I modified one of my spreadsheets to look for period 16, 32, and 48 bigrams with identical symbols. Basically to show all of the pivots. I marked the pivots that doranchak describes with red boxes.

I used the same message making spreadsheet as with the multi inscription rectangle experiment, but without transposition, to figure out if there are a remarkable count of period 16, 32, or 48 pivots in the 340. There are not. It was very easy to make messages with more pivots like these. Except that, obviously, it is not easy to make such pivots that all line up with each other. I agree that the pivots are very interesting and statistically significant. I just don’t know if they are the product of the cipher, some type of marker, or random. At this point I don’t see how they could be created with a transposition scheme that also creates period 15/19 repeats. And random is not likely. So maybe it is some type of marker.

It is interesting that there are three period 32 pivots in the bottom half of the message that share the same rows and are evenly spaced. Marked with a blue box. Perhaps there is a different transcription scheme for the top and bottom halves.

?

just thinking about how to combine pivots and 19S..The 19s work really well for skytail and the pivots for spirals .. I`m trying to figure if it could have been turned out in skytail and then re written out in a spiral format or visa versa. or would one counteract the other..Smokie can you do this with spreadsheets and see if the existing pivots come into play? That is to highlight the known pivots undo using spiral starting from the middle and working back out and then run through with period 19 skytail. Convoluted and difficult task I know. just ignore if you think it a waste of time. pensive I am..

Hey Mr. Lowe, how are you?

I have been thinking about the pivots and the period 15/19 repeats. So far I cannot think of a transposition scheme that could create both. Read up on transposition ciphers in some books or on websites. I have been doing that and working on getting the terminology right. I suggest that you do the same so that we can talk to each other better. I think that it is a very fun and interesting subject to talk about.

As the experiment above shows, it is likely that Zodiac used one big inscription rectangle, and transcribed the plaintext in a direction that was perpendicular to the direction of the plaintext. But it could be some other scheme. And prior experiments show that it is not very easy to make a message with so many period 15/19 repeats without transposition.

I think that you are talking about transcribing the plaintext in spiral directions. Perhaps that could create pivots. Perhaps portions of the message are transcribed horizontally, and other parts are transcribed vertically. Either way seems very complicated. We can think about it together if you want to. Check out some of the cryptography websites and look for schemes that could create either. We have plenty of time.

viewtopic.php?f=81&t=2916

I am well smokie.. Always busy.. Thanks for asking.. Whilst I don’t have doranchacs , glurks or your knowledge i only have a basic concept..it would take me ages to catch up to you, I will try but I will also sit in the back seat so I don’t disturb your great work.. I will read up more and follow your posts.
Cheers

I found a very simple explanation for the discrepancy between the count of period 39 repeats and mirrored period 29 repeats. It seems intuitive that mirroring the message will create the exact same number of period 15/19 repeats and period 29/39 repeats. But that is not so.

The period 29 repeats are created by mirroring the message. That’s because mirroring the message flips sections of 17 symbols horizontally, but does not write the message backwards.

I only have a few minutes this morning, but here are the two configurations of period 29 repeats. The first configuration occupies two rows in the message that are next to each other, and the second configuration occupies two rows in the message that are separated by a row.

So if you take any message and mirror it, some of the period 39 repeats are going to disappear, and some period 29 repeats are going to be created. Or possibly vice versa. And that is going to happen with any period that can occupy two rows separated by a row.

I have been scratching my head for a long time trying to think of a transposition scheme that could create period 15 and period 29 repeats. And written a lot of spreadsheets to try to duplicate that. But now I feel silly. The mirrored period 29 repeats are created by the mirroring process! That doesn’t mean that they aren’t a product of the cipher, but it means that they might not be. A very simple possible explanation for the mystery.

Upcoming posts will show a few examples in the 340, and explore what happens to the periods in general when mirroring a message in a 17 column grid.

Smokie, thanks for posting the results of your inscription rectangles analysis. When you created your test messages, did you encipher them too, or just inscribe and transcribe them?

I am also curious how the selection of column permutations (i.e., for columnar transposition) within the rectangles would affect the stats.

Still, even if columnar transposition was used, why would those pivots appear? If they are part of some spiral route, why do they appear "normally" (i.e., at period 1), when the ngram stats suggest the cipher text needs to be manipulated to unravel some kind of transposition?

Another thing I wondered is if the pivots themselves were operating at some periodic interval. You know what’s interesting? You just mentioned period 39 repeats, which is exactly the interval between the two pivots!

