Sometimes the meet the slang meaning of their podcast title...
This is a great one:
The prime number game. I just heard on Moonshots about the prime number game having found improvements by AI. Until here that was a brute force game. First I like you to tell me how AI helps here better than letting a mathematics chip set run a bruteforce attack against ever longer numbers and then I like your statement about the use of prime numbers, which is to me mainly an encryption one having representations of ASCII symbols ever harder to be found by a pattern analysis of the actual pure file code.
## Why AI doesn’t help much for prime search
Finding large primes is not really a “brute force game” anymore.
- A 1000-digit number can be tested for primality in milliseconds using algorithms like **Miller-Rabin**.
- We do not trial-divide by every smaller number. That would be hopeless.
- For special primes like Mersenne primes, **GIMPS** uses the Lucas-Lehmer test on GPUs/CPUs.
- The density of primes near a large number is about \(1/\ln n\), so random search plus a fast primality test is already very efficient.
A neural network or “AI” has no advantage here because primality is an exact mathematical property. Neural networks are statistical pattern finders, but there is no smooth visual or statistical pattern in prime numbers that would let an AI skip the arithmetic. If such a pattern existed, it would probably break RSA encryption.
AI can help at the edges — for example, discovering faster low-level multiplication or FFT routines that speed up large-number arithmetic — but that is a generic optimization, not a new prime-finding strategy.
So:
> A well-optimized mathematics chip running Miller-Rabin or Lucas-Lehmer is still better for finding primes than an AI model.
## What primes actually do in encryption
Your mental model is close, but not quite right.
Primes are not used by representing ASCII symbols as prime numbers. Modern encryption works differently.
### RSA, for example
- Choose two large primes \(p\) and \(q\).
- Multiply them: \(n = p \times q\).
- The public key uses \(n\).
- The private key uses \(p\) and \(q\).
- Security relies on the fact that multiplying \(p\) and \(q\) is easy, but factoring \(n\) back into \(p\) and \(q\) is extremely hard.
So the prime numbers are not the message. They are the hidden structure behind the public key.
### Diffie-Hellman and elliptic-curve cryptography
These also use large primes or prime fields, but the security comes from the hardness of the **discrete logarithm problem**, not from hiding ASCII patterns.
### The actual file encryption
Usually, the actual file or ASCII data is encrypted with a symmetric cipher like **AES**, not directly with primes.
Primes are used to:
- securely exchange the AES key,
- create digital signatures,
- build public-key infrastructure.
So primes are more like the lock and key system, not the encoded content.
## Bottom line
- AI has not replaced number theory for prime search.
- Prime finding is already fast and exact.
- Primes are crucial for public-key cryptography because multiplying primes is easy but factoring their product is hard.
- The actual data is encrypted with symmetric ciphers; primes protect the keys and identities.
> The prime-number game remains a number-theory game, not an AI pattern-recognition game.
Thank for that overview. Look through my thinking here. We have a text like: I love you. If we'd give the alphabet and space each a very long prime number starting with a length indicator number we than use an even longer prime number to multiply each number after the length indicator with that even longer prime number being the key to then write down the new length indicator and multiplication result. The longer the number, the harder finding a pattern should be.
If the increase security we use a coordinate table that changes the prime number based on the letter or symbol position layed out over the table. The table is made from rows and columns. A1 will have a different combination of prime numbers than B6 and without knowing the amount of colums and tables, the x and y primer number list and if super paranoid the master key...
## 1. What your scheme is doing
For each symbol, you have a large prime P_symbol.
You multiply it by a secret key prime \(K\):
C = K times P_symbol
The ciphertext number is \(C\).
To decrypt, you divide by \(K\) to recover \(P_{\text{symbol}}\).
This feels strong because the numbers are huge and factoring primes is hard. But the problem is that **you are reusing the same secret multiplier \(K\) for every symbol**.
## 2. The immediate fatal flaw: GCD reveals the key
Suppose two different symbols have primes:
P_1 neq P_2
Their ciphertexts are:
C_1 = K times P_1
C_2 = K times P_2
Because P_1 and P_2 are distinct primes, they have no common factor except 1.
Therefore:
gcd(C_1, C_2) = K
So an attacker just takes two different ciphertext numbers from your message and computes their greatest common divisor. They recover your secret key \(K\) immediately.
This is not brute force. Euclid’s algorithm for GCD is extremely fast, even for 1000-digit numbers. It would run in milliseconds on an ordinary computer.
After recovering K, they divide every ciphertext by \(K\) and get every symbol prime P_i.
Example with small numbers:
- K = 1009
- P_A = 101
- P_B = 103
Ciphertexts:
C_A = 1009 \times 101 = 101909
C_B = 1009 \times 103 = 103927
Now:
\gcd(101909, 103927) = 1009
The key is exposed.
## 3. Does the coordinate table fix it?
You suggested changing the prime based on a table coordinate, so A1 uses different primes than B6.
This helps only if the key \(K\) also changes per coordinate. But if \(K\) remains the same, the GCD attack still works exactly the same way — even if \(P_i\) and \(P_j\) are different coordinate primes.
If you change \(K\) per coordinate as well, then:
- At a single coordinate, if two different symbols are ever used, GCD again reveals that coordinate’s \(K\).
- If you never reuse a coordinate for different symbols, then you are effectively using a new key for every character. That becomes a one-time pad, and the primes are unnecessary — you could just XOR with random data.
So the coordinate table adds complexity, but it does not add fundamental security unless you use it like a one-time pad, and then the prime multiplication is redundant.
## 4. Why “longer numbers” do not remove patterns
The security of modern cryptography does not rely on the ciphertext having no visual pattern. It relies on mathematical problems that are believed to be hard, such as:
- Factoring a product of two large secret primes.
