Canonical cipher 3: Z13 vol.1. Entropy based solution of Z13

 An entropy-based solution of the Z13 based on a cascade consideration of ciphers

Comment: Originally proposed in May 2024 on Tom Voigt’s Tapatalk Zodiac Killer forum and message board, this solve has undergone revision that does not take away from the step-1 solution (Table III) nor the end solution (Table IV). The solution wishes to explain the process of step-2 better and contextualize with the solution proposed by Dr. Ryan Garlick. Moreover, it offers a step-3 for the validation of symbols order.

Z13 has been considered time and time again as per its short length which does not allow for an inclusion (and therefore discovery) of an internal key. Various suggestions have been provided and various solutions have been proposed, including some very plausible and interesting solutions by zodiac researchers and cipher scholars in this forum, such as cipher (comment: SolidSnake1990) and rando.

My approach, based on my signal processing/information theory background, is based in a direct confrontation of the Z13 on two simple criteria of hypothesis:

Criterion 1: Any chain of time-discrete symbols of information is bound to form a certain alphabet. This alphabet defines the alphabet length of the symbol series. The occurrences and frequencies of each alphabet variable produce the overall cipher length, but more importantly, define the entropy of the information piece (here: the cipher) based on the definition provided by Claude Shannon.

Criterion 2: Even though the Shannon-Fanno table is not as successful in achieving approximation of Entropy as the Huffman tree, it is to be employed because it provides the tiers of the symbols (cipher alphabets) within the confines of the Z13. It also allows for respective retrieval of information as per the previous ciphers. If the Z13 is not long enough to provide “key from within” then we must acquire external information. Since the alphabet variables are independent in their “making” and uncorrelated to the previous or next symbol (as opposed to DPCM methods, lets say), then we must not seek inside the Z13 for their interpretation, but to the previous ciphers. This validates the unified consideration of Z408-Z340-Z13 as a cascade of systems, where each sub-system consists of a table of statistical metrics and attributes. Indeed, Shannon’s entropy stands at the very centre of these metrics.

 The well known cipher length is 13:

Figure 1. Cipher length

However, the alphabet length is 8. Seven out of the eight unique symbols appear in Z340. They are:

A    E    N     (ZODIAC CROSS)   K     M     _I_

It is interesting to note that the H/8 symbol characterizes the very length of the cipher alphabet: for 8==H. The three occurrences of the “8-ball” could very well stand for 3xH and/or 3x8: a reference to the three sections of Z408. In a follow-up note I will try to offer further outliers towards that direction.

STEP-1: THE SEVEN SYMBOLS

 

Table I. Frequencies and Entropy of the Z13 Cipher Alphabet

We see that the total entropy is 2.873. This of course can be validated from [https://onlinetexttools.com/calculate-text-entropy]:

 

Figure 2. Fast-track calculation of entropy for a cipher alphabet of 8 that produces a cipher length of 13 based on the frequencies of Table I.

 

But if the symbols are uncorrelated and independent “in their making” and only their order is correlated to the desired output of the Z13 (the name), then we need to establish a liaison between the definitions/assignments of these alphabet variables from the previous sub-systems of the cascade. The directly previous is Z340. It is also referenced (in taunting manner) in the April 20, 1970 communique.

The results are shown in Table II.

 

 

 

 

Table II. Assignments from the previous sub-systems of the cipher-cascade

 

A direct feed of Table II assignments to the Z13 provides:

Table III. Z13 substitution: step-1

Α

Ε

Ν

ZODIAC

HBALL

K

HBALL

M

HBALL

_I_

N

A

M

D

R

E

A

Α

O

L

E

D

O

 

Comment: in Spring 2025 Dr Ryan Garlick presented his solve on “DREATATORPEDO”. This can be considered as a variant of step-1 evolving into an independent step2 where the 8-ball values can be assigned any given letter of the Latin alphabet pending on any given hypothesis. Dr Garlick’s solve assumes that the anchor-like symbol of position 10 equals “P” whereas I follow the respective Z340 symbol, with the key assigning a value of “L”.

 

 

 

 

 

 

 

 

STEP-2: THE 8-BALL

In my original proposal of May 2024 I offered a convoluted explanation that utilized the position of the 8-ball symbols (pos 5 -7 – 9) as a basis for the assignment of “J L N” values. Offering a direct schematic in Figure 3, we have:

Figure 3. Substitution of 8-ball values based on their position on the cipher

AEN {ZODIAC CROSS AS PUNCT.MARK 1} J K L M N {ANCHOR AS PUNCT.MARK 2} NAM

These are letters J, L, N respectively. Therefore, we have for Z13:

Table IV. Z13 substitution: step-2

Why the shift?

From an entropy-perspective, the-8ball is the strongest symbol in the Z13 (tier-1).

The second tier of Z13 is A N M. This the order as we traverse Z13 from L to R.

From the symbols in the tier-2, only “N” appears as a substitution letter for the-ball. The last triad places the “N” letter at position1 of the triad.

Following this “hint”, we shift JLN by the same manner, therefore NJL.

Thus Table IV is produced on this simple shift:

N is part of the tier-2 triad (entropy-wise).

It appears in first section (AEN), last section (NAM) and in the triad of letters for the 8ball substitution (JLN).

