> 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/17.-padding-oracle-attack.md).

# Padding Oracle Attack (Bleichenbacher Attack)

## 1. Overview

This is a **chosen-ciphertext attack** against RSA when **PKCS#1 v1.5 padding** is used.

You are given:

```
n, e, C
```

and access to an oracle:

```
Oracle(C') → tells whether decrypted message has valid padding
```

Goal:

```
Recover plaintext M without knowing d
```

***

## 2. PKCS#1 v1.5 Padding Format

Before encryption, message is padded:

```
EM = 0x00 || 0x02 || PS || 0x00 || M
```

Where:

```
PS = random non-zero bytes (at least 8 bytes)
```

So valid decrypted message must satisfy:

```
Starts with: 00 02
```

***

## 3. Core Idea

RSA is multiplicative:

```
C' = (C · s^e) mod n
```

After decryption:

```
M' = (M · s) mod n
```

We manipulate `s` and observe:

```
Does M' have valid padding?
```

***

## 4. Key Insight

Valid padding means:

```
2B ≤ M' < 3B
```

Where:

```
B = 2^(8*(k-2))   (k = byte length of n)
```

***

## 5. Attack Strategy

We search for `s` such that:

```
M' = (M · s) mod n falls in [2B, 3B)
```

Each successful oracle response:

```
narrows possible range of M
```

***

## 6. Interval Reduction

We maintain a set of intervals:

```
M ∈ [a, b]
```

Each valid `s` gives:

```
new constraints on M
```

We update intervals until:

```
interval size = 1 → M found
```

***

## 7. Step-by-Step Flow

### Step 1 — Blinding

Find first `s1` such that:

```
Oracle(C · s1^e mod n) = valid
```

***

### Step 2 — Narrow intervals

Compute:

```
M ∈ [2B/s, 3B/s]
```

Update bounds.

***

### Step 3 — Search next s

Find next `s` values that maintain valid padding.

***

### Step 4 — Repeat

Keep shrinking interval until:

```
M is uniquely determined
```

***

## 8. Simplified Python Structure

```python
def padding_oracle_attack(n, e, C, oracle):
    k = (n.bit_length() + 7) // 8
    B = 2 ** (8 * (k - 2))

    # Step 1: find s1
    s = 1
    while True:
        s += 1
        C_test = (C * pow(s, e, n)) % n
        if oracle(C_test):
            break

    # Initial interval
    intervals = [(2*B, 3*B)]

    # Loop until narrowed
    while True:
        new_intervals = []

        for a, b in intervals:
            # update bounds based on s
            # (simplified — real version more complex)
            new_intervals.append((a, b))

        intervals = new_intervals

        if len(intervals) == 1 and intervals[0][0] == intervals[0][1]:
            return intervals[0][0]
```

***

## 9. Oracle Example

```python
def oracle(C):
    M = pow(C, d, n)
    return (M >> (8*(k-1))) == 0x02
```

***

## 10. Why This Works

Because:

```
Padding creates structure in plaintext
```

Oracle leaks:

```
valid / invalid → 1 bit of structured information
```

Repeated queries:

```
fully recover M
```

***

## 11. When to Recognize

Look for:

```
Server responds differently for:
- valid padding
- invalid padding
```

Examples:

```
"Decryption error"
"Invalid padding"
HTTP 500 vs 200
```

***

## 12. Real-World Impact

This attack was used against:

```
SSL/TLS implementations (historically)
```

It is one of the most famous RSA attacks.

***

## 13. Limitations

```
Requires oracle access
Requires PKCS#1 v1.5 padding
Slow (many queries)
```

***

## 14. Defense

```
Use RSA-OAEP padding
Do not leak padding errors
Constant-time responses
```

***

## 15. Summary

```
Manipulate ciphertext with s^e
→ oracle reveals padding validity
→ shrink interval
→ recover M
```

***

## 16. Final Note

```
This is not a math attack
This is an implementation attack
```


---

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