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Password strength is a measure of the effectiveness of a password against guessing or brute-force attacks. In its usual form, it estimates how many trials an attacker who does not have direct access to the password would need, on average, to guess it correctly. The strength of a password is a function of length, complexity, and unpredictability.
Using strong passwords lowers overall risk of a security breach, but strong passwords do not replace the need for other effective security controls. The effectiveness of a password of a given strength is strongly determined by the design and implementation of the factors (knowledge, ownership, inherence). The first factor is the main focus in this article.
The rate at which an attacker can submit guessed passwords to the system is a key factor in determining system security. Some systems impose a time-out of several seconds after a small number (e.g. three) of failed password entry attempts. In the absence of other vulnerabilities, such systems can be effectively secured with relatively simple passwords. However the system must store information about the user passwords in some form and if that information is stolen, say by breaching system security, the user passwords can be at risk.
Passwords are created either automatically (using randomizing equipment) or by a human; the latter case is more common. While the strength of randomly chosen passwords against a brute-force attack can be calculated with precision, determining the strength of human-generated passwords is challenging.
Typically, humans are asked to choose a password, sometimes guided by suggestions or restricted by a set of rules, when creating a new account for a computer system or Internet Web site. Only rough estimates of strength are possible, since humans tend to follow patterns in such tasks, and those patterns can usually assist an attacker. In addition, lists of commonly chosen passwords are widely available for use by password guessing programs. Such lists include the numerous online dictionaries for various human languages, breached databases of plaintext and hashed passwords from various online business and social accounts, along with other common passwords. All items in such lists are considered weak, as are passwords that are simple modifications of them. For some decades, investigations of passwords on multi-user computer systems have shown that 40% or more are readily guessed using only computer programs, and more can be found when information about a particular user is taken into account during the attack.
Systems that use passwords for authentication must have some way to check any password entered to gain access. If the valid passwords are simply stored in a system file or database, an attacker who gains sufficient access to the system will obtain all user passwords, giving the attacker access to all accounts on the attacked system, and possibly other systems where users employ the same or similar passwords. One way to reduce this risk is to store only a cryptographic hash of each password instead of the password itself. Standard cryptographic hashes, such as the Secure Hash Algorithm (SHA) series, are very hard to reverse, so an attacker who gets hold of the hash value cannot directly recover the password. However, knowledge of the hash value lets the attacker quickly test guesses offline. Password cracking programs are widely available that will test a large number of trial passwords against a purloined cryptographic hash.
Improvements in computing technology keep increasing the rate at which guessed passwords can be tested. For example, in 2010, the Georgia Tech Research Institute developed a method of using GPGPU to crack passwords much faster. Elcomsoft invented the usage of common graphic cards for quicker password recovery in August 2007 and soon filed a corresponding patent in the US. As of 2011, commercial products are available that claim the ability to test up to 112,000 passwords per second on a standard desktop computer using a high-end graphics processor. Such a device will crack a 6 letter single-case password in one day. Note that the work can be distributed over many computers for an additional speedup proportional to the number of available computers with comparable GPUs. Special key stretching hashes are available that take a relatively long time to compute, reducing the rate at which guessing can take place. Although it is considered best practice to use key stretching, many common systems do not.
Another situation where quick guessing is possible is when the password is used to form a cryptographic key. In such cases, an attacker can quickly check to see if a guessed password successfully decodes encrypted data. For example, one commercial product claims to test 103,000 WPA PSK passwords per second.
If a password system only stores the hash of the password, an attacker can pre-compute hash values for common passwords variants and for all passwords shorter than a certain length, allowing very rapid recovery of the password once its hash is obtained. Very long lists of pre-computed password hashes can be efficiently stored using rainbow tables. This method of attack can be foiled by storing a random value, called a cryptographic salt, along with the hash. The salt is combined with the password when computing the hash, so an attacker precomputing a rainbow table would have to store for each password its hash with every possible salt value. This becomes infeasible if the salt has a big enough range, say a 32-bit number. Unfortunately, many authentication systems in common use do not employ salts and rainbow tables are available on the Internet for several such systems.
It is usual in the computer industry to specify password strength in terms of information entropy, measured in bits, a concept from information theory. Instead of the number of guesses needed to find the password with certainty, the base-2 logarithm of that number is given, which is the number of "entropy bits" in a password. A password with, say, 42 bits of strength calculated in this way would be as strong as a string of 42 bits chosen randomly, say by a fair coin toss. Put another way, a password with 42 bits of strength would require 242 (4,398,046,511,104) attempts to exhaust all possibilities during a brute force search. Thus, adding one bit of entropy to a password doubles the number of guesses required, which makes an attacker's task twice as difficult. On average, an attacker will have to try half of the possible passwords before finding the correct one.
