Exploiting Integer Arithmetic A Technical Deep Dive into A Small Game to Withdraw More Than 100
The seemingly innocuous challenge of "A Small Game to Withdraw More Than 100" serves as a compelling case study in the critical importance of secure software design, particularly concerning low-level programming languages like C and C++. At its core, this challenge is a classic manifestation of integer overflow and underflow vulnerabilities, compounded by flawed program logic. A thorough technical analysis reveals not just a simple bug, but a fundamental flaw in the trust relationship between a program's control flow and its underlying data representation. This discussion will deconstruct the game's probable mechanics, explore the exploitation techniques, delve into the assembly-level execution, and propose robust mitigation strategies.
**Deconstructing the Game's Logic**
We can infer the game's structure from its title and objective. A typical implementation might resemble the following C code snippet:
```c
#include
#include
int balance = 100;
void withdraw() {
int amount;
printf("Your current balance is: %d\n", balance);
printf("Enter amount to withdraw: ");
scanf("%d", &amount);
// The Flawed Logic
if (amount <= balance) {
balance -= amount;
printf("Withdrawal successful. New balance: %d\n", balance);
} else {
printf("Insufficient funds!\n");
}
}
int main() {
printf("Welcome to the withdrawal game!\n");
printf("Your goal is to make your balance exceed 100.\n");
while (balance > 0 && balance <= 100) {
withdraw();
}
if (balance > 100) {
printf("Congratulations! You won!\n");
} else {
printf("Game Over.\n");
}
return 0;
}
```
The program's logic appears sound at first glance. It initializes a global integer `balance` to 100. The `withdraw` function prompts the user for an `amount`, checks if it is less than or equal to the current `balance`, and if so, deducts the amount. The `main` function's loop continues as long as the balance is between 0 and 100 (inclusive), and victory is declared if the loop exits with a balance greater than 100.
The vulnerability lies in the interaction between the signed integer data type (`int`) and the subtraction operation. In most modern systems, a signed `int` is a 32-bit two's complement value, giving it a range of -2,147,483,648 to 2,147,483,647. The check `if (amount <= balance)` is a logical one, but the subsequent operation `balance -= amount;` is an arithmetic one. This discrepancy is the gateway to exploitation.
**The Exploitation Mechanism: Integer Underflow**
The winning condition is to achieve a `balance > 100`. Since we can only withdraw, how can subtracting money *increase* the balance? The answer is integer underflow.
Consider a scenario where the user inputs a very large positive integer, for example, `2,147,483,647` (which is 2^31 - 1). The program flow is as follows:
1. **Check:** `if (amount <= balance)`. Is 2,147,483,647 <= 100? This is false.
2. The `else` branch is executed: "Insufficient funds!".
3. The withdrawal fails, and the balance remains 100.
This does not help us. The key is to use a large *negative* number.
1. **Input:** Let the user input `-2,147,483,647`.
2. **Check:** `if (amount <= balance)`. Is -2,147,483,647 <= 100? This is unequivocally **true**.
3. **Execution:** The program executes `balance -= amount;`.
This becomes: `balance = balance - (-2,147,483,647)`
Which simplifies to: `balance = 100 + 2,147,483,647`
The result is 2,147,483,747.
This new value, 2,147,483,747, is indeed greater than 100, satisfying the winning condition and breaking the loop in `main()`. The program has been successfully exploited.
**Technical Deep Dive: Two's Complement and Assembly-Level View**
To fully grasp why this works, we must understand the two's complement system. In this system, the most significant bit (MSB) represents the sign (0 for positive, 1 for negative). The value of a negative number is obtained by inverting all the bits of its positive counterpart and adding one.
For a 32-bit system:
* The decimal number `100` is `0x00000064` in hex.
* The decimal number `-100` is `0xFFFFFF9C` in hex.
The critical operation `balance -= amount` compiles down to simple assembly instructions. Let's assume `balance` is at memory address `[ebp-0x4]` and `amount` is at `[ebp-0x8]`.
A non-malicious withdrawal of 50 would look like this:
```
mov eax, [ebp-0x4] ; EAX = balance (100)
sub eax, [ebp-0x8] ; EAX = EAX - amount (100 - 50 = 50)
mov [ebp-0x4], eax ; balance = EAX (50)
```
The `sub` instruction performs integer subtraction and sets the relevant flags (Carry, Overflow, Sign, Zero) in the EFLAGS register.
Now, for the malicious withdrawal of `-2,147,483,647`:
```
mov eax, [ebp-0x4] ; EAX = balance (100 = 0x00000064)
sub eax, [ebp-0x8] ; EAX = EAX - (-2147483647)
; This is 0x00000064 - 0x80000001
; In two's complement, subtracting 0x80000001 is the same as adding its positive equivalent.
; The positive of 0x80000001 is 0x7FFFFFFF.
; So, 0x00000064 + 0x7FFFFFFF = 0x80000063
mov [ebp-0x4], eax ; balance = 0x80000063 (decimal: 2,147,483,747)
```
The `sub` instruction operates purely on the bit patterns, oblivious to the semantic meaning we assign to them as signed integers. It has no inherent concept of "underflow" in the arithmetic sense; it only deals with carries and borrows. The result, `0x80000063`, is a valid bit pattern that, when interpreted as a signed integer, represents a large positive number, thus achieving the exploit.
**Advanced Exploitation and Variants**
The basic exploit is straightforward, but the vulnerability can manifest in more subtle ways.
1. **Integer Overflow instead of Underflow:** Imagine a game where you *deposit* money to reach a goal, and the balance is stored in an unsigned 32-bit integer (`unsigned int` with range 0 to 4,294,967,295). If the balance is 4,000,000,000 and you deposit 300,000,000, the operation `balance += amount` would result in 4,300,000,000. This exceeds the maximum value of 4,294,967,295, causing an integer overflow. The value would wrap around, becoming 4,300,000,000 - 4,294,967,296 = 5,032,704. An attacker could use this to manipulate the balance to a specific, desired value.
2. **Width-Conversion Vulnerabilities:** The program might read user input into a larger data type (e.g., a `long long`) before assigning it to the `int amount`. If the conversion check is inadequate, a value like 500,000,0000 (5 billion) that is valid for a `long long` would be truncated when stored in an `int`, potentially becoming a negative number due to two's complement representation and triggering the original underflow vulnerability.
3. **Arithmetic with Different Types:** If `balance` is an `unsigned int` and `amount` is an `int`, the comparison `if (amount <= balance)` becomes treacherous. In C/C++, when a signed and unsigned integer are compared, the signed value is implicitly converted to unsigned. A user input of `-1` (signed) becomes `0xFFFFFFFF` (unsigned, value 4,294,967,295), which is certainly not less than 100. This would fail the check. However, the subsequent operation `balance -= amount;` might still perform the subtraction with unexpected results, depending on compiler-specific behavior regarding mixed-type arithmetic. This highlights the danger of implicit type conversions.
**Mitigation Strategies: Building Robust Systems**
Preventing such vulnerabilities requires a defense-in-depth approach, combining secure coding practices, compiler tools, and language-level safeguards.
1. **Input Validation and Sanitization:** This is the first and most crucial line of defense. The program should rigorously validate all user input. For a withdrawal function, the only valid input is a positive integer within a sensible business range (e.g.,
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