> 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/cryptography/rsa/12.-short-pad-attack.md).

# Short Pad Attack (Low Exponent + Small Padding)

This attack applies when RSA uses a **small exponent (usually `e = 3`)** and messages are padded with a **small random value**.

```
M1 = m + r1
M2 = m + r2
```

where:

```
m  = original message (same)
r1, r2 = small padding values
```

Encryption:

```
C1 = (m + r1)^e mod n
C2 = (m + r2)^e mod n
```

Goal:

```
Recover m
```

## 2. Core Idea

The padding is **too small**, so:

```
Difference between messages is small
```

This allows us to:

```
Treat it like a related message attack
```

But instead of known relation:

```
M2 = M1 + k   (known k)
```

we have:

```
M2 = M1 + (r2 − r1)   (unknown but small)
```

## 3. Mathematical Setup

Let:

```
x = m
```

Then:

```
C1 = (x + r1)^e mod n
C2 = (x + r2)^e mod n
```

Define:

```
Δ = r2 − r1   (small)
```

So:

```
(x + r2) = (x + r1 + Δ)
```

## 4. Key Insight

```
Δ is small → brute-force possible
```

Once Δ is guessed:

```
M2 = M1 + Δ
```

Now it becomes:

```
Franklin–Reiter attack
```

## 5. Attack Strategy

1. Loop over small Δ
2. Assume:

```
M2 = M1 + Δ
```

3. Build:

```
f(x) = x^e − C1
g(x) = (x + Δ)^e − C2
```

4. Compute:

```
gcd(f, g)
```

5. If valid:

```
Recover x = m
```

## 6. Python (SageMath) Implementation

```python
# SageMath required

n = ...
e = 3

C1 = ...
C2 = ...

R.<x> = PolynomialRing(Zmod(n))

for delta in range(1, 1000):  # small padding search
    f = x^e - C1
    g = (x + delta)^e - C2

    h = f.gcd(g)

    if h.degree() == 1:
        m = -h[0]
        print("Found m:", m)
        break
```

## 7. Why This Works

```
Small padding → small Δ
Small Δ → brute-force feasible
After guess → becomes algebraic (Franklin-Reiter)
```

## 8. When to Use

Look for:

```
Same message encrypted twice
Small random padding
Same n and e
e = 3 (very common)
```

Hints like:

```
"short padding"
"random suffix"
"tiny noise added"
```

## 9. Important Notes

```
Works only if padding difference is small
Requires polynomial GCD → use Sage
Fails if padding is large or randomized properly
```

## 10. Example (Concept)

```
m = 100
r1 = 2 → M1 = 102
r2 = 5 → M2 = 105
```

```
Δ = 3
```

We brute Δ:

```
Try Δ = 3 → success
```

## 11. Complexity

```
O(small_padding_range)
```

Usually:

```
< 2^16 or smaller
```

## 12. Summary

```
Same message + small padding
→ guess Δ
→ reduce to Franklin-Reiter
→ recover m
```

## 13. Practice Problem

```
n = 18721
e = 3

C1 = 10652
C2 = 16021

Message padded with small values (difference < 50)
```

## Task

```
Recover original message m
```


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