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Science Section => Science General Discussion => Math and Computers => Topic started by: Baruch on March 23, 2019, 05:34:11 AM

In quantum mechanics, information is neither created nor destroyed (barring black holes). See my argument after the film, on why the title is a rhetorical question.
https://www.youtube.com/watch?v=HF9Dy6iB_4
In a cryptography situation, if you know the key, you can encrypt a message, and decrypt it too. The total process involves conservation of information. And QM supports this total model. But you really encrypted a message using a random key (in the sense that you don't know what it is) then you won't be able to decrypt the message, only encrypt it. If you look at the Shannon Information Entropy for a classical system, or the Von Neumann Information Entropy for a quantum system, you will see that the entropy (and thus the information) changes going from the plain text to the crypt text. The missing information is restored when decrypted by someone with the correct key. With asymmetric key exchange, you do have a situation where there is a public key and a private key. I encrypt my email with my private key (which subtly matches the public key), and using a third party key holder, you can use my public key to decrypt my email. Without knowing my private key. Again, a more sophisticated situation, but because the plain text is restored from the crypt text, information is conserved over the total process.

Firstly i should express that i havent got even a clue about about what you are talking about?
What do you mean by information? it is something like "the sun rises in the east" or "birds fly, fish swim"...
if the information you mean in the universe is this, the mission for us is to discover that

Firstly i should express that i havent got even a clue about about what you are talking about?
What do you mean by information? it is something like "the sun rises in the east" or "birds fly, fish swim"...
if the information you mean in the universe is this, the mission for us is to discover that
Well, the OP is a rhetorical question ... turns out, within the limits outlined, cryptography is consistent with quantum mechanics being complete.
I will give you an elementary lesson ...
Shannon Entropy is for classical systems ...
https://www.youtube.com/watch?v=R4OlXb9aTvQ
This whole series (this is part 12) is the best intro to Information Theory ever. It really helped when I first studied this 4 years ago.

In quantum mechanics, information is neither created nor destroyed (barring black holes). See my argument after the film, on why the title is a rhetorical question.
In a cryptography situation, if you know the key, you can encrypt a message, and decrypt it too. The total process involves conservation of information. And QM supports this total model. But you really encrypted a message using a random key (in the sense that you don't know what it is) then you won't be able to decrypt the message, only encrypt it. If you look at the Shannon Information Entropy for a classical system, or the Von Neumann Information Entropy for a quantum system, you will see that the entropy (and thus the information) changes going from the plain text to the crypt text. The missing information is restored when decrypted by someone with the correct key. With asymmetric key exchange, you do have a situation where there is a public key and a private key. I encrypt my email with my private key (which subtly matches the public key), and using a third party key holder, you can use my public key to decrypt my email. Without knowing my private key. Again, a more sophisticated situation, but because the plain text is restored from the crypt text, information is conserved over the total process.
Did you just say that you are falsifying who you are using 3rd party encryption?

Nope. That is a different discussion. Here is how PKI keys are used asymmetrically. You know a reliable third party key holder. You create a key pair, one private, the other public. You send your public key, along with your verified ID to the third party key holder. Other people do the same. So if I want to exchange encrypted mail with another party, I encrypt my outgoing meal with someone else's public key. The receiving party gets my public key from the third party key holder, as well. Once the receiving party gets my email, they have their private key all to themselves, and use it to decrypt my message, that was encrypted with their public key. The other party, sending encrypted mail to me, does the same thing symmetrically. They encrypt their email using my public key, and when I receive it, I use my private key, that only I have, to decrypt the message sent to me. Basically blind sharing of keys.
Unfortunately the majority of asymmetrical key exchange, is based on the factoring of dual prime numbers (an integer made up of just two large primes multiplied together). Quantum computing is uniquely able to factor dual prime numbers. Quantum computing is an analog/digital hybrid, useful for certain hard problems, but not all, and not practical for ordinary computing. Once you have factored the large prime, you can read an asymmetrical PKI (which might be 256 bits long).

