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+---- Thread: How does RSA work? (/showthread.php?tid=6307079)
How does RSA work? - rjusticemi - Aug 31, 2025
I know how asymmetric encryption works; it works by exchanging public keys, and when someone wants to send a message, the message is encrypted with the recipient's public key, which can only be decrypted with the recipient's private key. I know RSA is one of the most popular asymmetric encryption algorithms, but I wanna know more about how it works
When I search "how does RSA work" on Google, it doesn't actually tell me how the RSA algorithm itself works, but it just tells me the basics of how asymmetric encryption in general works (what I just explained above).
Can someone simply explain how RSA works, and the process of encrypting and decrypting messages using the RSA algorithm?
RE: How does RSA work? - DramaKing - Aug 31, 2025
(Aug 31, 2025 01:27 AM)rjusticemi Wrote: I know how asymmetric encryption works; it works by exchanging public keys, and when someone wants to send a message, the message is encrypted with the recipient's public key, which can only be decrypted with the recipient's private key. I know RSA is one of the most popular asymmetric encryption algorithms, but I wanna know more about how it works
When I search "how does RSA work" on Google, it doesn't actually tell me how the RSA algorithm itself works, but it just tells me the basics of how asymmetric encryption in general works (what I just explained above).
Can someone simply explain how RSA works, and the process of encrypting and decrypting messages using the RSA algorithm?
At a basic level, RSA involves generating two large prime numbers and calculating keys from those. Information on the key generation process is easy to find. https://www.geeksforgeeks.org/dsa/how-to-generate-large-prime-numbers-for-rsa-algorithm/
RE: How does RSA work? - h4ckn4xus - Sep 10, 2025
RSA uses two keys: a public key (anyone can use it to send you messages) and a private key (you keep it secret to read those messages).
Choose two big prime numbers:
Let’s call them p and q.
Example: p = 11, q = 13 (small numbers just for illustration).
Compute n = p × q:
n = 11 × 13 = 143
n is part of both the public and private keys.
Compute Euler’s totient function φ(n) = (p−1) × (q−1):
φ(n) = (11−1) × (13−1) = 10 × 12 = 120
Choose a public exponent e:
e must be coprime with φ(n) (no common factors besides 1).
Example: e = 7 (gcd(7, 120) = 1)
Compute the private exponent d:
d is the multiplicative inverse of e modulo φ(n)`
Solve for d such that (d × e) % φ(n) = 1
Here, d = 103 because (103 × 7) % 120 = 1
Now you have:
Public key: (n = 143, e = 7)
Private key: (n = 143, d = 103)
ENCRYPTION:
To send a message m (as a number smaller than n):
Take the message m → example: m = 9
Encrypt it using the public key (n, e):
c = 9^7 % 143
Calculate:
9^7 = 4782969
4782969 % 143 = 48
So, the ciphertext c = 48. This is what you send.
DECRYPT:
The receiver uses the private key (n, d) to decrypt:
m = 48^103 % 143
Using modular arithmetic tricks (you usually use software for big numbers), the result is m = 9, recovering the original message.
RESUME:
1-> Generate keys using large primes → public and private key.
2 -> Encrypt: c = m^e % n with the public key.
3 -> Decrypt: m = c^d % n with the private key.
The security of RSA comes from the fact that factoring n into p and q is extremely hard if p and q are large (hundreds of digits).
RE: How does RSA work? - ▄︻デ══━一一一一━ - Sep 10, 2025
(Sep 10, 2025 09:52 AM)h4ckn4xus Wrote: RSA uses two keys: a public key (anyone can use it to send you messages) and a private key (you keep it secret to read those messages).
Choose two big prime numbers:
Let’s call them p and q.
Example: p = 11, q = 13 (small numbers just for illustration).
Compute n = p × q:
n = 11 × 13 = 143
n is part of both the public and private keys.
Compute Euler’s totient function φ(n) = (p−1) × (q−1):
φ(n) = (11−1) × (13−1) = 10 × 12 = 120
Choose a public exponent e:
e must be coprime with φ(n) (no common factors besides 1).
