B1VeO8bSWzGC wcHXYLP7OZQTDl7do p23M0iqeurQUIJ1j2 bQv3xkPyAcsWVtOXz d0STBUCFV7PGumnw! lER"4u5vQ"fFxgOFw EFOREREASONEDWITHY OUBUTYOUHAVEPROVED YOURSELFUNWORTHYOF MYCONDESCENSIONREM EMBERTHATIHAVEPOWE RYOUBELIEVEYOURSEL FMISERABLEBUTICANM AKEYOUSOWRETCHEDTH ATTHELIGHTOFDAYWIL LBEHATEFULTOYOUYOU AREMYCREATORBUTIAM YOURMASTEROBEYTHEH OUROFMYIRRESOLUTIO NISPASTANDTHEPERIO DOFYOURPOWERISARRI VEDYOURTHREATSCANN OTMOVEMETODOANACTO FWICKEDNESSBUTTHEY mG!W4f5noA!KjOLQH 8SJBhPTI0UCx8X1VJ S7!pgvO8Yu!q2bDAy 3rPOPcszMdtJ7!x7G OTyfB64ZUekFH5mP! n0No!uWzgp12qrO3l 6PQOV0!4sPtux1Xiu CbSKOu#hP!T5IOcyJ zLi7Yn8joJ6PdfDU2 epq8brsktMm4J53JO cl0nhgNfAP1BjVoSx !k62yZ3pK48OqWLP0 J7ir!zsOP!T1UuuQt uOdmCVPn!GO2olp!5 euJ3XbcD6QY4jx0!d PHZ8!Jq1JWIrX2OPs Oe5g3ABbyY0c1SfCT 7!6UVtdhPMSiOTPD4 FOmPUnQzN2VeObST! PER9"89F3"JUuku8"
Of these ciphers only 157 had an IOC >= 2200 and at the same time a peak at P19.
Ah yes, I see. Kind of misinterpreted that.
Here are some results with some randomization in the key cycling:
Total Ciphers Processed: 296189 Ciphers with IOC >= 2200 AND bigram peak P19: 100% cyclic : 157 (0.053%) 20% random : 282 (0.095%) 50% random : 887 (0.299%) 100% : 5245 (1.77%) Total Ciphers Processed: 296189 Ciphers with IOC >= 2200 (no checks for period bigram peaks): 100% cyclic : 771 (0.26%) 20% random : 1564 (0.528%) 50% random : 5389 (1,819%) 100% random : 31481 (10,628%)
Since the IOC is of independed on transposition it must therefore be considered separately. It is true, however, that increasingly random cycles also increase the IOC.
Higher ioc should increase bigram repeats.
Can you explain this to me in more detail, please? I don’t see the connection. Maybe I’m missing something.
It still seems to me that the IOC of homophonic encryption depends very much on the effectiveness of the key used.
The 9×9 magic square idea is meanwhile a minor one to me, but I don’t want to give it up yet.
Higher ioc should increase bigram repeats.
Can you explain this to me in more detail, please? I don’t see the connection. Maybe I’m missing something.
Consider the string "0123456789". It has 0 bigram repeats and a raw ioc of 0. The minimum for both.
Versus the string "0000000000". It has 9 bigram repeats and a raw ioc of 90. The maximum for both.
Another thing that increases bigram repeats are repeating sequences. Language works like that.
The string "CABACABCB" has 2 bigram repeats.
Versus "ABCABCABC" which has 5 bigram repeats.
Therefore both a higher ioc and a more sequential plaintext and/or ciphertext (such as sequential homophonic substitution) increase bigrams. These concepts are fundamentals.
A little test:
Higher ioc increases bigrams:
10,000 ciphers, raw ioc target 1800, 25% random cycles. Average bigrams: 28.19
10,000 ciphers, raw ioc target 2200, 25% random cycles. Average bigrams: 37.29 <—
More sequential increases bigrams:
10,000 ciphers, raw ioc target 2200, 50% random cycles. Average bigrams: 36.88
10,000 ciphers, raw ioc target 2200, 0% random cycles. Average bigrams: 37.64 <—
Thank you for the detailed explanation! I now understand why a higher IOC increases the number of bigrams.
More sequential increases bigrams:
10,000 ciphers, raw ioc target 2200, 50% random cycles. Average bigrams: 36.88
10,000 ciphers, raw ioc target 2200, 0% random cycles. Average bigrams: 37.64 <—
My measurements show a slightly different result.
I can only confirm your statement at 10% and 25% randomness. In this case, the number of bigrams actually decreases compared to a fully cyclical substitution.
At just over 50%, however, it seems to be rising again. With 100% random cycles even more bigrams are obtained than with perfect cyclical encryption. Are you familiar with this behaviour? I can provide precise measurement data and a description of my procedure if you are interested.
The higher the randomization of cycles, the higher the IOC. Together with your statement this means: randomly selected cycles creates more bigrams.
Or do I have a thinking error here? My measurements seem to be correct.
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To be more specific "More sequential increases bigrams independent of changes in ioc". As you can see I specified a raw ioc target of 2200 for both tests in that comparison.
I am trying to describe these as 2 separate phenomena that produce repetitive phenomena:
1) Higher ioc = more bigrams/repeats of varying kind.
2) More sequential = more bigrams/repeats of varying kind independent of changes in ioc.
The higher the randomization of cycles, the higher the IOC. Together with your statement this means: randomly selected cycles creates more bigrams.
Indeed.
Hi Jelberg,
These are really interesting findings and the possibilities described show many similarities to z340 and further letters.
The only problem with these things is that z340 is not square, but rectangular. But as you wrote, a grille can also consist of 6×6. Unfortunately 9×9 is not possible (for a rotating grille).
If you look at the picture at the end of your post, you can basically see four quadrants, of which the lower left one is even rotated. I noticed this a long time ago and ran some tests in which I performed all combinations of Rotation/Flip/Mirror for all possible quadrant sizes (really very, very many ciphers were created). Unfortunately without result. But what I’ve only just noticed about this picture: In the middle of the four sections is a +. For z340 this is the same if you take line 10 as the center. And the lowermost quadrants are a bit higher than the uppermost. Just like in z340.
Whether Grilles produce Period 15 or Period 19, I do not know. Never tested that.
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