Nope. That is a different discussion. Here is how PKI keys are used asymmetrically. You know a reliable third party key holder. You create a key pair, one private, the other public. You send your public key, along with your verified ID to the third party key holder. Other people do the same. So if I want to exchange encrypted mail with another party, I encrypt my outgoing meal with someone else's public key. The receiving party gets my public key from the third party key holder, as well. Once the receiving party gets my email, they have their private key all to themselves, and use it to decrypt my message, that was encrypted with their public key. The other party, sending encrypted mail to me, does the same thing symmetrically. They encrypt their email using my public key, and when I receive it, I use my private key, that only I have, to decrypt the message sent to me. Basically blind sharing of keys.
Unfortunately the majority of asymmetrical key exchange, is based on the factoring of dual prime numbers (an integer made up of just two large primes multiplied together). Quantum computing is uniquely able to factor dual prime numbers. Quantum computing is an analog/digital hybrid, useful for certain hard problems, but not all, and not practical for ordinary computing. Once you have factored the large prime, you can read an asymmetrical PKI (which might be 256 bits long).
Thank you for mansplaining the dual prime numbers multiples. GAGobvious!
I'll wait for the practical application.

Thank you for mansplaining the dual prime numbers multiples. GAGobvious!
I'll wait for the practical application.
Supposedly the dual prime numbers technique (used to public key distribution) is vulnerable to quantum computers. Not everything is. But this is a hugely important target. It is yet to be seen that quantum computing can be used in a practical way to break typical public keys.

Supposedly the dual prime numbers technique (used to public key distribution) is vulnerable to quantum computers. Not everything is. But this is a hugely important target. It is yet to be seen that quantum computing can be used in a practical way to break typical public keys.
I agree that quantum computers could make passwords obsolete. I'll go mostly offline then. But there will be other ways to protect stuff, so maybe not.

I agree that quantum computers could make passwords obsolete. I'll go mostly offline then. But there will be other ways to protect stuff, so maybe not.
I hear lemon juice is a useful invisible ink ;)

It does exist solutions against cryptanalysis with a quantum computer.
Officially , it doesn't exist one with a sufficient calculation power to get a RSA private key with a size of 4096 bit (Maximum size for the majority of RSA implementation) from the public key.
Shor's algorithm can be theoretically used with a quantum computer (With a sufficient calculation power) to break RSA and others asymetric cipher based on integer factorization like Benaloh , Blum–Goldwasser , Damgård–Jurik , Goldwasser–Micali , Naccache–Stern , Okamoto–Uchiyama , Rabin , SchmidtSamoa etc...
We speak often of RSA when we are speaking of publickey cryptography but we forget RSA isn't the one PK cipher in the world.
We have many solutions against quantum computers.
A list of solutions against quantum cryptanalysis : https://en.wikipedia.org/wiki/PostQuantum_Cryptography_Standardization
Onetime pad is an unpractical solution against quantum computers to resist to these but if we use it under some conditions (Never use the same key , Key size is the same than message send , The generation of key must to be aleatory) it can't be broke : https://en.wikipedia.org/wiki/Onetime_pad
McEliece cryptosystem is immune against Shor's algorithm but I doubt it will still resist to quantum computers as I'm certain new algorithms will appear : https://en.wikipedia.org/wiki/McEliece_cryptosystem
Benaloh : https://en.wikipedia.org/wiki/Benaloh_cryptosystem
Blum–Goldwasser : https://en.wikipedia.org/wiki/Blum%E2%80%93Goldwasser_cryptosystem
Damgård–Jurik : https://en.wikipedia.org/wiki/Damg%C3%A5rd%E2%80%93Jurik_cryptosystem
Goldwasser–Micali : https://en.wikipedia.org/wiki/Goldwasser%E2%80%93Micali_cryptosystem
Integer Factorization : https://en.wikipedia.org/wiki/Integer_factorization
Naccache–Stern : https://en.wikipedia.org/wiki/Naccache%E2%80%93Stern_cryptosystem
Postquantum cryptography : https://en.wikipedia.org/wiki/Postquantum_cryptography
Rabin : https://en.wikipedia.org/wiki/Rabin_cryptosystem
RSA : https://en.wikipedia.org/wiki/RSA_(cryptosystem)
SchmidtSamoa : https://en.wikipedia.org/wiki/SchmidtSamoa_cryptosystem
Shor's algorithm : https://en.wikipedia.org/wiki/Shor%27s_algorithm
Okamoto–Uchiyama : https://en.wikipedia.org/wiki/Okamoto%E2%80%93Uchiyama_cryptosystem

