## Money may grow on trees

2015-08-25 18:25

By Fernando Arnaboldi

Sometimes when buying something that costs \$0.99 USD (99cents) or \$1.01 USD (one dollar and one cent), you may pay an even dollar. Eitheryou or the cashier may not care about the remaining penny, and so one of youtakes a small loss or profit.

Rounding at the cash register is a common practice, just asit is in programming languages when dealing with very small or very largenumbers. I will describe here how an attacker can make a profit when dealingwith the rounding mechanisms of programming languages.

Lack of precision innumbers

IEEE 754 standard has defined floating point numbers for morethan 30 years. The requirements that guided the formulation of the standard forbinary floating-point arithmetic provided for the development of very high-precisionarithmetic.

The standard defines how operations with floating pointnumbers should be performed, and also defines standard behavior for abnormaloperations. It identifies five possible types of floating point exceptions:invalid operation (highest priority), division by zero, overflow, underflow,and inexact (lowest priority).

We will explore what happens when inexact floating pointexceptions are triggered. The rounded result of a valid operation can be differentfrom the real (and sometimes infinitely precise) result and certain operationsmay go unnoticed.

Rounding

Rounding takes an exact number and, if necessary, modifiesit to fit in the destination's format. Normally, programming languages do not alertfor the inexact exception and instead just deliver the rounded value. Nevertheless,documentation for programming languages contains some warnings about this:

To exemplify this matter, this is how the number 0.1 looksinternally in Python:

The decimal value 0.1 cannot be represented precisely whenusing binary notation, since it is an infinite number in base 2. Python willround the previous number to 0.1 before showing the value.

Salami slicing thedecimals

Salami slicing refers to a series of small actions that willproduce a result that would be impossible to perform all at once. In this case,we will grab the smallest amount of decimals that are being ignored by the programminglanguage and use that for profit.

Let's start to grab some decimals in a way that goesunnoticed by the programming language. Certain calculations will trigger more obviousdifferences using certain specific values. For example, notice what happenswhen using v8 (Google's open source JavaScript engine) to add the values 0.1plus 0.2:

Perhaps we could take some of those decimals withoutJavaScript noticing:

But what happens if we are greedy? How much can we takewithout JavaScript noticing?

That's interesting; we learned that it is even possible to take more than whatwas shown, but no more than 0.00000000000000008 from that operation beforeJavaScript notices.

I created a sample bank application to explain this, pennies.js:

// This is used to wire money
function wire(deposit, money, withdraw) {
account[deposit] += money;
account[withdraw] -= money;
if(account[withdraw]<0)
return 1; // Error! The account can not have a negative balance
return 0;
}

// This is used to print the balance
function print_balance(time) {
print("--------------------------------------------");
print(" The balance of the bank accounts is:");
print(" Account 0: "+account+" u\$s");
print(" Account 1: "+account+" u\$s (profit)");
print(" Overall money: "+(account+account))
print("--------------------------------------------");
if(typeof time !== 'undefined') {
print(" Estimated daily profit: "+( (60*60*24/time) * account ) );
print("--------------------------------------------");
}
}

// This is used to set the default initial values
function reset_values() {
account = initial_deposit;
account = 0; // I will save my profit here
}

account = new Array();
initial_deposit = 1000000;
profit = 0
print("n1) Searching for the best profit");
for(i = 0.000000000000000001; i < 0.1; i+=0.0000000000000000001) {
reset_values();
wire(1, i, 0); // I will transfer some cents from the account 0 to the account 1
if(account==initial_deposit && i>profit) {
profit = i;
//   print("I can grab "+profit.toPrecision(21));
} else {
break;
}
}
print("   Found: "+profit.toPrecision(21));
print("n2) Let's start moving some money:");

reset_values();

start = new Date().getTime() / 1000;
for (j = 0; j < 10000000000; j++) {
for (i = 0; i < 1000000000; i++) {
wire(1, profit, 0); // I will transfer some cents from the account 0 to the account 1
}
finish = new Date().getTime() / 1000;
print_balance(finish-start);
}

Theattack against it will have two phases. In the first phase, we will determinethe maximum amount of decimals that we are allowed to take from an accountbefore the language notices something is missing. This amount is related to thevalue from which we are we taking: the higher the value, the higher the amountof decimals. Our Bank Account 0 will have \$1,000,000 USD to start with, and wewill deposit our profits to a secondary Account 1:

Due to the decimals being silently shifted to Account 1, thebank now believes that it has more money than it actually has.

Another possibility to abuse the loss of precision is whathappens when dealing with large numbers. The problem becomes visible when usingat least 17 digits.

Now, the sample attack application will occur on a cryptocurrency, fakecoin.js:

// This is used to wire money
function wire(deposit, money, withdraw) {
account[deposit] += money;
account[withdraw] -= money;
if(account[withdraw]<0) {
return 1; // Error! The account can not have a negative balance
}
return 0;
}

// This is used to print the balance
function print_balance(time) {
print("-------------------------------------------------");
print(" The general balance is:");
print(" Crypto coin:  "+crypto_coin);
print(" Crypto value: "+crypto_value);
print(" Crypto cash:  "+(account*crypto_value)+" u\$s");
print(" Account 0: "+account+" coins");
print(" Account 1: "+account+" coins");
print(" Overall value: "+( ( account + account ) * crypto_value )+" u\$s")
print("-------------------------------------------------");
if(typeof time !== 'undefined') {
seconds_in_a_day = 60*60*24;
print(" Estimated daily profit: "+( (seconds_in_a_day/time) * account * crypto_value) +" u\$s");
print("-------------------------------------------------n");
}
}

// This is used to set the default initial values
function reset_values() {
account = initial_deposit;
account = 0; // profit
}

// My initial money
initial_deposit = 10000000000000000;
crypto_coin = "Fakecoin"
crypto_value = 0.00000000011;

// Here I have my bank accounts defined
account = new Array();

print("n1) Searching for the best profit");
profit = 0
for (j = 1; j < initial_deposit; j+=1) {
reset_values();
wire(1, j, 0); // I will transfer a few cents from the account 0 to the account 1
if(account==initial_deposit && j > profit) {
profit = j;
} else {
break;
}
}
print("   Found: "+profit);
reset_values();
start = new Date().getTime() / 1000;
print("n2) Let's start moving some money");
for (j = 0; j < 10000000000; j++) {
for (i = 0; i < 1000000000; i++) {
wire(1, profit, 0); // I will transfer my 29 cents from the account 0 to the account 1
}
finish = new Date().getTime() / 1000;
print_balance(finish-start);
}

We will buy 10000000000000000 units of Fakecoin (with each coin valuedat \$0.00000000011 USD) for a total value of \$1,100,000 USD. We will transferone Fakecoin at the time from Account 0 to a second account that will hold ourprofits (Account 1). JavaScript will not notice the missing Fakecoin from Account0, and will allow us to profit a total of approximately \$2,300 USD per day:

Depending on the implementation, a hacker can used either floatnumbers or large numbers to generate money out of thin air.

Conclusion

In conclusion, programmers can avoid inexact floating pointexceptions with some best practices:
• If you rely on values that use decimal numbers,use specialized variables like BigInteger in Java.
• Do not allow values larger than 16 digits to bestored in variables and truncate decimals, or
• Use libraries that rely on quadruple precision (thatis, twice the amount of regular precision).

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