ABSTRACT

Encryption plays an essential role in protecting the privacy of electronic information against
threats from a variety of potential attackers. In so doing, modern cryptography employs a combi-
nation of conventional or symmetric cryptographic systems for encrypting data and public key or
asymmetric systems for managing the keys used by the symmetric systems. Assessing the strength
required of the symmetric cryptographic systems is therefore an essential step in employing cryp-
tography for computer and communication security.

Technology readily available today (late 1995) makes brute-force attacks against cryptographic
systems considered adequate for the past several years both fast and cheap. General purpose
computers can be used, but a much more efficient approach is to employ commercially available
Field Programmable Gate Array (FPGA) technology. For attackers prepared to make a higher
initial investment, custom-made, special-purpose chips make such calculations much faster and
significantly lower the amortized cost per solution.

As a result, cryptosystems with 40-bit keys offer virtually no protection at this point against
brute-force attacks. Even the U.S. Data Encryption Standard with 56-bit keys is increasingly
inadequate. As cryptosystems often succumb to `smarter' attacks than brute-force key search, it
is also important to remember that the keylengths discussed here are the minimum needed for
security against the computational threats considered.

Fortunately, the cost of very strong encryption is not significantly greater than that of weak
encryption. Therefore, to provide adequate protection against the most serious threats --- well-
funded commercial enterprises or government intelligence agencies --- keys used to protect data
today should be at least 75 bits long. To protect information adequately for the next 20 years in
the face of expected advances in computing power, keys in newly-deployed systems should be at
least 90 bits long.

1. Encryption Plays an Essential Role in Protecting the Privacy of Electronic Information.

1.1 There is a need for information security.

As we write this paper in late 1995, the development of electronic commerce and the Global In-
formation Infrastructure is at a critical juncture. The dirt paths of the middle ages only became
highways of business and culture after the security of travelers and the merchandise they carried
could be assured. So too the information superhighway will be an ill-traveled road unless informa-
tion, the goods of the Information Age, can be moved, stored, bought, and sold securely. Neither
corporations nor individuals will entrust their private business or personal data to computer net-
works unless they can assure their information's security.

Today, most forms of information can be stored and processed electronically. This means a wide
variety of information, with varying economic values and privacy aspects and with a wide variation
in the time over which the information needs to be protected, will be found on computer networks.
Consider the spectrum:

• Electronic Funds Transfers of millions or even billions of dollars, whose short term security is
  essential but whose exposure is brief;
• A company's strategic corporate plans, whose confidentiality must be preserved for a small
  number of years;
• A proprietary product (Coke formula, new drug design) that needs to be protected over its
  useful life, often decades; and
• Information private to an individual (medical condition, employment evaluation) that may
  need protection for the lifetime of the individual.

1.2 Encryption can provide strong confidentiality protection.

Encryption is accomplished by scrambling data using mathematical procedures that make it ex-
tremely difficult and time consuming for anyone other than authorized recipients --- those with
the correct decryption keys --- to recover the plain text. Proper encryption guarantees that the
information will be safe even if it falls into hostile hands.

Encryption --- and decryption --- can be performed by either computer software or hardware.
Common approaches include writing the algorithm on a disk for execution by a computer central
processor; placing it in ROM or PROM for execution by a microprocessor; and isolating storage
and execution in a computer accessory device (smart card or PCMCIA card).

The degree of protection obtained depends on several factors. These include: the quality of the
cryptosystem; the way it is implemented in software or hardware (especially its reliability and the
manner in which the keys are chosen); and the total number of possible keys that can be used to
encrypt the information. A cryptographic algorithm is considered strong if:

1. There is no shortcut that allows the opponent to recover the plain text without using brute
force to test keys until the correct one is found; and

2. The number of possible keys is sufficiently large to make such an attack infeasible.

The principle here is similar to that of a combination lock on a safe. If the lock is well designed
so that a burglar cannot hear or feel its inner workings, a person who does not know the combination
can open it only by dialing one set of numbers after another until it yields.

The sizes of encryption keys are measured in bits and the difficulty of trying all possible keys
grows exponentially with the number of bits used. Adding one bit to the key doubles the number
of possible keys; adding ten increases it by a factor of more than a thousand.

There is no definitive way to look at a cipher and determine whether a shortcut exists. Nonethe-
less, several encryption algorithms --- most notably the U.S Data Encryption Standard (DES) ---
have been extensively studied in the public literature and are widely believed to be of very high
quality. An essential element in cryptographic algorithm design is thus the length of the key, whose
size places an upper bound on the system's strength.