Ah, someone who is more than a recreational mathematician ;)
With circular functions, you have the truly periodic and the almost periodic. The truly periodic corresponds to a pure timbre, the simplest being a single pure tone (sine wave). If the ratio of at least one irregular component among the purely periodic ones, meet the number theory criteria of not being rational, not even algebraic eg square root of two, but is transcendental "pi/e/etc" ... then even after an infinite expansion of digits, the odd component will never align (harmonize). Such is the theory of almostperiodic functions, and how it becomes harder and harder to derive an accurate fourier analysis of the nonharmonics. Such is "percussive" sound, which can be partly harmonic (tympany) or nearly pure noise (true random number).
Consider not a true one time pad, but one that is arbitrarily close to a one time pad. Then the matter of being hard to decrypt is simply a parameter, that one can extend as necessary, as decryption gets better (an arms race). What if you have a key that isn't periodic (has a recurring depth), but is algebraic or even transcendental. In that case you can extend the key system indefinitely, without running into recurrence (which weakens the encryption). You can't as a practical matter use a true nonrational number, one approximates a nonrational number with a close enough rational number eg you use however many digits of "e" as necessary to meet the message size. The true key being incommensurable, while the practical key is a many digit rational approximation.
So what quantum computer can calculate the last digit of square root of two, or of "pi"? There is no free lunch on general problem solving, because of the "halting" problem. But specific classes of problems can be solved in closed form. We will have to wait to see, a practical demonstration that the multiplication of two large prime numbers is easy to factor or not. There are always surprises. To what extent is mathematics actually empirical not analytical?

A challenge ...
One can try to decrypt a known crypt message with a known method but unknown key (Bletchley Park), but how about when you have the plain text and crypt text but not the other two components ...
Please reverse engineer the following plain text plus crypt text for unknown method/key ...
THE0RAIN0IN0SPAIN0FALLS0MAINLY0ON0THE0PLAIN
L5CRAGQIPVPVW9MTD827I6Q1ROVO0C8K51XJ416JXN0
This is a symmetric key block encryption (single key, but encrypt method is inverse of decrypt method). Same number of characters in plain text and crypt text. Same symbol table used for both. One could start with a statistical analysis. However a simple symbol frequency analysis would fail, even with a longer message, because a given plain text symbol isn't converted to the same crypt text symbol, but varies with position in the chain ...
This is generated from a nearly periodic one time pad (one time in that it can be set to recompute via computer clock over whatever short period you need to make a new key. Enigma was changed once per day at midnight. The method can be shown statistically to be as good as AES with message lengths greater than 40 characters (shorter messages are easier to guess, which is to say effective depth is less). For example, a one character message, in a 36 symbol alphabet (as used in this example) can be correctly found with at most 36 guesses, and on average 18 guesses. With 40 character messages, given the nearly random key generation, you need worst case 36^40~1.8x10^62 guesses, or on average half that. Some claim, with favorable problems, a quantum computer at best can achieve the average on every try (which would be quite an accomplishment) but still insufficient to break my 40 character message.

Ah, someone who is more than a recreational mathematician ;)
With circular functions, you have the truly periodic and the almost periodic. The truly periodic corresponds to a pure timbre, the simplest being a single pure tone (sine wave). If the ratio of at least one irregular component among the purely periodic ones, meet the number theory criteria of not being rational, not even algebraic eg square root of two, but is transcendental "pi/e/etc" ... then even after an infinite expansion of digits, the odd component will never align (harmonize). Such is the theory of almostperiodic functions, and how it becomes harder and harder to derive an accurate fourier analysis of the nonharmonics. Such is "percussive" sound, which can be partly harmonic (tympany) or nearly pure noise (true random number).
Consider not a true one time pad, but one that is arbitrarily close to a one time pad. Then the matter of being hard to decrypt is simply a parameter, that one can extend as necessary, as decryption gets better (an arms race). What if you have a key that isn't periodic (has a recurring depth), but is algebraic or even transcendental. In that case you can extend the key system indefinitely, without running into recurrence (which weakens the encryption). You can't as a practical matter use a true nonrational number, one approximates a nonrational number with a close enough rational number eg you use however many digits of "e" as necessary to meet the message size. The true key being incommensurable, while the practical key is a many digit rational approximation.
So what quantum computer can calculate the last digit of square root of two, or of "pi"? There is no free lunch on general problem solving, because of the "halting" problem. But specific classes of problems can be solved in closed form. We will have to wait to see, a practical demonstration that the multiplication of two large prime numbers is easy to factor or not. There are always surprises. To what extent is mathematics actually empirical not analytical?
I'm not a Mathematician. Not even a recreational one.
I think quantum computers will maybe be able a day to break a cryptosystem like this.
We will certainly discover new mathematical algorithms (Maybe it does already exist secret or top secret algorithms owned by some state actors).
But before ask if quantum computers will be able to do so a moment or another.
I think we must use a sufficient encryption process until the time the information we want to kept won't be necessary to be secret.
An info that must to be a secret indefinitely (For example an industrial secret that is the formula of a beverage. In this case encryption would be necessary even if the secret can be partially or fully discovered by reverse engineering with chemical analysis) isn't the same as an info that must be a secret for one week as the fact a secret meeting will take place to reduce the risk of someone spying the meeting.
What was said in this meeting can be fully secret , partially secret or not secret.
If this meeting is about the secret project of companies for the next five years. Information must to be protected for the next five years.
If this meeting is about public and secret projects of companies for the next ten years. Information about public project can be released fully or partially released and info about secret projects must to be protected for the next ten years. The same for nonreleased info on public projects.
If this meeting is about public projects for the next two years. Information can be released fully or partially and nonreleased info must to be protected for the next two years.
Even if a project is public it doesn't means all details are public because some of these need to be secret in some situations.
If a project is secret. The public must not known its existence and therefore the content should not be known to the public.
Few secrets need to be kept indefinitely.
An industrial process can need to be a forever a secret but the fact a video game is being created don't need to be kept indefinitely as the game will be released unless the project is abandoned.
If this ciphertext and plain text is really from you ? I can't find it over Internet. I can suppose you're the author of this.
THE0RAIN0IN0SPAIN0FALLS0MAINLY0ON0THE0PLAIN
L5CRAGQIPVPVW9MTD827I6Q1ROVO0C8K51XJ416JXN0