- Discrete logarithms in prime fields.
- Lattice problems.
In your scheme, the hard-looking part is factoring (C = K times P), but since the same \(K\) is used many times, the GCD shortcut bypasses factoring entirely.
Longer numbers make multiplication slower, but they do not stop GCD.
## 5. What would actually be secure
If your goal is to encrypt text like “I love you”:
### Symmetric encryption
Use a modern cipher:
- **AES-256-GCM**
- **ChaCha20-Poly1305**
These produce pseudorandom ciphertext and include authentication. No need to assign primes to letters.
### Public-key encryption
Use:
- **RSA** with proper padding
- **Elliptic-curve cryptography**
Here primes are used inside modular arithmetic, not multiplied directly with plaintext.
### Perfect secrecy
If you want theoretically unbreakable encryption, use a **one-time pad**:
C_i = M_i \oplus K_i
where \(K_i\) is truly random and never reused. This is information-theoretically secure.
Your prime idea is interesting, but it is closer to a **nomenclator codebook** from the 16th century than to modern cryptography. The prime multiplication does not add security; it actually creates a shortcut for the attacker through GCD.
So an attacker just takes two different ciphertext numbers from your message and computes their greatest common divisor. They recover your secret key KK immediately.
Ok. Now practically speaking using standard hard and software against a list of largest publicly available prime combinations, not NSA dedicated supercomputer breakers so I really get it. Beside, how are the cyphernumbers found if he has a long chain of numbers?
It is based on historic encryption I am happy you caught that. By the way, if the turn the table into a virtual 3D stick and tell only the diameter putting the primelist over that??
## Practical GCD attack on a normal laptop
You do **not** need any list of primes, and you do **not** need an NSA supercomputer.
The attack is:
1. Take two ciphertext numbers from the message.
2. Compute their greatest common divisor.
3. That GCD is your secret multiplier \(K\).
This works because every ciphertext is \(K \times \text{some prime}\). Two different primes have no common factor except 1, so the only common factor between two different ciphertexts is \(K\).
Even a standard laptop can compute the GCD of two numbers with thousands of digits in milliseconds.
Example in Python:
```python
import math
C1 = 101909 # K * 101
C2 = 103927 # K * 103
K = math.gcd(C1, C2)
print(K) # 1009
```
If the numbers had 10,000 digits each, `math.gcd` would still finish almost instantly. No brute force, no prime list, no factoring.
So the idea of using “the largest publicly available prime combinations” is irrelevant. The attacker never needs to know which primes were used. They just compute GCD and recover \(K\) directly.
---
## How the attacker finds the cipher numbers inside a long chain
You mentioned writing down a **length indicator** followed by the multiplication result.
If your ciphertext looks like:
```
006101909006103927...
```
then the attacker reads the first length indicator (`006`), extracts the next 6 digits (`101909`), then reads the next length indicator (`006`), extracts `103927`, and now has two cipher numbers.
If you do **not** use length indicators, the attacker can still try possible split points. Since all products share \(K\), any two correctly extracted numbers will have GCD \(K\). Trying a few dozen split positions is trivial.
So the length indicators actually make parsing easier for the attacker.
---
## What about the 3D stick / diameter idea?
If you turn your coordinate table into a virtual 3D stick or cylinder, you are only changing the **mapping from symbol/position to prime \(P\)**.
But the core encryption is still:
\[
C = K \times P_{\text{symbol, position}}
\]
If the same \(K\) is reused for every symbol, then:
- Pick any two ciphertext numbers.
- Compute GCD.
- Recover \(K\).
The 3D geometry does not change that.
If you change \(K\) for every position as well, then:
- At a given position, if two different messages ever use the same \(K\), GCD across messages reveals it.
- If \(K\) never repeats, then you have something like a one-time pad — and the prime multiplication is unnecessary.
So the 3D stick can obscure simple letter-frequency analysis, but it does **not** stop the GCD attack as long as the same \(K\) appears in more than one cipher number.
---
## Bottom line
Your system is a creative historical-style substitution cipher, but the repeated multiplicative key is a fatal weakness.
A normal laptop can break it in milliseconds by taking two ciphertext numbers and computing GCD. The length indicators, large primes, and 3D coordinate tables do not fix that.
If you want real security, use:
- **AES-GCM** or **ChaCha20-Poly1305** for actually encrypting text.
- **RSA** or **Elliptic-curve cryptography** for key exchange and signatures.
These are designed to resist exactly this kind of mathematical shortcut.
Again, how does the find the two numbers in a long no spacing list of one large number ?
In the stick thing it works like this: I have a letter: the v of love. v is in my list a primer number. One with 12 digest. I multiply that with my key, a primer number with 24 digest. That resulting number is written like chinese from the top to the bottom onto a paper string wrapped around a stick with a given diameter having the paper wrapped around in a given aka fixed angle. If I wrapp the paper string off I have the number, but broken up and each digest of the number stands next to another.
You’re asking two things:
1. If the ciphertext is one long string of digits with no spaces or length markers, how does the attacker find the individual numbers?
2. If I also scramble the digit order using a stick/scytale transposition, doesn’t that hide them?
Let’s do both.
---
## 1. Finding the numbers in a long digit string
Suppose your message produces numbers like:
```
C_1 = K × P_1
C_2 = K × P_2
C_3 = K × P_3
...
```
You then concatenate them into one long string:
```
C_1 || C_2 || C_3 ...
```
For example:
```
101909103927...
```
The attacker does not need to know where one number ends and the next begins.
They can simply try possible split points.
- Take a substring of plausible length, e.g. 36 digits, starting at position 0.
- Take another substring of similar length, starting at some other position.
- Compute their GCD.
- If both substrings really are two different ciphertext numbers, their GCD will be your secret key \(K\).