Throughout this paper, we will assume that there are no shortcuts and treat the length of the
key as representative of the cryptosystem's workfactor --- the minimum amount of effort required
to break the system. It is important to bear in mind, however, that cryptographers regard this as
a rash assumption and many would recommend keys two or more times as long as needed to resist
brute-force attacks. Prudent cryptographic designs not only employ longer keys than might appear
to be needed, but devote more computation to encrypting and decrypting. A good example of this
is the popular approach of using triple-DES: encrypting the output of DES twice more, using a
total of three distinct keys.

Encryption systems fall into two broad classes. Conventional or symmetric cryptosystems -
those in which an entity with the ability to encrypt also has the ability to decrypt and vice versa
- are the systems under consideration in this paper. The more recent public key or asymmetric
cryptosystems have the property that the ability to encrypt does not imply the ability to decrypt.
In contemporary cryptography, public-key systems are indispensable for managing the keys of
conventional cryptosystems. All known public key cryptosystems, however, are subject to shortcut
attacks and must therefore use keys ten or more times the lengths of those discussed here to achieve
the an equivalent level of security.

Although computers permit electronic information to be encrypted using very large keys, ad-
vances in computing power keep pushing up the size of keys that can be considered large and thus
keep making it easier for individuals and organizations to attack encrypted information without
the expenditure of unreasonable resources.

1.3 There are threats from a variety of potential attackers.

Threats to confidentiality of information come from a number of directions and their forms depend
on the resources of the attackers. `Hackers,' who might be anything from high school students to
commercial programmers, may have access to mainframe computers or networks of workstations.
The same people can readily buy inexpensive, off-the-shelf, boards, containing Field Programmable
Gate Array (FPGA) chips that function as `programmable hardware' and vastly increase the ef-
fectiveness of a cryptanalytic effort. A startup company or even a well-heeled individual could
afford large numbers of these chips. A major corporation or organized crime operation with `seri-
ous money' to spend could acquire custom computer chips specially designed for decryption. An
intelligence agency, engaged in espionage for national economic advantage, could build a machine
employing millions of such chips.

1.4 Current technology permits very strong encryption for effectively the same cost as weaker encryption.

It is a property of computer encryption that modest increases in computational cost can produce
vast increases in security. Encrypting information very securely (e.g., with 128-bit keys) typically
requires little more computing than encrypting it weakly (e.g., with 40-bit keys). In many applica-
tions, the cryptography itself accounts for only a small fraction of the computing costs, compared
to such processes as voice or image compression required to prepare material for encryption.

One consequence of this uniformity of costs is that there is rarely any need to tailor the strength
of cryptography to the sensitivity of the information being protected. Even if most of the infor-
mation in a system has neither privacy implications nor monetary value, there is no practical or
economic reason to design computer hardware or software to provide differing levels of encryption
for different messages. It is simplest, most prudent, and thus fundamentally most economical, to
employ a uniformly high level of encryption: the strongest encryption required for any information
that might be stored or transmitted by a secure system.

2. Readily Available Technology Makes Brute-Force Decryption Attacks Faster and Cheaper

The kind of hardware used to mount a brute-force attack against an encryption algorithm depends
on the scale of the cryptanalytic operation and the total funds available to the attacking enterprise.
In the analysis that follows, we consider three general classes of technology that are likely to be
employed by attackers with differing resources available to them. Not surprisingly, the cryptana-
lytic technologies that require larger up-front investments yield the lowest cost per recovered key,
amortized over the life of the hardware.

It is the nature of brute-force attacks that they can be parallelized indefinitely. It is possible
to use as many machines as are available, assigning each to work on a separate part of the prob-
lem. Thus regardless of the technology employed, the search time can be reduced by adding more
equipment; twice as much hardware can be expected to find the right key in half the time. The
total investment will have doubled, but if the hardware is kept constantly busy finding keys, the
cost per key recovered is unchanged.

At the low end of the technology spectrum is the use of conventional personal computers or
workstations programmed to test keys. Many people, by virtue of already owning or having access
to the machines, are in a position use such resources at little or no cost. However, general purpose
computers --- laden with such ancillary equipment as video controllers, keyboards, interfaces, mem-
ory, and disk storage --- make expensive search engines. They are therefore likely to be employed
only by casual attackers who are unable or unwilling to invest in more specialized equipment.

A more efficient technological approach is to take advantage of commercially available Field
Programmable Gate Arrays. FPGAs function as programmable hardware and allow faster imple-
mentations of such tasks as encryption and decryption than conventional processors. FPGAs are
a commonly used tool for simple computations that need to be done very quickly, particularly
simulating integrated circuits during development.

FPGA technology is fast and cheap. The cost of an AT&T ORCA chip that can test 30 million
DES keys per second is $200. This is 1,000 times faster than a PC at about one-tenth the cost!
FPGAs are widely available and, mounted on cards, can be installed in standard PCs just like sound cards, modems, or extra memory.