I hear lemon juice is a useful invisible ink ;)
An idiot tried to rob a bank after putting lemon juice all over his face, thinking that because it could be used to make invisible ink it would also make his face invisible!
Why a bank robber thought covering himself in lemon juice would help him get away with it (https://qz.com/986221/whatknowitallsdontknowortheillusionofcompetence/)
:P :D

Voicjit  yes, that is a string run thru "my" system. Gets harder the longer the message, usually it gets easier to break with length, because you get better statistics. But statistics won't help much. I had a profession hacker try to break it with his best tools (I gave him only the crypt text, nothing else) and he failed ;)
Basically decryption is looking for correlation patterns. If you work against that in particular, you can make a system as hard as you like (but that isn't enough to make it commercially viable). Basically you first run statistics on individual characters, then on doubles, then on triples until you get to the total length of the message. So the first set of stats have N values equal to the total length, and the final stat has just one value. That is looking at letters just sequentially (there are other choices, like every other letter).

Soon we'll have quantum cryptography (https://en.wikipedia.org/wiki/Quantum_cryptography), which will be inherently unbreakable.

Soon we'll have quantum cryptography (https://en.wikipedia.org/wiki/Quantum_cryptography), which will be inherently unbreakable.
Correct. The government will have it, you and I will not. But there probably aren't to many organizations, other than national ones, who need that kind of protection. Corporations and individuals can deal with breakable encryption, just like the old wiretap days of rotary phones.

Correct. The government will have it, you and I will not. But there probably aren't to many organizations, other than national ones, who need that kind of protection. Corporations and individuals can deal with breakable encryption, just like the old wiretap days of rotary phones.
Any "unbreakable" password will be broken. Ita quod umquam fuerit non habet.

Any "unbreakable" password will be broken. Ita quod umquam fuerit non habet.
Not if, but when. Decoding Enigma from 1940, in 1960 is too late. It only has to be sufficiently hard to break. In self education, I have come to understand this principle (known by many others) in my own experience.
Please reverse engineer my challenge or ...