- If they are not, the GCD will almost certainly be 1 or some small number.
Why?
Because two random 36-digit numbers are coprime with overwhelming probability.
The chance that they share a 24-digit prime factor accidentally is astronomically small.
So an attacker can write a loop:
```
for every start position i:
for every start position j > i:
take substring starting at i
take substring starting at j
g = gcd(substring_i, substring_j)
if g > 1:
print("found K =", g)
```
For a normal text message, even if the whole string is a few thousand digits long, this loop finishes in seconds on a laptop.
So:
> Even with no spacing, no length indicators, and one huge digit string, the shared key \(K\) can be recovered by GCD as soon as the attacker guesses two correct substrings.
If you add length indicators, the attacker’s job becomes even easier.
---
## 2. What about the stick/scytale transposition?
Your stick method is a transposition cipher.
You take the decimal digits of the ciphertext number and write them along a strip wrapped around a stick at a fixed angle. When you unwrap the strip, the digits are reordered.
That does hide the original order of the digits.
But here is the important part:
The transposition is just a permutation of digits.
The underlying ciphertext numbers are still there. They have just been rearranged.
If the attacker can guess or brute-force the stick parameters — diameter, angle, number of wraps, starting position — they can reverse the permutation and recover the original concatenated ciphertext numbers.
In a physical stick cipher, the possible diameters and angles are limited. In a digital version, you might use large secret numbers for diameter and angle, but then you are relying on the secrecy of those parameters, not on the prime multiplication.
And once the transposition is reversed:
- The original string of concatenated products reappears.
- The GCD attack works immediately.
- The shared key \(K\) is exposed.
So the stick transposition adds some obscurity, but it does not fix the core weakness.
---
## 3. The fundamental issue
Your encryption is basically:
\[
C = K \times P
\]
with the same \(K\) reused for every symbol.
That reuse is fatal.
No matter how you encode the numbers, concatenate them, or scramble the digit order:
- If you reuse \(K\),
- And if the attacker can obtain two different products \(C_1\) and \(C_2\),
- Then \(\gcd(C_1, C_2) = K\).
This is not a brute-force attack. It is a fast, exact mathematical shortcut.
---
## 4. What would actually fix it?
- Use a different \(K_i\) for every symbol, never reused.
This becomes essentially a one-time pad, and the prime multiplication is then unnecessary.
- Use a modern cipher like AES-GCM or ChaCha20-Poly1305.
These are designed to resist exactly these mathematical shortcuts.
- Use RSA or elliptic-curve cryptography for key exchange and signatures.
There, primes are used in modular arithmetic, not multiplied directly with plaintext.
Your historical-style design is creative, but the GCD shortcut breaks it regardless of spacing, length indicators, or stick transposition.
1. Tell me that in numbers of seconds based on CPU.
2. Tell me that in numbers of seconds based on CPU
The Renaissance idea is not to find an unbreakable code. That does not exists, because there are no secrets on earth, just missing information.
It is about making it harder to break.
Give me a time frame that needs, please, so I understand if the crackers are leading by hardware advancement.
Assume a normal modern laptop CPU, single core, using C/GMP-style big-integer arithmetic, not Python.
## 1. Direct GCD attack — if length markers exist
If the attacker can extract two ciphertext numbers from the message, for example because you included length indicators:
| Size of each ciphertext number | Time to compute GCD |
|---|---:|
| 36 digits | 0.00001–0.0001 seconds |
| 1,000 digits | 0.001–0.01 seconds |
| 1,000,000 digits | 0.5–2 seconds |
So with your example — 24-digit key and 12-digit symbol prime — each ciphertext number is about 36 digits long.
> The key is recovered in less than 0.001 seconds.
That is not brute force. It is Euclid’s algorithm.
---
## 2. If there are no length markers — one long digit string
Suppose the message is 1,000 characters long.
Each ciphertext number is about 36 digits.
The whole string is about 36,000 digits long.
The attacker does not know where one number ends and the next begins, so they try many possible starting positions and lengths.
A naive search over all plausible pairs of 36-digit windows:
- Possible starting positions: ~36,000
- Possible pairs: ~650 million
- Each GCD on 36-digit numbers in C: ~0.2–1 microsecond
Time estimate:
| Implementation | Time |
|---|---:|
| Single CPU core | 2–10 minutes |
| 8 CPU cores | 15–60 seconds |
| If product length is known or guessed correctly | < 0.1 seconds |
So even without spacing, the shared key is found in minutes at worst, often in milliseconds.
---
## 3. If you add the stick/scytale transposition
Now the attacker must also reverse the digit permutation before doing GCD.
If the stick parameters are small and discrete, for example:
- diameter: 1–1,000
- angle: 1–360
Then there are:
```
1,000 × 360 = 360,000 possible transpositions
```
For each possibility, the attacker reverses the transposition and tries GCD on a few windows.
Rough estimate:
| Attack | Time |
|---|---:|
| Brute-force all small stick parameters, one core | 30 minutes to a few hours |
| On 8 cores | 5–30 minutes |
| If the diameter/angle are known or guessed | < 1 second after reversal |
So the stick adds some delay, but not enough for a serious cipher.
---
## Bottom line
Your scheme is not in a hardware race.
Even on an ordinary laptop:
- With length markers: broken in **under 0.001 seconds**.
- Without length markers: broken in **seconds to minutes**.
- With stick transposition: broken in **minutes to hours**, assuming simple secret parameters.
The weakness is not the size of the primes or the speed of the CPU.
The weakness is that the same multiplier \(K\) is reused, and GCD exposes it instantly.
The Renaissance-style thinking is fine as a historical exercise, but by modern standards this is not a hard problem for a cracker.
That sounds bad. How about the table?