Random passwords consist of a string of symbols of specified length taken from some set of symbols using a random selection process in which each symbol is equally likely to be selected. The symbols can be individual characters from a character set (e.g., the ASCII character set), syllables designed to form pronounceable passwords, or even words from a word list (thus forming a passphrase).
The strength of random passwords depends on the actual entropy of the underlying number generator; however, these are often not truly random, but pseudo random. Many publicly available password generators use random number generators found in programming libraries that offer limited entropy. However most modern operating systems offer cryptographically strong random number generators that are suitable for password generation. It is also possible to use ordinary dice to generate random passwords. See stronger methods. Random password programs often have the ability to ensure that the resulting password complies with a local password policy; for instance, by always producing a mix of letters, numbers and special characters.
For passwords generated by a process that randomly selects a string of symbols of length, L, from a set of N possible symbols, the number of possible passwords can be found by raising the number of symbols to the power L, i.e. NL. Increasing either L or N will strengthen the generated password. The strength of a random password as measured by the information entropy is just the base-2 logarithm or log2 of the number of possible passwords, assuming each symbol in the password is produced independently. Thus a random password's information entropy, H, is given by the formula
|Symbol set||Symbol count N||Entropy per symbol H|
|Arabic numerals (0–9) (e.g. PIN)||10||3.322 bits|
|hexadecimal numerals (0–9, A–F) (e.g. WEP keys)||16||4.000 bits|
|Case insensitive Latin alphabet (a–z or A–Z)||26||4.700 bits|
|Case insensitive alphanumeric (a–z or A–Z, 0–9)||36||5.170 bits|
|Case sensitive Latin alphabet (a–z, A–Z)||52||5.700 bits|
|Case sensitive alphanumeric (a–z, A–Z, 0–9)||62||5.954 bits|
|All ASCII printable characters except space||94||6.555 bits|
|All ASCII printable characters||95||6.570 bits|
|All extended ASCII printable characters||218||7.768 bits|
|Binary (0–255 or 8 bits or 1 byte)||256||8.000 bits|
|Diceware word list||7776||12.925 bits|
To find the length, L, needed to achieve a desired strength H, with a password drawn randomly from a set of N symbols, one computes
The following table uses this formula to show the required lengths of truly randomly generated passwords to achieve desired password entropies for common symbol sets:
|Desired password entropy H||Arabic numerals||Hexadecimal||Case insensitive Latin alphabet||Case insensitive alphanumeric||Case sensitive Latin alphabet||Case sensitive alphanumeric||All ASCII printable characters||All extended ASCII printable characters||Diceware word list|
|8 bits (1 byte)||3||2||2||2||2||2||2||2||1|
|32 bits (4 bytes)||10||8||7||7||6||6||5||5||3|
|40 bits (5 bytes)||13||10||9||8||8||7||7||6||4|
|64 bits (8 bytes)||20||16||14||13||12||11||10||9||5|
|80 bits (10 bytes)||25||20||18||16||15||14||13||11||7|
|96 bits (12 bytes)||29||24||21||19||17||17||15||13||8|
|128 bits (16 bytes)||39||32||28||25||23||22||20||17||10|
|160 bits (20 bytes)||49||40||35||31||29||27||25||21||13|
|192 bits (24 bytes)||58||48||41||38||34||33||30||25||15|
|224 bits (28 bytes)||68||56||48||44||40||38||35||29||18|
|256 bits (32 bytes)||78||64||55||50||45||43||39||33||20|
People are notoriously poor at achieving sufficient entropy to produce satisfactory passwords. According to one study involving half a million users, the average password entropy was estimated at 40.54 bits. Some stage magicians exploit this inability for amusement, in a minor way, by divining supposed random choices (of numbers, say) made by audience members.
Thus, in one analysis of over 3 million eight-character passwords, the letter "e" was used over 1.5 million times, while the letter "f" was used only 250,000 times. A uniform distribution would have had each character being used about 900,000 times. The most common number used is "1", whereas the most common letters are a, e, o, and r.
Users rarely make full use of larger character sets in forming passwords. For example, hacking results obtained from a MySpace phishing scheme in 2006 revealed 34,000 passwords, of which only 8.3% used mixed case, numbers, and symbols.
