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You’re probably familiar with some of the most common requirements for creating passwords. A mix of upper and lowercase letters is a simple example. These are known as password constraints. They’re rules for how you must construct a password. If your password must be at least eight characters long, contain lower case, uppercase, numbers and symbol characters, then you have one length, and four character set constraints.

Password constraints eliminate a number of both good and bad passwords. I had never heard anyone ask “how many potential passwords, good and bad, are eliminated?” And so I began searching for the answer. The results were surprising. If you want to know the precise number of possible 8-character passwords there are if all of the character sets must be used, then the equation looks something like this.

A serious limitation of this approach is that it tells you nothing about the effects of each constraint alone or relative to other constraints. (I’m also not sure if there were supposed to be four consecutive ∑s or if the mathematician was stuttering.)

We choose to use a Monte Carlo simulation to analyze the mathematical impact of the various combinations of constraints. A Monte Carlo simulation uses a statistical analysis approach that provides a close approximation of the answer, while also providing the flexibility to quickly and easily measure the impact of each constraint and combination of constraints.

## A look at minimum length limits

To start, let’s look at the impact of an eight-character length constraint alone. There are 95^8 possible combinations of 8 characters. 26 uppercase letters + 26 lowercase letters + 10 numerals + 33 symbols = 95 characters. For a length of 8 characters, we have 95˄8 possible passwords.

If a password must be at least 8 characters long, then there are also about 70.6 trillion otherwise viable passwords you are not allowed to use (95+(95^2 ) +(95^3 ) +(95^4 ) +(95^5)+(95^6 )+(95^7)). That’s a good thing. It means you can’t use 95 one character passwords, 9,025 two character passwords, and so on. Almost 70 trillion of those passwords you cannot use are seven characters long. This is a great and wholly intended effect of a password length constraint.

The problem with a lack of constraints is that people will use a very small set of all possible passwords, which invariably includes passwords that are incredibly easy to guess. In the analysis of over one million leaked passwords, it was found that 30.8 percent passwords eight to 11 characters long contained only lowercase letters, and 43.9 percent contained only lowercase letters and numbers.  In fact, to perform a primitive brute force attack against an eight-character password containing only lower case letters, it’s only necessary to try about 209 billion character combinations. That does not take a computer very long to crack. And, as we know from analyzing large numbers of passwords, it’s likely to contain one of the most popular ten thousand passwords.

To beef up security, we begin to add character constraints. But, in doing so, we decrease the number of possible passwords; both good and bad.

Just by requiring both uppercase and lowercase letters, more than 15 percent of all possible 8-character combinations have been eliminated as possible passwords. This means that 1QV5#T&|cannot be a password because there are no lowercase letters. Compared to Darnrats,which meets the constraint requirements, 1QV5#T&|is a fantastic password. But you cannot use it. Superior passwords that cannot be used are acceptable collateral damage in the battle for better security. “Corndogs” is acceptable, but “fruit&veggies” is not. This clearly is not a battle for lower cholesterol.

## As constraints pile up, possibilities shrink

If a password must be exactly eight characters long and contain at least one lower case letter, at least one uppercase letter and at least one symbol, we are getting close to one-in-five combinations of 8 characters that are not allowable as passwords. Still, the effect of constraints on 12 and 16 character passwords is negligible. But that is all about to change… you can count on it.

Are you required to use a password that is at least eight characters long, has lower and uppercase letters, number and symbols? Just requiring a number to be part of a password removes over 40 percent of 8-character combinations from the pool of possible passwords. Even though you can use lowercase and uppercase letters, and you can use symbols, if one of the characters in your password must be a number then there are far fewer great passwords that you can use. If a 16 character long password must have a number, then 13 times more potential passwords have become illegal as a result of that one constraint than the combined constraints of lower and uppercase letters and symbols caused. More than one-in-four combinations of 12 characters can no longer become a passwords either.

You might have noticed that there is little effect on the longer passwords. Frequently there is also very little value in imposing constraints on long passwords. This is because each additional character in a password grows the pool of passwords exponentially. There are 6.5 million times as many combinations of 16 character pass words using only lowercase letters than there are of eight character passwords using all four character sets. That means that “toodlesmypoodles” is going to be a whole lot harder to crack than “I81B@gle”

## Long and simple is better than short and hard

People tend to be very predictable. There are more symbols (than there are in any other characters set. Theoretically that means that symbols are going to do the most to make a password strong, but 80 percent of the time it is going to be one of the top five most frequently used symbols, and 95 percent of the time is will be one of the top 10 most frequently used symbols.

Analysis of two million compromised passwords showed that about one in 14 passwords start with the number one, however for those that started with the number one, 75 percent of them ended with a number as well.

The use of birthdays and names, for example, make it much easier to quickly crack many passwords.

## Password strength: It’s length, not complexity that matters

As covered above, all four character sets (95 characters) in an eight character password allow for about 6.634 quadrillion different password possibilities. But a 16 character password with only lowercase letters has about 43.8 sextillion possible passwords. That means that there are well over 6.5 million times more possible passwords for 16 consecutive lowercase letters than for any combination of eight characters regardless of how complex the password is.

My great password is “cats and hippos are friends!”, but I can’t use it because of constraints – and because I just told you what it is.

For years password experts have been advocating for the use of simple passphrases over complex passwords because they are stronger and simpler to remember. I’d like to throw a bit of gasoline on to the fire and tell you, those 95^8 combinations of characters are only  half that many when you tell me I have to use uppercase, lowercase, numbers, and symbols.

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