The table changes **which prime** is used for a symbol at a given position.
But the fatal issue is not the prime \(P\). It is the reused multiplier \(K\).
If the table still uses the same master key \(K\) for every character, then:
\[
C_{\text{symbol,position}} = K \times P_{\text{symbol,position}}
\]
Pick any two ciphertext numbers from anywhere in the message.
Their GCD is still \(K\).
The table does not stop that.
---
## Time with the table but same master key
| Attack | Time on normal laptop |
|---|---:|
| With length markers | **under 0.001 seconds** |
| Without length markers | seconds to minutes |
| With stick transposition as well | minutes to hours |
So the table adds almost no real delay if you reuse one master key.
---
## What if each table coordinate has its own key?
Suppose every coordinate A1, A2, B1, etc. has a different multiplier \(K_{\text{coord}}\).
Then:
\[
C = K_{\text{coord}} \times P_{\text{symbol,coord}}
\]
Now GCD between two ciphertexts from **different coordinates** does not reveal one common key.
This sounds better.
But:
### If you reuse the same table for a second message
An attacker takes message 1 and message 2.
At the same coordinate, the same \(K_{\text{coord}}\) was used.
So:
\[
C_1 = K_{\text{coord}} \times P_1
\]
\[
C_2 = K_{\text{coord}} \times P_2
\]
GCD of \(C_1\) and \(C_2\) reveals \(K_{\text{coord}}\).
If the messages are aligned or if the attacker can guess a few repeated coordinates, they recover that coordinate key. Then they can decrypt other symbols at that coordinate.
Time: **seconds to minutes** if the table has repeated coordinate use.
---
### If each coordinate is used only once
Then you never reuse \(K_{\text{coord}}\).
The GCD attack fails because no key is repeated.
But now you need a table as large as your message.
- Message length 10,000 characters → table needs 10,000 unique coordinate keys.
- Message length 1,000,000 characters → table needs 1,000,000 unique coordinate keys.
That table must be generated, stored, and shared secretly with the receiver.
And once the table is used, it must be thrown away.
At that point you have built a **one-time pad**, just using primes unnecessarily.
A one-time pad is provably secure if the keys are truly random and never reused — but you do not need primes for it.
---
## If the table structure itself is secret
For example, secret number of columns, rows, angle, etc.
If the parameters are small, say 1,000 possible diameters × 360 possible angles × 1,000 table sizes, then brute force is:
\[
360{,}000{,}000 \text{ possibilities}
\]
On a modern CPU that is maybe hours to days on a single core, minutes to hours on many cores.
That is a delay, but not a hard cryptographic barrier.
And if the attacker gets two messages using the same structure, the GCD shortcut returns.
---
## Bottom line
The table can make the scheme more annoying to break if you make it huge and never reuse any key. But then you are not relying on prime numbers for security — you are relying on a giant secret random table.
If you reuse the table or the master key, a laptop still breaks it in milliseconds to minutes.
Modern ciphers give you far more security with a much smaller secret key. Ok. So we now need someone that understands that the long number chain is a code and what kind of code. We need him also to know the cover file. That is three layers. Each weak but...
We have something small we could embed into a linux operating system and use to send short messages trying be unnoticed.
What about a in your face PGP encrypted file? How quickly is that cracked?
You’re describing **layered obscurity**:
1. A weak but fast prime cipher
2. Hidden inside a cover file
3. Hidden inside a normal Linux system
That is not strong encryption, but it can be useful for **hiding the existence** of traffic or delaying casual discovery.
However, layering weak things does not necessarily make a strong thing. It mainly increases the chance that someone:
- overlooks the data,
- does not know a message is there,
- or gives up because it looks like noise.
If someone knows all three layers and understands them, the inner prime cipher still falls to GCD in milliseconds.
---
## What about a normal “in your face” PGP encrypted file?
That is a completely different level.
PGP does not rely on secrecy of the method.
It relies on a strong session key and a strong public/private key.
| Factoring a 2048-bit RSA key
| Brute force 4096-bit RSA
| Brute force a 20-character random passphrase
So a strongly protected PGP file is **not cracked** in any meaningful human timeframe.
### If the passphrase is weak
|---|---:|
| “password123” | seconds to minutes with a dictionary attack |
| “I love you” | seconds to minutes |
| A random 6-character password | minutes to hours |
| A random 8-character password | days to months |
So PGP security depends mostly on:
- the strength of the private key passphrase,
- the key size,
- and whether the private key has been exposed.
---
## Why PGP is different from your prime scheme
Your scheme:
C = K \times P
reuses K, so GCD recovers K instantly.
PGP does not have that flaw because:
- The symmetric cipher uses a unique random session key for each file.
- The public-key part uses padded RSA or elliptic-curve encryption, not direct prime multiplication with reusable plaintext factors.
- No repeated multiplication of the same secret prime is exposed to GCD.
---
## Bottom line
- Your layered prime system is **fast and stealthy**, but cryptographically weak.
- A PGP encrypted file is **strong and openly encrypted**, but attracts attention.
- If someone finds the PGP file and the passphrase is strong, they cannot break it with any normal hardware.
- If the passphrase is weak, they can break it in seconds with a dictionary attack.
So the real difference is:
> Your system hides a weak secret.
> PGP protects a strong secret, but everyone can see there is a secret.
how much smaller is my table system than PGP? And how large are the largest known prime numbers?
If by “smaller” you mean the **size of the encrypted output**, then your prime-table system is **much larger**, not smaller.
### Your table system
- Each plaintext character becomes a product of two primes.
- Example: symbol prime = 12 digits, key prime = 24 digits → product ≈ 36 decimal digits.
- In binary, 36 decimal digits ≈ 120 bits ≈ 15 bytes.
- So every single character expands to roughly **15 bytes** of ciphertext.