The full strength associated with using the entire ASCII character set (numerals, mixed case letters and special characters) is only achieved if each possible password is equally likely. This seems to suggest that all passwords must contain characters from each of several character classes, perhaps upper and lower case letters, numbers, and non-alphanumeric characters. In fact, such a requirement is a pattern in password choice and can be expected to reduce an attacker's "work factor" (in Claude Shannon's terms). This is a reduction in password "strength". A better requirement would be to require a password NOT to contain any word in an online dictionary, or list of names, or any license plate pattern from any state (in the US) or country (as in the EU). If patterned choices are required, humans are likely to use them in predictable ways, such a capitalizing a letter, adding one or two numbers, and a special character. This predictability means that the increase in password strength is minor when compared to random passwords.
Using this scheme, an eight-character human-selected password without upper case letters and non-alphabetic characters is estimated to have 18 bits of entropy. The NIST publication concedes that at the time of development, little information was available on the real world selection of passwords.
Later research into human-selected password entropy using newly available real world data has demonstrated that the NIST scheme does not provide a valid metric for entropy estimation of human-selected passwords. The June 2017 revision of SP 800-63 (Revision 3) drops this approach.
Because national keyboard implementations vary, not all 94 ASCII printable characters can be used everywhere. This can present a problem to an international traveler who wished to log into remote system using a keyboard on a local computer. See keyboard layout. Many hand held devices, such as tablet computers and smart phones, require complex shift sequences to enter special characters.
Authentication programs vary in which characters they allow in passwords. Some do not recognize case differences (e.g., the upper-case "E" is considered equivalent to the lower-case "e"), others prohibit some of the other symbols. In the past few decades, systems have permitted more characters in passwords, but limitations still exist. Systems also vary in the maximum length of passwords allowed.
As a practical matter, passwords must be both reasonable and functional for the end user as well as strong enough for the intended purpose. Passwords that are too difficult to remember may be forgotten and so are more likely to be written on paper, which some consider a security risk. In contrast, others argue that forcing users to remember passwords without assistance can only accommodate weak passwords, and thus poses a greater security risk. According to Bruce Schneier, most people are good at securing their wallets or purses, which is a "great place" to store a written password.
The minimum number of bits of entropy needed for a password depends on the threat model for the given application. If key stretching is not used, passwords with more entropy are needed. RFC 4086, "Randomness Requirements for Security", presents some example threat models and how to calculate the entropy desired for each one. Their answers vary between 29 bits of entropy needed if only online attacks are expected, and up to 96 bits of entropy needed for important cryptographic keys used in applications like encryption where the password or key needs to be secure for a long period of time and stretching isn't applicable. A 2010 Georgia Tech Research Institute study based on unstretched keys recommended a 12-character random password, but as a minimum length requirement.
The upper end is related to the stringent requirements of choosing keys used in encryption. In 1999, an Electronic Frontier Foundation project broke 56-bit DES encryption in less than a day using specially designed hardware. In 2002, distributed.net cracked a 64-bit key in 4 years, 9 months, and 23 days. As of October 12, 2011, distributed.net estimates that cracking a 72-bit key using current hardware will take about 45,579 days or 124.8 years. Due to currently understood limitations from fundamental physics, there is no expectation that any digital computer (or combination) will be capable of breaking 256-bit encryption via a brute-force attack. Whether or not quantum computers will be able to do so in practice is still unknown, though theoretical analysis suggests such possibilities.
Guidelines for choosing good passwords are typically designed to make passwords harder to discover by intelligent guessing. Common guidelines advocated by proponents of software system security include:
Some guidelines advise against writing passwords down, while others, noting the large numbers of password protected systems users must access, encourage writing down passwords as long as the written password lists are kept in a safe place, not attached to a monitor or in an unlocked desk drawer.
The possible character set for a password can be constrained by different web sites or by the range of keyboards on which the password must be entered.
As with any security measure, passwords vary in effectiveness (i.e., strength); some are weaker than others. For example, the difference in weakness between a dictionary word and a word with obfuscation (i.e., letters in the password are substituted by, say, numbers — a common approach) may cost a password cracking device a few more seconds; this adds little strength. The examples below illustrate various ways weak passwords might be constructed, all of which are based on simple patterns which result in extremely low entropy, allowing them to be tested automatically at high speeds.:
There are many other ways a password can be weak, corresponding to the strengths of various attack schemes; the core principle is that a password should have high entropy (usually taken to be equivalent to randomness) and not be readily derivable by any "clever" pattern, nor should passwords be mixed with information identifying the user. On-line services often provide a restore password function that a hacker can figure out and by doing so bypass a password. Choosing hard-to-guess restore password questions can further secure the password.