The last triad places N at pos1. We follow that logic for the final arrangement of the 8-ball substitutes: JLN becomes NJL.

J is in the middle. Two clusters of six-symbols in either side.

The name produced reads: Andrea J. Odell

 

STEP 3: READ SEQUENCE / PRINT OF MESSAGE

Comment: The read sequence compels us to devise (or try to grasp) a self-indulgent teletype system as it may have been introduced by the killer. The reason is that the order of the plaintext must follow a systemic method based on rules of conduct.

1.        No symbols must be left unread/not accessed.

2.        Read sequence begins from right to left (accessing the alphabet from Z to A).

3.        Boundary points: first (far-right) and last (far-left) points of message block.

4.        Each “8-ball” can be seen as carrying content and added functionality as a “Paragraph/Change order of symbol read” break point. A variant of the “Paragraph” command in the ASCII system (code: 244).

5.        Next read routine begins with shifted order at the next “8-ball” break point. Each break point is to the nearest/next left available.

6.        All symbols can be accessed only once (clear content after read/print).

7.        The ZODIAC CROSS is a superior break point (“end-point”) which carries content and added functionality: it activates “ETB” and “EOT” (“End of transmission block / end of transmission”). It is the last, as it is the killer’s signature, as in the letters.

FIRST READ ROUTINE

SECOND READ ROUTINE

THIRD READ ROUTINE

 

FOURTH READ ROUTINE

For the current read routine, the change of reading order means we read/access symbols from left to right from the furthermost boundary point on the left.

We read the final “DREA”.

The ZODIAC CROSS carries the content “A” and then activates the “end-point” functionality.

So, the process ends (ASCII: ETB/EOT/).

The last symbol is the ZODIAC CROSS (the killer’s signature) with content “A”, the first letter of the Latin alphabet, which is read as the last, validating the original “right to left” order of read (from “Z” / ZODIAC as first to “A” as last / ZODIAC CROSS). The message ends with the last symbol being the ZODIAC CROSS, as in the killer’s letters.

The final printout is:

ODELL   “Paragraph/Change order of symbol read”

J.               “Paragraph/Change order of symbol read”

AN            “Paragraph/Change order of symbol read”

DREA       “End of transmission block / end of transmission”

 

The name produced reads: Andrea J. Odell

Andrea Jeanette Odell, born in 1958 and d. 1959 (see Figure 5) is genealogically linked to all seven Zodiac canonical victims, as well as non-canonical victims and victims from notable murder cases. And what was the purpose of the killer? In my opinion, to validate his most importance of all the “Zodiac semiotics”: genealogy and murder. But even this would have to be yet another “treasure hunt” of riddles.

Comment: this solution does not dispute the existence of many other solutions, including Dr Garlick’s or any other for that matter.

The plausibility of each proposal, including the many deeply interesting proposals of Solidsnake1990 who has put time and effort to highlight through his rational proposals a systemic pool of significant data.

The very length of the cipher means no key is inherent. However, on this approach I can offer a robust key built on the Z340 key for seven of the eight symbols and a logical function for the substitution of the 8-ball which can have a singular expression, applicable to all occurrences of the 8-ball throughout the cipher. This provides a higher resilience, in my opinion, compared to Dr Garlick’s method (Figure 4).

Figure 4. My proposal vs Dr Garlick’s on the common foundation of Z340 key: a comparative approach

 

A singular function for the 8-ball substitution: POSITION RIGHT SHIFTED BY E (5)

1st  8ball: Position 5 à Right shift by 5 à 5+5=10 == J

2nd 8ball: Position 7 à Right shift by 5 à 7+5=12 == L

3rd 8ball: Position 9 à Right shift by 5 à 9+5=14  == N

FINAL KEY

Z13 key= {Z340 key} AND {8ball: POSITION RIGHT SHIFT BY E}

 

CONCLUSION

 In this proposal, resilience lies in the following foundation:

1.        The systemic sequence of Z408àZ340àZ13 reflects a logical cascade.

2.        Seven of the eight symbols are “fed” directly from the Z340 based on this cascade.

3.        The multi-variate 8-ball hints at a higher functionality from a systemic perspective. The substitution of J L N assumes the middle section as an alphabet sequence proper, where the content (J L N) are the positions of the symbols (5-7-9) shifted by “E” (dominant Z408 element). The fact-checking lies in the logical sequence of “J K L M N”.

4.        Final placement (order) of 8-ball content “J L N” is based on the shift of the N in the last triad (NAM) at position-1. This is based on identifying the “N” as being in the tier-2 class on an entropy-based classification of symbols. It is the only one of the tier-2 symbols that is part of the proposed “8-ball” content triad.

5.        The final placement of the plaintext content is justified and indeed validates a rational teletype system on ASCII basis where fundamental “reading logic” obeys the rules of conduct without overlapped reads or missed symbols. The printout gives the name “ODELL J. AN DREA”.

6.        The Zodiac Cross is also a symbol with added teletype functionality as it offers ETB/EOT command. Thus, the last symbol is the killer’s signature, as in letters.

7.        The method offers a singular key based for the most part on the Z340 key (cascade logic flow).

8.        A name leading to a further “riddle” is compatible with the killer’s taunting method.

Figure 5. Obituary of Andrea J. (Jeanette) Odell

SDS

Version1: May 5, 2024

Version 2 (revised): April 23, 2025