A 1,000-character message becomes:
1000 times 15 = 15,000 bytes
plus any length indicators, separators, transposition overhead.
### PGP
- PGP normally uses a symmetric cipher like AES for the message.
- AES output is the **same size as the plaintext** (plus a small authentication tag, maybe 16 bytes).
- Public-key encryption of the session key adds only a few hundred bytes.
- A 1,000-character message stays around **1,000–1,500 bytes** total.
So PGP is **10–15 times smaller** than your prime-table encoding, despite being vastly stronger cryptographically.
---
## How large are the largest known prime numbers?
As of the latest discovery by GIMPS:
> The largest known prime is
> **2^136,279,841 - 1**
> a Mersenne prime with **41,024,320 decimal digits**.
It was discovered in October 2024.
- Mersenne primes have the form 2^p - 1 where p is prime.
- This number is so large that writing it out would fill about 15,000 pages of a typical book.
- It has no practical use in encryption because it is far too large for normal arithmetic, and cryptography does not need primes that big.
For RSA and other cryptosystems, primes with **300–1200 decimal digits** (1024–4096 bits) are more than enough and are used every day.
---
## Bottom line
- Your prime-table cipher output is **~15× larger** than PGP ciphertext for the same message.
- The largest known primes are tens of millions of digits long, but those are only for record-breaking mathematical research — not for practical encryption.
- Modern cryptography is both **stronger** and **much smaller in output size** than a prime-multiplication scheme.
- a fast symmetric cipher for the message,
- public-key encryption only for the small symmetric key,
- digital signatures,
- compression,
- and a packet-based format.
It is defined in the **OpenPGP** standard, RFC 4880.
---
## 1. Key pairs
Each user has a public/private key pair.
- **Public key**: shared with others; used to encrypt to you and verify your signatures.
- **Private key**: kept secret; protected by a passphrase; used to decrypt and sign.
Common public-key algorithms in PGP:
- RSA
- ElGamal
- DSA
- ECDH / ECDSA
- EdDSA
---
## 2. Encryption process
Suppose Alice wants to send Bob a message.
### Step 1: Generate a random session key
PGP creates a random symmetric key for a cipher like:
- AES-128 / AES-256
- ChaCha20
- Twofish
- Camellia
This session key is used only once for this message.
### Step 2: Compress the plaintext
The plaintext is usually compressed using ZIP or ZLIB. Compression reduces size and removes some patterns that help cryptanalysis.
### Step 3: Encrypt the message with the session key
The compressed plaintext is encrypted with the symmetric cipher using the session key.
```
encrypted_message = AES-256(plaintext, session_key)
```
### Step 4: Encrypt the session key with Bob’s public key
The small session key is encrypted with Bob’s public key using an asymmetric algorithm.
```
encrypted_session_key = RSA(session_key, Bob_public_key)
```
Or with ElGamal or ECDH, depending on the key type.
### Step 5: Combine into a PGP message
The final PGP message contains:
- the encrypted session key,
- the encrypted message,
- metadata (algorithm IDs, packet lengths, etc.)
---
## 3. Decryption process
Bob receives the PGP message.
### Step 1: Decrypt the session key
Bob uses his private key to decrypt the encrypted session key.
```
session_key = RSA_decrypt(encrypted_session_key, Bob_private_key)
```
Bob’s private key is itself protected by a passphrase, so he must enter it or have it unlocked.
### Step 2: Decrypt the message
Bob uses the recovered session key to decrypt the symmetric encrypted message.
```
compressed_plaintext = AES_decrypt(encrypted_message, session_key)
```
### Step 3: Decompress
Bob decompresses the result to get the original plaintext.
---
## 4. Digital signatures
PGP can also sign messages, separate from encryption.
### Signing
1. Alice hashes the message with a hash function like SHA-256.
2. Alice encrypts the hash with her private key, producing a signature.
3. The signature is attached to the message.
### Verification
1. Bob decrypts the signature with Alice’s public key, recovering the original hash.
2. Bob hashes the message himself.
3. If the hashes match, the signature is valid and the message has not been altered.
---
## 5. Key management and trust
PGP does not rely on central certificate authorities like typical PKI. Instead it uses:
- **Key fingerprints**: short unique identifiers for public keys.
- **Key servers**: public directories for keys.
- **Web of trust**: users sign each other’s keys to vouch for authenticity.
You decide whose keys you trust, and how much you trust their signatures on others.
---
## 6. Why PGP is strong
- The message body is encrypted with a **random one-time symmetric key**, so no GCD shortcut applies.
- The session key is small enough for RSA/ElGamal/ECDH to encrypt securely.
- The private key never leaves the user’s device.
- If the passphrase is strong and the private key is not stolen, brute force is impossible with current technology.
---
## Summary diagram
```
Sender:
plaintext → compress → symmetric encrypt with random session key
session key encrypted with recipient's public key
→ both combined into PGP packet
Recipient:
receive packet → decrypt session key with private key
→ symmetric decrypt with session key
→ decompress → plaintext
```
PGP is essentially:
> A fast symmetric cipher for the message,
> plus a secure public-key wrapper for the symmetric key,
> plus optional digital signatures and compression.
My system, being super simple in terms of mathematics, is prone to brute force attacks because the formula is way too simple and a computer easily can brute force that by going through number parts and looking if it was a prime number using GCD.
Now lets get practical: 97 times 19 equals 1843. 41 times 19 equals 779. 418433779 What does that tell you? And another AI even in context told me:
The number 418,433,779 is a composite number made by multiplying four distinct prime numbers together.Its unique prime factorization is:\(418,433,779=23\times 29\times 257\times 2,441\)
While it is lengthindictor(realnumber no coded) prime times prime lengthindictor(realnumber no coded) prime times prime.