In December, 2012, William Cheswick wrote an article published in ACM magazine that included the mathematical possibilities of how easy or difficult it would be to break passwords that are constructed using the commonly recommended, and sometimes followed, standards of today. In his article, William showed that a standard eight character alpha-numeric password could withstand a brute force attack of ten million attempts per second, and remain unbroken for 252 days. Ten million attempts each second is the acceptable rate of attempts using a multi-core system that most users would have access to. A much greater degree of attempts, at the rate of 7 billion per second, could also be achieved when using modern GPUs. At this rate, the same 8 character alpha-numeric password could be broken in approximately 30 seconds. Increasing the password complexity to a 13 character alpha-numeric password increases the time needed to crack it to more than 900,000 years at 7 billion attempts per second. This is, of course, assuming the password does not use a common word that a dictionary attack could break much sooner. Using a password of this strength reduces the obligation to change it as often as many organizations require, including the U.S. Government, as it could not be reasonably broken in such a short period of time.
A password policy is a guide to choosing satisfactory passwords. It is intended to:
For example, password expiration is often covered by password policies. Password expiration serves two purposes:
The hardest passwords to crack, for a given length and character set, are random character strings; if long enough they resist brute force attacks (because there are many characters) and guessing attacks (due to high entropy). However, such passwords are typically the hardest to remember. The imposition of a requirement for such passwords in a password policy may encourage users to write them down, store them in PDAs or cellphones, or share them with others as a safeguard against memory failure. While some people consider each of these user resorts to increase security risks, others suggest the absurdity of expecting users to remember distinct complex passwords for each of the dozens of accounts they access. For example, in 2005, security expert Bruce Schneier recommended writing down one's password:
Simply, people can no longer remember passwords good enough to reliably defend against dictionary attacks, and are much more secure if they choose a password too complicated to remember and then write it down. We're all good at securing small pieces of paper. I recommend that people write their passwords down on a small piece of paper, and keep it with their other valuable small pieces of paper: in their wallet.
The following measures may increase acceptance of strong password requirements, if carefully used:
Password policies sometimes suggest memory techniques to assist remembering passwords:
Computer users are generally advised to "never write down a password anywhere, no matter what" and "never use the same password for more than one account." However, an ordinary computer user may have dozens of password-protected accounts. Users with multiple accounts needing passwords often give up and use the same password for every account. When varied password complexity requirements prevent use of the same (memorable) scheme for producing high-strength passwords, oversimplified passwords will often be created to satisfy irritating and conflicting password requirements. A Microsoft expert was quoted as saying at a 2005 security conference: "I claim that password policy should say you should write down your password. I have 68 different passwords. If I am not allowed to write any of them down, guess what I am going to do? I am going to use the same password on every one of them."
If passwords are written down, they should never be kept in obvious places such as address books, Rolodex files, under drawers or keyboards, or behind pictures. Perhaps the worst, but all too common, location is a Post-It note on the computer monitor. Better locations are a safe deposit box or a locked file approved for information of sensitivity comparable to that protected by the password. Most locks on office file cabinets are far from adequate. Software is available for popular hand-held computers that can store passwords for numerous accounts in encrypted form. Another approach is to encrypt by hand on paper and remember the encryption method and key. And another approach is to use a single password or slightly varying passwords for low-security accounts and select distinctly separate strong passwords for a smaller number of high-value applications such as for online banking.
Another effective approach for remembering multiple passwords is to memorize a single "master" password and use software to generate a new password for each application, based on the master password and the application's name. This approach is used by Stanford's PwdHash, Princeton's Password Multiplier, and other stateless password managers. In this approach, protecting the master password is essential, as all passwords are compromised if the master password is revealed, and lost if the master password is forgotten or misplaced.
A reasonable compromise for using large numbers of passwords is to record them in a password manager program, which include stand-alone applications, web browser extensions, or a manager built into the operating system. A password manager allows the user to use hundreds of different passwords, and only have to remember a single password, the one which opens the encrypted password database. Needless to say, this single password should be strong and well-protected (not recorded anywhere). Most password managers can automatically create strong passwords using a cryptographically secure random password generator, as well as calculating the entropy of the generated password. A good password manager will provide resistance against attacks such as key logging, clipboard logging and various other memory spying techniques.