It occurs to me that I’ve never checked the other periods for the presence of pivots. Has anyone done this yet? Might be an interesting "side quest".

I’m looking forward to your continuing investigations.

Smokie, thanks for posting the results of your inscription rectangles analysis. When you created your test messages, did you encipher them too, or just inscribe and transcribe them?

I inscribed the plaintext messages into the rectangles from the leftmost rectangle to the rightmost rectangle, left to right, top down. Then I transcribed into a 17 x 20 grid by pulling the plaintext from each column, top to bottom, leftmost column to rightmost column. Then I encoded. The results show that it is a lot easier to make 70 period 15 repeats with fewer, larger rectangles ( likely just one ) as compared to more, smaller rectangles.

I am also curious how the selection of column permutations (i.e., for columnar transposition) within the rectangles would affect the stats.

I don’t know. It seems to me that rearranging the columns in a keyed columnar transposition would destroy the period 1 repeats, so that after transcription the period 15/19 repeats would not be created.

Still, even if columnar transposition was used, why would those pivots appear? If they are part of some spiral route, why do they appear "normally" (i.e., at period 1), when the ngram stats suggest the cipher text needs to be manipulated to unravel some kind of transposition? Another thing I wondered is if the pivots themselves were operating at some periodic interval. You know what’s interesting? You just mentioned period 39 repeats, which is exactly the interval between the two pivots! It occurs to me that I’ve never checked the other periods for the presence of pivots. Has anyone done this yet? Might be an interesting "side quest".

Well, the pivots may appear because the message is in a 17 column grid. If you make the message into an 18 column grid, or whatever, then they wouldn’t be there. But… I wonder about whether other pivots would appear if the shape of the message was changed. I have not checked the other periods for the presence of pivots. Interesting idea. I will take a look at the period 39 relationship between the pivots. That is very interesting…

I am really excited about my new period 39/29 discrepancy explanation. It may help me to see the message in a different way. It may be that for different analyses, the message should be drafted in a 15 or 19 column grid so that the repeats all line up vertically. That way when it is mirrored, the stats don’t change. And that might help with the heatmap or solving efforts. More posts coming soon.

EDIT: I just looked at the pivots again and indeed, they are made up of period 39 repeats. That is amazing!!!

Earlier in the thread we discussed that period 2 bigrams, after using a 19 column or row inscription rectangle, would show up as period 38 repeats after transcription. But there are quite a few more period 39 repeats in the message than there are period 38 repeats. We discussed that adding a couple of plaintext during transcription could cause a shift from period 38 to period 39 repeats. That was one possible explanation.

Regardless of how the period 39 repeats got there, it seems that a transposition scheme could increase the likelihood of pivots that occur at certain periods. I wonder if other, smaller pivots occur at other periods. Or even if there are other smaller pivots at period 39 intervals?

Wonder what effect a skytail 39 (or38) will do with the pivots?

Smokie, could you remind me why period 39 is significant? Here are my counts, for any period resulting in bigram count greater than or equal to 25 (the number of repeats in the unmodified Z340):

===Z340, normal===
37: 19
30: 5, 65, 74, 158
29: 39, 54, 82, 135
28: 85, 115
27: 16, 28, 29, 34, 62
26: 4, 21, 45, 128, 133, 140, 159
25: 1, 3, 10, 27, 56, 61, 97, 122, 134, 151, 153, 157

===Z340, horizontally mirrored===
41: 15
34: 29
33: 110
31: 104
30: 144
29: 1, 5, 128
28: 49
27: 3, 18, 34, 150
26: 6, 85, 138, 140
25: 57, 58, 79, 95, 97, 125, 152

First number is the bigram repeat count, and the list of numbers after are the periods that have that count.

Notice that period 39 (boldfaced above) has a count of 29, but 8 other periods have the same or better repeated bigram counts.

Wonder what effect a skytail 39 (or38) will do with the pivots?

A scytale 19 should result in a lot of period 19 repeats, and a few less but a lot of period 38 repeats. That is because period 1 and period 2 repeats in a message that is not transposed will have high counts. Words in the English language have letters that are often next to each other, and also letters that are often separated by one other letter.

A scytale 19 should increase the probability of "pivots" at period 19 and period 38. They would look like "pivots", but they would really just be period 19 or period 38 repeats that happen to occur next to each other. But I don’t know by how much the probability would increase.

Doranchak, for starters, see: viewtopic.php?f=81&t=2617&start=940. Period 39 has high count, and more than period 38. We were thinking that nulls added during transcription could have caused that. I will have to search through the 100 page thread to find more detailed information and post again.