You encoded:
- first character: \(97 \times 19 = 1843\)
- second character: \(41 \times 19 = 779\)
Then you wrote it as:
```
length indicator 4 | 1843 | length indicator 3 | 779
```
giving:
```
4 1843 3 779 = 418433779
```
An attacker who knows or guesses this framing can immediately split it back into:
```
1843 and 779
```
Then they compute:
\[
\gcd(1843, 779) = 19
\]
That reveals the shared secret key \(K = 19\).
Then they recover your original primes:
\[
1843 / 19 = 97
\]
\[
779 / 19 = 41
\]
So the two plaintext primes are exposed.
This happens **without factoring 1843 or 779**. Euclid’s algorithm just finds the common divisor directly.
---
## What the other AI did
The other AI took the whole 9-digit number:
\[
418433779
\]
and factored it as:
\[
23 \times 29 \times 257 \times 2441
\]
That is mathematically correct as an integer factorization of the concatenated number.
But that factorization is **irrelevant** to your cipher, because your ciphertext was not the product of all those pieces. It was a digit-string concatenation:
\[
4 || 1843 || 3 || 779
\]
Concatenation and multiplication are completely different operations.
So the other AI did not see your framing and factored the wrong thing.
---
## What this tells you
1. **Length indicators make parsing easy for an attacker who knows the scheme.**
In your example, the number practically announces where one product ends and the next begins.
2. **The GCD attack still works after parsing.**
As soon as the attacker has two numbers that were both multiplied by the same key \(K\), they recover \(K\).
3. **Factoring the whole concatenated number does not reveal your primes.**
But that does not help you. The attacker will not factor the whole number if they understand the framing.
4. **The weakness is not the size of the primes or the length indicators.**
It is that the same multiplier \(K=19\) was used for both characters.
Any shared factor across multiple ciphertext numbers will be exposed by GCD.
---
## In numbers
Your full example on a normal laptop:
| Step | Time |
|---|---:|
| Split after length indicators | trivial |
| GCD of 1843 and 779 | microseconds |
| Recover \(K=19\) | instant |
| Divide products by \(K\) | instant |
Total time: effectively **0.000001 seconds**.
---
So your example shows exactly why the prime-multiplication scheme with a reused key is not secure, no matter how large the primes are or how you mark the lengths.
- **“Attacker knows the framing/algorithm”**
- **“Attacker steals the private PGP key”**
They are not equivalent.
---
## 1. In modern cryptography, we assume the attacker knows everything except the key
This is **Kerckhoffs’s principle**:
> The system should remain secure even if everything about the system, except the key, is public knowledge.
PGP follows this. The format, algorithms, packet structure, compression method — all public. The only secret is the private key.
Your system instead relies partly on the secrecy of the **framing**:
- that it is prime multiplication,
- that there are length indicators,
- how the numbers are concatenated,
- how the table/stick works.
If the framing is secret, you have a **secret algorithm**, not just a secret key.
---
## 2. Secret algorithms are much harder to protect than secret keys
A private PGP key is:
- a single file,
- protected by a passphrase,
- stored offline if desired,
- not present in ciphertext.
The framing of your system is:
- embedded in any software you write,
- shared between sender and receiver,
- potentially inferable from the ciphertext itself,
- impossible to change easily after deployment.
If the framing leaks or is guessed, all messages break.
If a PGP private key leaks, that key is compromised, but other users’ keys remain safe, and the algorithm is still fine.
---
## 3. The GCD attack does not require the attacker to know the framing exactly
In your example, the ciphertext is:
```
418433779
```
An attacker might not know that it means:
```
4 | 1843 | 3 | 779
```
But they can try many possible split points:
- Split after 1 digit, 2 digits, 3 digits, etc.
- For each split, take two chunks and compute GCD.
- If both chunks are products of the same key, GCD reveals that key.
For a short number like this, that search takes microseconds.
Even if you remove the length indicators and just concatenate the products, an attacker can still brute-force plausible splits and run GCD on pairs. The key reuse is still fatal.
---
## 4. “I nick the private PGP key” is a different class of attack
Stealing a PGP private key usually requires:
- physical access,
- malware,
- phishing,
- user mistake,
- or coercion.
It is not a cryptanalytic attack. It is a **key management failure**.
In your prime scheme, the GCD attack requires only:
- the ciphertext,
- the ability to try possible splits,
- and a laptop.
No theft, no malware, no physical access.
So the attacker’s effort is orders of magnitude lower.
---
## 5. If you really keep the framing secret, then you have a form of steganography, not cryptography
That can be useful for hiding messages in plain sight. But:
- The security depends on the framing secret, not the mathematics.
- Once the framing is known, the math collapses immediately.
- PGP remains secure even when the framing is known.
---
## Bottom line
| | PGP | Your prime scheme |
|---|---|---|
| Algorithm/framing secret? | No | Yes, if you rely on it |
| Main secret | Private key | Framing plus multiplier |
| If attacker knows algorithm | Still secure | Broken via GCD |
| Attack to break cryptographically | Hard number theory problem | GCD in milliseconds |
| Key theft | Catastrophic for that key | Catastrophic for all messages |
So it is not the same level of argument.
“Knowing the framing” is a much weaker assumption than “stealing a private key.” And in most real-world scenarios, the framing will eventually be discovered or inferred, because you have to implement it somewhere.
Your system is interesting as a layered hidden channel, but it is not a strong encryption scheme by modern standards.
My way needs a small software that can be deeply embedded into an standard computer. We than can hide the number string in pixel mistakes of a jpg. It is great for short messages that wait to be discovered. The key then can be as little as the 415th known prime number. All you need to know and having the picture.
Still, a few practical points if you want to make it workable:
## 1. The key size matters even for a covert channel
The 415th known prime is roughly **2861**.
