> For the complete documentation index, see [llms.txt](https://alham-rizvi.gitbook.io/alhamrizvi/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://alham-rizvi.gitbook.io/alhamrizvi/picoctf-writeups/picoctf/cryptography/b00tl3grsa4.md).

# b00tl3grsa4

### The Challenge

Connect via netcat and receive:

```
c: 7821284413607156999099037941922718048546728118501714272754771531395454643151598402388981985486801198547064160448354343192049702172597137904962684049730561514562954160260555429640805667950809463084835189307739372171319177103480437837078192632150029075584424462537181384488291190637736218814332379653805074526802285414240329671605717588215896850
n: 9879962505429438587327716228010413610187726687076936201862843964715927081688957105676422143610974577253586418350462983077401072435906147211091220678606751124707241014836299352086515681960133377148052245298519266399792554702601503772160669040949324189657930936491310231878947582680294977822434251894991564050215801208332193134930374674608505677
e: 65537
```

***

### Why Standard RSA Decryption Fails

Normal RSA assumes `n = p × q` (two primes). Decryption requires:

```
φ(n) = (p-1)(q-1)
d = e⁻¹ mod φ(n)
m = c^d mod n
```

Here `n` is 343 digits long and can't be factored by trial division. But unlike standard RSA, this `n` is actually a product of **many small primes** — 34 of them! This makes it vulnerable because tools like factordb and alpertron can break it quickly.

***

### The Vulnerability — Too Many Factors

Standard RSA uses exactly 2 large primes. This `n` uses **34 small primes**, all around 10 digits each:

```
n = 8603565937 × 8635146629 × 8660513317 × ... × 16873275253
```

Small primes are easy to find using **ECM (Elliptic Curve Method)** factorization. The security of RSA comes from using two *huge* primes — small ones offer no protection.

***

### Solution

#### Step 1 — Connect and grab the values

```bash
nc fickle-tempest.picoctf.net 57484
```

Copy `c`, `n`, and `e`.

#### Step 2 — Factor n using Alpertron

Go to **<https://www.alpertron.com.ar/ECM.HTM>**

Paste `n` in the Value box and click **Factor**.

In about 3 seconds it returns all 34 prime factors AND — crucially — **Euler's totient φ(n)** directly:

```
φ(n) = 9879962477328754116973777764013580703505702722284524485956731509359599386015813859615735068213797065665803992307497112624191113661855910950399801471484915576710565840360877525764509091261718011153685744369465609699611735670197797367791318782970419983861521168743529376153488393126743775449005101718624908833935581093200414111604251033600000000
```

No need to multiply out `(p-1)(q-1)` for all 34 factors — alpertron does it for you.

#### Step 3 — Decrypt

```python
c = 7821284413607156999099037941922718048546728118501714272754771531395454643151598402388981985486801198547064160448354343192049702172597137904962684049730561514562954160260555429640805667950809463084835189307739372171319177103480437837078192632150029075584424462537181384488291190637736218814332379653805074526802285414240329671605717588215896850
n = 9879962505429438587327716228010413610187726687076936201862843964715927081688957105676422143610974577253586418350462983077401072435906147211091220678606751124707241014836299352086515681960133377148052245298519266399792554702601503772160669040949324189657930936491310231878947582680294977822434251894991564050215801208332193134930374674608505677
e = 65537

phi = 9879962477328754116973777764013580703505702722284524485956731509359599386015813859615735068213797065665803992307497112624191113661855910950399801471484915576710565840360877525764509091261718011153685744369465609699611735670197797367791318782970419983861521168743529376153488393126743775449005101718624908833935581093200414111604251033600000000

d = pow(e, -1, phi)
m = pow(c, d, n)
h = hex(m)[2:]
if len(h) % 2: h = '0' + h
print(bytes.fromhex(h).decode())
```

***

### Output

```
picoCTF{too_many_fact0rs_3023548}
```

***

### Key Takeaways

RSA security depends on **two large primes**. Using many small primes instead breaks it in two ways:

1. **Small primes are easy to find** — ECM factorization cracks them in seconds
2. **More factors = smaller individual primes** — the more factors, the weaker each one is

The flag says it all — `too_many_fact0rs`. Stick to two large primes or you're done.

***

**Flag:** `picoCTF{too_many_fact0rs_3023548}`


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