FPGA technology may be optimal when the same tool must be used for attacking a variety
of different cryptosystems. Often, as with DES, a cryptosystem is sufficiently widely used to
justify the construction of more specialized facilities. In these circumstances, the most cost-effective
technology, but the one requiring the largest initial investment, is the use of Application-Specific
Integrated Circuits (ASICs). A $10 chip can test 200 million keys per second. This is seven times
faster than an FPGA chip at one-twentieth the cost.

Because ASICs require a far greater engineering investment than FPGAs and must be fabricated
in quantity before they are economical, this approach is only available to serious, well-funded oper-
ations such as dedicated commercial (or criminal) enterprises and government intelligence agencies.

3. 40-Bit Key Lengths Offer Virtually No Protection

Current U.S. Government policy generally limits exportable mass market software that incorporates
encryption for confidentiality to using the RC2 or RC4 algorithms with 40-bit keys. A 40-bit key
length means that there are 2 40 possible keys. On average, half of these (2 39 ) must be tried to
find the correct one. Export of other algorithms and key lengths must be approved on a case by
case basis. For example, DES with a 56-bit key has been approved for certain applications such as
financial transactions.

The recent successful brute-force attack by two French graduate students on Netscape's 40-
bit RC4 algorithm demonstrates the dangers of such short keys. These students at the Ecole
Polytechnique in Paris used `idle time' on the school's computers, incurring no cost to themselves
or their school. Even with these limited resources, they were able to recover the 40-bit key in a few
days.

There is no need to have the resources of an institution of higher education at hand, however.
Anyone with a modicum of computer expertise and a few hundred dollars would be able to attack
40-bit encryption much faster. An FPGA chip --- costing approximately $400 mounted on a card
--- would on average recover a 40-bit key in five hours. Assuming the FPGA lasts three years and
is used continuously to find keys, the average cost per key is eight cents.

A more determined commercial predator, prepared to spend $10,000 for a set-up with 25 ORCA
chips, can find 40-bit keys in an average of 12 minutes, at the same average eight cent cost. Spending
more money to buy more chips reduces the time accordingly: $300,000 results in a solution in an
average of 24 seconds; $10,000,000 results in an average solution in 0.7 seconds.

As already noted, a corporation with substantial resources can design and commission custom
chips that are much faster. By doing this, a company spending $300,000 could find the right 40-bit
key in an average of 0.18 seconds at 1/10th of a cent per solution; a larger company or government
agency willing to spend $10,000,000 could find the right key on average in 0.005 seconds (again
at 1/10th of a cent per solution). (Note that the cost per solution remains constant because we
have conservatively assumed constant costs for chip acquisition --- in fact increasing the quantities
purchased of a custom chip reduces the average chip cost as the initial design and set-up costs are
spread over a greater number of chips.)

These results are summarized in Table I. 

4. Even DES with 56-Bit Keys Is Increasingly Inadequate

4.1 DES is no panacea today.

The Data Encryption Standard (DES) was developed in the 1970s by IBM and NSA and adopted
by the U.S. Government as a Federal Information Processing Standard for data encryption. It was
intended to provide strong encryption for the government's sensitive but unclassified information.
It was recognized by many, even at the time DES was adopted, that technological developments
would make DES's 56-bit key exceedingly vulnerable to attack before the end of the century.

Today, DES may be the most widely employed encryption algorithm and continues to be a
commonly cited benchmark. Yet DES-like encryption strength is no panacea. Calculations show
that DES is inadequate against a corporate or government attacker committing serious resources.
The bottom line is that DES is cheaper and easier to break than many believe.

As explained above, 40-bit encryption provides inadequate protection against even the most
casual of intruders, content to scavenge time on idle machines or to spend a few hundred dollars.
Against such opponents, using DES with a 56-bit key will provide a substantial measure of security.
At present, it would take a year and a half for someone using $10,000 worth of FPGA technology
to search out a DES key. In ten years time an investment of this size would allow one to find a
DES key in less than a week.

The real threat to commercial transactions and to privacy on the Internet is from individuals
and organizations willing to invest substantial time and money. As more and more business and
personal information becomes electronic, the potential rewards to a dedicated commercial predator
also increase significantly and may justify the commitment of adequate resources.

A serious effort --- on the order of $300,000 --- by a legitimate or illegitimate business could
find a DES key in an average of 19 days using off-the-shelf technology and in only 3 hours using a
custom developed chip. In the latter case, it would cost $38 to find each key (again assuming a 3
year life to the chip and continual use). A business or government willing to spend $10,000,000 on
custom chips, could recover DES keys in an average of 6 minutes, for the same $38 per key.