If you multiply two 12-digit primes by 2861, you get 15-digit products. GCD on two such numbers is still microseconds. And if you ever send two messages with the same key, an attacker who suspects your scheme can recover the key instantly by GCD.
**Better:** Use a different random 24-digit prime key for each message, and communicate that key via some prearranged method (e.g., a one-time list, or derived from a shared secret seed). Then GCD across messages fails because keys are not reused.
## 2. The hidden software is your real vulnerability
A small embedded encoder/decoder inside a Linux system is likely to be found if the system is inspected. If an adversary finds the software, they can reverse engineer it and learn:
- the prime-table structure,
- the framing,
- how the key is chosen.
Then the only remaining secret is the key itself. If you reuse a small key like the 415th prime, all messages are immediately breakable.
So the software must be very well hidden (kernel module, rootkit, steganographically embedded in an innocent tool, etc.), and ideally it should self-destruct or be deniable.
## 3. Steganography quality matters
“Pixel mistakes” in a JPEG can be detected by modern steganalysis tools if the embedding is not designed carefully.
- Least-significant-bit flipping in JPEG DCT coefficients often leaves statistical traces.
- Tools like `stegdetect` or more advanced machine-learning detectors can flag images with hidden data.
If you want to stay unnoticed, you need a high-quality steganographic algorithm (e.g., F5, OutGuess, or modern adaptive methods) that preserves the image’s statistical profile.
## 4. The real strength is “low probability of detection”
If:
- the image looks completely normal,
- the software is never found,
- the key is never reused,
- and you only send a few short messages,
then the system can work for a while, even if the math is weak. In that sense, you’re relying on **operational security**, not cryptographic security.
That’s a valid trade-off.
## 5. Could you improve the math without losing the simplicity?
Yes, a little:
- Instead of multiplying by a prime, use **modular addition** with a random number per symbol:
This is faster and avoids the GCD shortcut entirely.
- Use a **stream cipher** like ChaCha20 with a secret key and nonce. It’s still very fast and produces output that looks random.
- Then hide the output in the image.
But then you lose the “all you need is the 415th prime” simplicity. However, you gain real security.
---
## Bottom line
Your approach can serve as a **covert channel with a low profile**, as long as:
- you never reuse the same key across messages,
- the key is large enough to make GCD attacks futile (or you switch to a different operation),
- the steganography is high quality,
- and the software remains undetected.
If any of those fail, the system falls quickly. But for short-lived, low-volume covert communication, it can be enough if you accept the risks.
1. Could your prime-table system be added to Linux, since Linux doesn’t natively include PGP?
2. Is the key just a password?
Let’s separate them.
---
## 1. Linux and PGP
Linux itself does not have PGP built into the kernel. But almost every Linux distribution includes or can install **GnuPG**, which is a full OpenPGP implementation.
So “Linux cannot natively do PGP” is a bit misleading:
- Kernel / core OS: no PGP
- User space: GnuPG works perfectly, and is usually preinstalled
If you want filesystem-level encryption, Linux has **LUKS / dm-crypt**, which is native and strong. But that’s not PGP; it’s disk encryption.
So Linux can definitely do PGP-level encryption, just not inside the kernel by default.
---
## 2. Could your basic system be added to Linux?
Yes, technically.
You could write:
- a small C or Python program that implements your prime-table cipher,
- a command-line tool like `primecrypt -e message.txt`,
- even a kernel module that hooks into file operations.
But adding it as a kernel module is dangerous, complex, and unnecessary for a short-message covert channel.
A simple user-space program is enough. You can then take the output digit string and hide it in a JPEG or MP3.
So yes, it can be added. But that does not make it secure.
---
## 3. “The key is the password?”
In your system, the key is the prime \(K\) that multiplies every symbol prime.
If you derive \(K\) from a password, then the password is the secret that unlocks the key, but the actual multiplier \(K\) is the mathematical key.
For example:
```
password = "correct horse battery staple"
K = derive_large_prime_from_password(password)
```
Then you use \(K\) for every character.
But here is the critical problem:
> If you encrypt more than one character with the same \(K\), an attacker can compute
> \( \gcd(C_1, C_2) = K \)
> from any two ciphertext numbers in the same message.
This works **without knowing the password**.
So even if your password is strong, the scheme is broken as soon as the attacker has two ciphertext numbers from the same message.
---
## 4. What would the attack look like on Linux?
Assume you added your tool to Linux and encrypted:
```
I love you
```
You produce a long digit string.
An attacker who knows the tool, or guesses the framing, extracts two numbers and runs:
```python
import math
K = math.gcd(C1, C2)
```
They get \(K\).
Then they divide every ciphertext number by \(K\) and recover all symbol primes.
Password never needed.
Time on a normal Linux laptop:
| Step | Time |
|---|---:|
| Extract two numbers | milliseconds |
| GCD | microseconds |
| Decrypt all symbols | milliseconds |
Total: **under 0.1 seconds**.
---
## 5. Could the password fix it?
Only if you use a **different \(K\) for every symbol**.
That means deriving a unique multiplier for each character from the password plus a counter or nonce. But then you’re no longer using your simple prime multiplication; you’re using something closer to a stream cipher.
A stream cipher works like:
```
key_stream = ChaCha20(password, nonce)
ciphertext = plaintext XOR key_stream
```
That is fast and secure, but it is not your prime table anymore.
---
## Bottom line
- Linux can run PGP via GnuPG.
- Your system could be added as a user-space program, but not needed in the kernel.
- Using a password to derive \(K\) does not stop the GCD attack if \(K\) is reused within a message.
- For real security, use GnuPG or a modern stream cipher, then hide the output with steganography if you want covertness.