An idiot tried to rob a bank after putting lemon juice all over his face, thinking that because it could be used to make invisible ink it would also make his face invisible!
Why a bank robber thought covering himself in lemon juice would help him get away with it (https://qz.com/986221/whatknowitallsdontknowortheillusionofcompetence/)
:P :D
Is this a joke ?
Voicjit  yes, that is a string run thru "my" system. Gets harder the longer the message, usually it gets easier to break with length, because you get better statistics. But statistics won't help much. I had a profession hacker try to break it with his best tools (I gave him only the crypt text, nothing else) and he failed ;)
Basically decryption is looking for correlation patterns. If you work against that in particular, you can make a system as hard as you like (but that isn't enough to make it commercially viable). Basically you first run statistics on individual characters, then on doubles, then on triples until you get to the total length of the message. So the first set of stats have N values equal to the total length, and the final stat has just one value. That is looking at letters just sequentially (there are other choices, like every other letter).
THE0RAIN0IN0SPAIN0FALLS0MAINLY0ON0THE0PLAIN
L5CRAGQIPVPVW9MTD827I6Q1ROVO0C8K51XJ416JXN0
What can I say about your challenge ? It's important to know what we already know before do a cryptanalysis work because what we know will help us.
1.We know the plaintext (In blue) and the cipher text (In grey).
2.Each of these have 43 characters.
3.We don't know the cryptosystem used but its author say us it's a symmetric block cipher.
4.We don't know the number of blocks used.
5.We don't know the size of a block.
6.We don't know the mode of operation used (CBC , CFB , CCM , CTR , CTS , CWC , EAX , ECB , GCM , IACBC , IAPM , KFB , LRW , OCB , OFB , PCBC , SGCM , TCB , XCBC , XEX , XTS etc...).
7.We don't know the initialization vector.
8.Does the initialization vector use a cryptographic nonce ?
9.Does the cryptosystem use a key schedule ?
10.How deep is avalanche effect ? (Refer to confusion and diffusion)
11.Does it use a keystream ?
12.Does it use a whitening transformation ?
13.Does it use a one waycompression function ?
14.Does it use a one way function ?
15.Does it use a PBox ?
16.Does it use a SBox ?
17.How product cipher is made if there are one ?
18.What is architecture used (Feistel network , Lai–Massey scheme , Substitutionpermutation network etc...) ?
19.We don't know the size of the key but we can suppose it does only use Latin characters or / and numbers. If it does use Latin characters we can suppose there are no case sensitivity.
20.Does this cipher have bijective , injective or subjective functions ?
21.How many rounds used ?
If we find the answer to one or more of these questions we can resolve the problem more easily.
Frequency analysis of the plaintext (Unuseful for cryptanalysis but can help to understand more about how the cipher operate) :
There are 8 numbers and this is still the digit 0.
There are 35 letters but many of these are repetitions.
T = Appear 1 time as the first character and first letter , Appear a second time as the thirtiethfive character and twentiethseven letter
H = Appear 1 time as the second character and second letter , Appear a second time as the thirtiethsix character and twentietheight letter
E = Appear 1 time as the third character and third letter , Appear a second time as the thirtiethsix character and twentiethnine letter
0 = Appear 8 times (Fourth character , Ninth character , Twelfth character , Eighteenth character , twentiethfour character , thirtiethone character , thirteenthfour character , thirteentheight character)
R = Appear 1 time as the fifth character and fourth letter
A = Appear 1 time as the sixth character and fifth letter , Appear a second time as the fifteenth character and twelfth letter , Appear a third time as the twentieth character and sixteenth letter , Appear a fourth time as the twentiethsix character and twentieth letter , Appear a fifth time as the fortiethone character and fortiethtwo letter
I = Appear 1 time as the seventh character and sixth letter , Appear a second time as the tenth character and eight letter , Appear a third time as the sixteenth character and thirteenth letter , Appear a fourth time as the twentiethseven character and twentiethone letter , Appear a fifth time as the fortiethtwo character and fortieththree letter
N = Appear 1 time as the eighth character and seventh letter , Appear a second time as the eleventh character and ninth letter , Appear a third time as the seventeenth character and fourteenth letter , Appear a fourth time as the twentietheight character and twentiethtwo letter , Appear a fifth time as the thirtieththree character and twentiethsix letter , Appear a sixth time as the fortieththree character and fortiethfour letter
S = Appear 1 time as the thirteenth character and tenth letter , Appear a second time as the twentieththree character and nineteenth letter
P = Appear 1 time as the fourteenth character and eleventh letter , Appear a second time as the thirtiethnine character and fortieth letter
F = Appear 1 time as the nineteenth character and fifteenth letter
L = Appear 1 time as the twentiethone character and seventeenth letter , Appear a second time as the twentiethtwo character and eighteenth letter , Appear a third time as the twentiethnine character and twentieththree , Appear a fourth time as the fortieth character and fortieth one letter