EDIT: Also, period 39 is the same as period 29 mirrored when the period is separated by one row.

EDIT 2:

My theory is that period 39 is actually period 38 that shifted away from its position due to an unknown misalignment problem. The same can be observed with the mirrored 340, where it is period 15 and 29 (should have been 15 and 30). It is you who showed that period 2 repeats transposed to period 38 (with misalignment) have a much higher chance of being disturbed.

checked the period 19 skytale and its influence on the pivots.. nothing to report that stands out.

Doranchak, for starters, see: viewtopic.php?f=81&t=2617&start=940. Period 39 has high count, and more than period 38. We were thinking that nulls added during transcription could have caused that. I will have to search through the 100 page thread to find more detailed information and post again.

EDIT: Also, period 39 is the same as period 29 mirrored when the period is separated by one row.

OK – But then, why are we ignoring the three periods (54, 82, 135) that have the same number of repeated bigrams as period 39, and the 5 periods (5, 65, 74, 158, 19) that have more repeated bigrams as period 39?

I’m just curious if there was some aspect of period 39 that distinguishes it from the other periods. I presume the bigrams in those other periods could just as well be formed by similar misalignments / shifts / mirroring.

Doranchak ..I don’t think smokie was attaching any more significance to the 39s other than the pivots related to them?

OK – But then, why are we ignoring the three periods (54, 82, 135) that have the same number of repeated bigrams as period 39, and the 5 periods (5, 65, 74, 158, 19) that have more repeated bigrams as period 39? I’m just curious if there was some aspect of period 39 that distinguishes it from the other periods. I presume the bigrams in those other periods could just as well be formed by similar misalignments / shifts / mirroring.

Good questions. I am also interested in higher periods, and will have to take a look at them. I have a spreadsheet that makes a bigram match heatmap for any message that I make so that I can see what is going on. The bigger the period, generally the lower the count. So I should take a look at those to see if they are abnormally high for such high periods.

The count of period 39 matches isn’t particularly high. But it is more than the count for period 38, which doesn’t make any sense. I made smokie12, a 19 row route transposition, with two skipped symbols. There was a spike at period 37 instead of period 38. Similar to the 340 spike at 39 instead of 38.

Find this quote on page 43, and then skim through the posts to the bottom of page 46.

In image B, at the bottom you can see the smokie12. I don’t remember it exactly but the smokie12 is also a misaligned transposition. Interestingly enough this has also caused the relfectionary peaks of 1 and 2 (that are 3 and 4) to shift away from their expected position. Because peak 3 should be at 38 (19 times 2) but is at 37 instead and peak 4 is at 31 instead of 30. The same shift can be observed in the 340 and I believe this is evidence of the misalignment of its scheme since these shifts don’t show up in the clean tests. Now I’m wondering if comparing these peaks (15 versus 29) will get us anywhere.

See what I’m getting at. And why this is so exciting. Because the bigram period 15, 19 thing is a very, very significant observation. But we haven’t solved the 340 with that hypothesis yet. So I’ve came to wonder why. And now it seems likely that there is misalignment because of the peak shift (which can also clearly be seen in the smoke12). So I’d like to you to compare bigram period 15 repeats versus bigram period 29 repeats and find out why the repeats are peaking at 29 and not at 30 (for the 340). I think your spreadsheet approach and knack for detail are ideal for this task. Take your time, find out as much as you can. And if you can’t at least we’ll know that comparing these peaks will get us nowhere. I don’t know how to compare them also, you’ll have to come up with something, if possible.

It’s just a theory, of course. But using nulls during transcription is an actual technique used to foil solutions of transposed messages. Skim through the actual scanned book pages 19 – 22, which discusses route transpositions and the use of nulls:

https://www.nsa.gov/public_info/_files/ … 078809.pdf

That’s why I find period 39 for the pivots very interesting. With or without skips or nulls, it seems to me that a route transposition should increase the probability of creating pivots that occur close to each other and at periods that are multiples of the number of rows or columns in the inscription rectangle. That is because the high frequency plaintext are not diffused and show up as period x repeats, and the period x repeats show up parallel to each other in the final message. The period 39 repeats in the 340 that line up in pivots are parallel to each other. If there are nulls that cause a shift from period 38 to period 39, then they could occur in the same rows as the pivots, causing the spike at period 39 instead of period 38.

Sorry it took so long to get back with a complete answer. I have been very burned out lately. Not on the Zodiac 340, but just tired in general.

Just checking in with the cipher gang here, making sure I didn’t miss any important break-throughs or potential solves Glad to see you all are so tenacious