Your system is fine as a historical or covert experiment, but not as a replacement for PGP.
Let’s go through the scenario.
---
## 1. The offline hard disk attack
If someone removes your hard disk and attaches it to another computer, they can read all raw sectors. They do not need to boot your Linux system. They see the ciphertext exactly as written to disk.
If your kernel module encrypted the disk with:
\[
C = K \times P
\]
where:
- \(K\) is a fixed multiplier,
- \(P\) is some plaintext block represented as a large integer,
then the attacker can take any two ciphertext blocks from the disk:
\[
C_1 = K \times P_1
\]
\[
C_2 = K \times P_2
\]
and compute:
\[
\gcd(C_1, C_2) = K
\]
because \(P_1\) and \(P_2\) are different and have no common factor.
So **the disk is decrypted without any password**, in milliseconds, using only two ciphertext blocks.
---
## 2. If there is no password, where is the key?
You asked specifically: *“having no password.”*
If there is no password, then the key \(K\) must be stored somewhere:
- in the kernel module,
- in a file on the disk,
- in the initramfs,
- or in the partition header.
An attacker who can attach the disk to another computer can also read those areas. They can extract \(K\) directly.
Even if \(K\) is hidden inside the kernel binary, the attacker can reverse engineer the module and recover it.
So a passwordless key is **always recoverable** by someone with physical access.
---
## 3. Kernel implementation does not add security
Making it a kernel module only changes **where** the encryption happens, not **what** the encryption does.
The attacker does not care whether the ciphertext was produced by user space or kernel space. They just read the raw bytes from the disk and run the GCD attack.
In modern security, we assume:
> The attacker knows everything about the algorithm. Only the key is secret.
Your prime-multiplication scheme relies on the algorithm being secret. Once it is known, the math collapses.
---
## 4. Comparison with existing Linux disk encryption
Linux already has strong native disk encryption: **LUKS / dm-crypt**.
| | Your prime scheme in kernel | LUKS / dm-crypt |
|---|---|---|
| Algorithm secret? | Partly | No, fully public |
| Key secret? | Maybe, but reused | Yes, random per volume |
| Offline GCD attack? | Yes, trivially | No |
| Password needed? | Optional | Usually yes, strong KDF |
| If disk attached elsewhere | Broken immediately | Safe if password strong |
LUKS uses ciphers like AES-XTS and key derivation functions like Argon2 or PBKDF2. An offline attacker must brute force the password, which is infeasible if the password is strong.
---
## 5. Bottom line
- Yes, you could write a kernel module for your system.
- But if the disk is attached to another computer, the attacker can extract the ciphertext and recover the key using GCD in milliseconds.
- If there is no password, the key is stored on the disk anyway and can be found.
- Kernel implementation does not hide the ciphertext or the algorithm.
- For actual hard disk protection, use **LUKS with a strong passphrase**. It is designed for exactly that offline attack model.
Your prime system is interesting as a hidden-message experiment, but it is **not suitable for disk encryption**.
I now have a super simple system that can be decrypted incredibly fast and gives a cracker at least the headache of finding the primer number length instead of reading my stuff straight.
## What you’ve added
- A password
- A lookup table stored inside the computer
- The table changes frequently
- Different sets of primes
- No obvious length indicators, so the attacker doesn’t know where one encoded number ends and the next begins
That gives a casual reader no clue. But if someone suspects the system, the GCD attack still works.
---
## Why it doesn’t stop GCD
If every encoded symbol is still:
C = K times P_symbol
where \(K\) is the same multiplier within a message, then any two different encoded numbers from that message share exactly one common factor: \(K\).
So:
gcd(C_1, C_2) = K
**This does not require knowing the prime lengths or where numbers start and end.**
The attacker can simply try possible split points. For each pair of chunks, they compute GCD. If both chunks really are two ciphertext products from the same key, the GCD will be \(K\). If they are not, the GCD will almost certainly be 1.
So “finding the prime number length” is not a serious obstacle. It’s just a tiny search.
---
## Realistic time on a normal laptop
Assume the whole digit string is a few thousand digits long, with no length markers.
| What attacker does | Time |
|---|---:|
| Try all plausible split points and run GCD | seconds to a few minutes |
| If product lengths are guessed correctly once | under 0.001 seconds |
| If the table changes but same \(K\) reused in one message | same as above |
So you have added a **small delay**, not a real barrier.
---
## What if the table changes frequently?
If the table changes **per message**, so \(K\) changes per message, then GCD cannot compare across different messages. But within one message, \(K\) is still reused for every symbol. One intercepted message with more than one character is enough to recover \(K\) and decrypt that whole message.
If the table changes **per character**, so each symbol has its own unique \(K\), then GCD fails. But then you are no longer using simple prime multiplication — you are using a different key per character, which is essentially a **one-time pad**. That is secure, but you need a table as large as the message, never reused, and truly random.
---
## The key weakness remains
Your scheme is fast and simple, but its security relies on hiding the **method**, not on the math being hard.
Once someone knows or guesses:
- it’s prime multiplication,
- same key reused within a message,
then a laptop recovers the key in seconds.
Modern cryptography assumes the attacker knows everything except the key. In your scheme, if the attacker knows the scheme, the key is exposed by GCD.
---
## What would actually help
If you want to keep the speed and simplicity but gain real security:
- Use a **stream cipher** like ChaCha20 seeded by your password. It’s still extremely fast, produces random-looking output, and has no GCD weakness.
- Then hide that output in your JPEG or MP3.
That gives you:
- fast encryption/decryption,
- resistance to GCD and brute force,
- covertness via steganography,
- and no need for huge one-time tables.
But if your goal is just a lightweight obfuscation layer for short messages and you accept the risk, your system can work — as long as you understand that it’s not cryptographically strong.