Example: e = 7 (gcd(7, 120) = 1)
Compute the private exponent d:
d is the multiplicative inverse of e modulo φ(n)`
Solve for d such that (d × e) % φ(n) = 1
Here, d = 103 because (103 × 7) % 120 = 1
Now you have:
Public key: (n = 143, e = 7)
Private key: (n = 143, d = 103)
ENCRYPTION:
To send a message m (as a number smaller than n):
Take the message m → example: m = 9
Encrypt it using the public key (n, e):
c = 9^7 % 143
Calculate:
9^7 = 4782969
4782969 % 143 = 48
So, the ciphertext c = 48. This is what you send.
DECRYPT:
The receiver uses the private key (n, d) to decrypt:
m = 48^103 % 143
Using modular arithmetic tricks (you usually use software for big numbers), the result is m = 9, recovering the original message.
RESUME:
1-> Generate keys using large primes → public and private key.
2 -> Encrypt: c = m^e % n with the public key.
3 -> Decrypt: m = c^d % n with the private key.
The security of RSA comes from the fact that factoring n into p and q is extremely hard if p and q are large (hundreds of digits).
RSA Encryption - More Simplified
Quote:RSA uses two keys – a public one for encryption, and a private one for decryption.
Example:
Code
p = 11
q = 13
n = p × q = 143
φ(n) = (p−1)(q−1) = 120
e = 7 (coprime with 120)
d = 103 (since (e×d) mod φ(n) = 1)Encrypt: c = m^e mod n
→ Example:
Code
m = 9 ⇒ c = 9^7 mod 143 = 48Decrypt: m = c^d mod n
→
Code
m = 48^103 mod 143 = 9Summary:
- Public key: (n=143, e=7)
- Private key: (n=143, d=103)
- Secure when using large primes + padding (e.g., OAEP)
Keep it padded, keep it safe. Always stay sharp ▄︻テ══━一一一
RE: How does RSA work? - KAPALfx - Sep 10, 2025
RSA encryption is based on modular arithmetic. You start by generating two large primes, multiply them to form n, and then choose an exponent e as part of the public key. To encrypt, you raise the message to the power of e mod n. To decrypt, you use the private exponent d to reverse the process. The difficulty of breaking RSA comes from factoring n back into its prime factors.
RE: How does RSA work? - rjusticemi - Sep 10, 2025
(Sep 10, 2025 09:52 AM)h4ckn4xus Wrote: RSA uses two keys: a public key (anyone can use it to send you messages) and a private key (you keep it secret to read those messages).
Choose two big prime numbers:
Let’s call them p and q.
Example: p = 11, q = 13 (small numbers just for illustration).
Compute n = p × q:
n = 11 × 13 = 143
n is part of both the public and private keys.
Compute Euler’s totient function φ(n) = (p−1) × (q−1):
φ(n) = (11−1) × (13−1) = 10 × 12 = 120
Choose a public exponent e:
e must be coprime with φ(n) (no common factors besides 1).
Example: e = 7 (gcd(7, 120) = 1)
Compute the private exponent d:
d is the multiplicative inverse of e modulo φ(n)`
Solve for d such that (d × e) % φ(n) = 1
Here, d = 103 because (103 × 7) % 120 = 1
Now you have:
Public key: (n = 143, e = 7)
Private key: (n = 143, d = 103)
ENCRYPTION:
To send a message m (as a number smaller than n):
Take the message m → example: m = 9
Encrypt it using the public key (n, e):
c = 9^7 % 143
Calculate:
9^7 = 4782969
4782969 % 143 = 48
So, the ciphertext c = 48. This is what you send.
DECRYPT:
The receiver uses the private key (n, d) to decrypt:
m = 48^103 % 143
Using modular arithmetic tricks (you usually use software for big numbers), the result is m = 9, recovering the original message.
RESUME:
1-> Generate keys using large primes → public and private key.
2 -> Encrypt: c = m^e % n with the public key.
3 -> Decrypt: m = c^d % n with the private key.
The security of RSA comes from the fact that factoring n into p and q is extremely hard if p and q are large (hundreds of digits).
Thanks for the explanation, this explains everything and is quite easy to understand