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# 03. Breaking RSA
## Breaking RSA — Basic Attacks
RSA security depends on the difficulty of factoring large numbers from cryptography. If this assumption fails, RSA can be broken.
## 1. Factorization Attack (Classic)
### Idea
```
If n = p × q can be factored → RSA is broken
```
### Steps
```
1. Factor n → get p, q
2. Compute φ(n) = (p − 1)(q − 1)
3. Compute d ≡ e⁻¹ mod φ(n)
4. Decrypt M = C^d mod n
```
#### Given
```
n = 55
e = 3
C = 10
```
#### Solution
```
n = 5 × 11
φ(n) = 40
d = 27
M = 10^27 mod 55 = 10
```
### Python Code
```python
from sympy import factorint, mod_inverse
n = 55
e = 3
C = 10
# Factor n
factors = factorint(n)
p, q = list(factors.keys())
# Compute phi
phi = (p - 1) * (q - 1)
# Compute d
d = mod_inverse(e, phi)
# Decrypt
M = pow(C, d, n)
print("Message:", M)
```
## 2. When Does This Work?
### Works When
```
n is small
n has small factors
poor key generation
```
### Fails When
```
n is very large (2048-bit)
primes are strong and random
```
## 3. Key Insight
```
RSA is secure only because factoring n is hard
```
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