At the very high end, an organization --- presumably a government intelligence agency --- willing
to spend $300,000,000 could recover DES keys in 12 seconds each! The investment required is large but not unheard of in the intelligence community. It is less than the cost of the Glomar Explorer,
built to salvage a single Russian submarine, and far less than the cost of many spy satellites. Such
an expense might be hard to justify in attacking a single target, but seems entirely appropriate
against a cryptographic algorithm, like DES, enjoying extensive popularity around the world.

There is ample evidence of the danger presented by government intelligence agencies seeking
to obtain information not only for military purposes but for commercial advantage. Congressional
hearings in 1993 highlighted instances in which the French and Japanese governments spied on
behalf of their countries' own businesses. Thus, having to protect commercial information against
such threats is not a hypothetical proposition.

4.2 There are smarter avenues of attack than brute force.

It is easier to walk around a tree than climb up and down it. There is no need to break the window
of a house to get in if the front door is unlocked.

Calculations regarding the strength of encryption against brute-force attack are worst case
scenarios. They assume that the ciphers are in a sense perfect and that attempts to find shortcuts have failed. One important point is that the crudest approach --- searching through the keys -  is
entirely feasible against many widely used systems. Another is that the keylengths we discuss are
always minimal. As discussed earlier, prudent designs might use keys twice or three times as long
to provide a margin of safety.

4.3 The analysis for other algorithms is roughly comparable.

The above analysis has focused on the time and money required to find a key to decrypt information
using the RC4 algorithm with a 40-bit key or the DES algorithm with its 56-bit key, but the results
are not peculiar to these ciphers. Although each algorithm has its own particular characteristics, the
effort required to find the keys of other ciphers is comparable. There may be some differences as the
result of implementation procedures, but these do not materially affect the brute-force breakability
of algorithms with roughly comparable key lengths.

Specifically, it has been suggested at times that differences in set-up procedures, such as the
long key-setup process in RC4, result in some algorithms having effectively longer keys than others.
For the purpose of our analysis, such factors appear to vary the effective key length by no more
than about eight bits.

5. Appropriate Key Lengths for the Future - A Proposal

Table I summarizes the costs of carrying out brute-force attacks against symmetric cryptosystems
with 40-bit and 56-bit keys using networks of general purpose computers, Field Programmable Gate
Arrays, and special-purpose chips.

It shows that 56 bits provides a level of protection --- about a year and a half --- that would be
adequate for many commercial purposes against an opponent prepared to invest $10,000. Against an
opponent prepared to invest $300,000, the period of protection has dropped to the barest minimum
of 19 days. Above this, the protection quickly declines to negligible. A very large, but easily
imaginable, investment by an intelligence agency would clearly allow it to recover keys in real time.

What workfactor would be required for security today? For an opponent whose budget lay
in the $10 to 300 million range, the time required to search out keys in a 75-bit keyspace would
be between 6 years and 70 days. Although the latter figure may seem comparable to the `barest
minimum' 19 days mentioned earlier, it represents --- under our amortization assumptions --- a cost
of $19 million and a recovery rate of only five keys a year. The victims of such an attack would
have to be fat targets indeed.

Because many kinds of information must be kept confidential for long periods of time, assessment
cannot be limited to the protection required today. Equally important, cryptosystems --- especially
if they are standards --- often remain in use for years or even decades. DES, for example, has
been in use for more than 20 years and will probably continue to be employed for several more. In
particular, the lifetime of a cryptosystem is likely to exceed the lifetime of any individual product
embodying it.

A rough estimate of the minimum strength required as a function of time can be obtained
by applying an empirical rule, popularly called `Moore's Law,' which holds that the computing
power available for a given cost doubles every 18 months. Taking into account both the lifetime
of cryptographic equipment and the lifetime of the secrets it protects, we believe it is prudent to
require that encrypted data should still be secure in 20 years. Moore's Law thus predicts that the
keys should be approximately 14 bits longer than required to protect against an attack today.

Bearing in mind that the additional computational costs of stronger encryption
are modest, we strongly recommend a minimum key-length of 90 bits for symmetric
cryptosystems.

It is instructive to compare this recommendation with both Federal Information Processing
Standard 46, The Data Encryption Standard (DES), and Federal Information Processing Standard
185, The Escrowed Encryption Standard (EES). DES was proposed 21 years ago and used a 56-bit
key. Applying Moore's Law and adding 14 bits, we see that the strength of DES when it was
proposed in 1975 was comparable to that of a 70-bit system today. Furthermore, it was estimated
at the time that DES was not strong enough and that keys could be recovered at a rate of one per
day for an investment of about twenty-million dollars. Our 75-bit estimate today corresponds to
61 bits in 1975, enough to have moved the cost of key recovery just out of reach. The Escrowed
Encryption Standard, while unacceptable to many potential users for other reasons, embodies a
notion of appropriate key length that is similar to our own. It uses 80-bit keys, a number that lies
between our figures of 75 and 90 bits.

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