M = Appear 1 time as the twentiethfive character and nineteenth letter
Y = Appear 1 time as the thirtieth character and twentiethfour letter
O = Appear 1 time as the thirtiethtwo character and twentiethfive letter
15 characters are used in the plaintext (1 digit and 14 letters) and the characters that appear more often are the next : 0NAILTHESPRFMYO
Frequency analysis of the ciphertext (Unuseful for cryptanalysis but can help to understand more about how the cipher operate) :
There are 9 numbers but many of these are repetitions.
There are 34 letters but many of these are repetitions.
L = Appear 1 time as the first character and first letter
5 = Appear 1 time as the second character and first digit , Appear a second time as the thirtieththree character and tenth digit
C = Appear 1 time as the third character and second letter , Appear a second time as the thirtieth character and twentiethtwo letter
R = Appear 1 time as the fourth character and third letter , Appear a second time as the twentiethfive character and eighteen letter
A = Appear 1 time as the fifth character and fourth letter
G = Appear 1 time as the sixth character and fifth letter
Q = Appear 1 time as the seventh character and sixth letter , Appear a second time as the twentieththree character and seventeenth letter
I = Appear 1 time as the eight character and seventh letter , Appear a second time as the twentiethone character and sixteenth letter
P = Appear 1 time as the ninth character and eight letter , Appear a second time as the eleventh character and tenth letter
V = Appear 1 time as the tenth character and ninth letter , Appear a second time as the twelfth character and eleventh letter , Appear a third time as the twentiethseven character and twentieth letter , Appear a fourth time as the twentiethseven character and twentieth letter
W = Appear 1 time as the thirteenth character and twelfth letter
9 = Appear 1 time as the fourteenth character and second digit
M = Appear 1 time as the fifteenth character and thirteenth letter
T = Appear 1 time as the sixteenth character and fourteenth letter
D = Appear 1 time as the seventeenth character and fifteenth letter
8 = Appear 1 time as the eighteenth character and third digit , Appear a second time as the thirtiethone character and ninth digit
2 = Appear 1 time as the nineteenth character and fourth digit
7 = Appear 1 time as the twentieth character and fifth digit
6 = Appear 1 time as the twentiethtwo character and sixth digit , Appear a second time as the thirtiethnine character and fourteenth digit
1 = Appear 1 time as the twentiethfour character and seventh digit , Appear a second time as the thirtiethfour character and eleventh digit , Appear a third time as the thirtietheight character and thirteenth digit
O = Appear 1 time as the twentiethsix character and nineteenth letter , Appear a second time as the twentietheight character and twentiethone letter
0 = Appear 1 time as the twentiethnine character and eight digit , Appear a second time as the fortieththree character and fifteenth digit
K = Appear 1 time as the thirtiethtwo character and twentieththree letter
X = Appear 1 time as the thirtiethfive character and twentiethfour letter , Appear a second time as the fortiethone character and twentiethsix letter
J = Appear a second time as the thirtiethsix character and twentiethfive letter , Appear a second time as the fortieth character and twentiethfive letter
4 = Appear 1 time as the thirtiethseven character and twelfth digit
N = Appear 1 time as the fortiethtwo character and twentiethseven letter
27 characters are used in the ciphertext (9 digits and 18 letters) and the characters that appear more often are the next : V15CRQIP86O0XJLAGW9MTD27K4N
Who can say if my frequency analysis of the plaintext and ciphertext is accurate ?

It would appear we have a professional in our midst!!
There are exactly 36 possible characters, 09 and AZ
The arithmetic is modular (of course). Aside from the key itself, it is mod36 (of course).
The key (almost pure random) is not mod36. That would create unnecessary "depth" via aliasing.
The key is muxed with the message, by base36 addition, but dynamically, not statically.
There is in addition, when converting character back and forth with base36 … a constant relettering vector (sticker per Enigma) for all 36 characters.
The initial offset is simply the effect of adding the first key character to the first message character, there is no other offset.

It would appear we have a professional in our midst!!
There are exactly 36 possible characters, 09 and AZ
The arithmetic is modular (of course). Aside from the key itself, it is mod36 (of course).
The key (almost pure random) is not mod36. That would create unnecessary "depth" via aliasing.
The key is muxed with the message, by base36 addition, but dynamically, not statically.
There is in addition, when converting character back and forth with base36 … a constant relettering vector (sticker per Enigma) for all 36 characters.
The initial offset is simply the effect of adding the first key character to the first message character, there is no other offset.
Why do you say I'm a professional ?
I know only a few things in crypto. I don't know the mathematical details in each concept.

Why do you say I'm a professional ?
I know only a few things in crypto. I don't know the mathematical details in each concept.
Take it as praise.