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Figure 2-7 The brute force attack. If you know that the key is a number between 1 and 100,000,000,000, you try each number in turn until a number produces something that s not gibberish
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It usually takes very little time to try a key. The attacker can probably write a program that tries many keys per second. Eventually, the attacker could try every possible number between 0 and 100 billion, but that may not be necessary. Once the correct key is found, there s no need to search any more. On average, the attacker will try half of all possible keys in our example, 50 billion keys before finding the correct one. Sometimes it takes more time, sometimes less, but, on average, about half the possible keys must be tried.
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How long would it take an attacker to try 50 billion keys Three years Three days Three minutes Suppose you want to keep your secret safe for at least three years, but it takes an attacker only three minutes to try 50 billion values. Then what do you do You choose a bigger range. Instead of finding a number between 0 and 100 billion, you find a number between 0 and 100 billion billion billion billion. Now the attacker will have to try, on average, many more keys before finding the right one. This concept of the range of possible keys is known as key size. Gold is measured in troy ounces, atoms are measured in moles, and cryptographic keys are measured in bits. If someone asks, How big is that key the answer might be 40 bits, 56 bits, 128 bits, and so on. A 40-bit key means that the range of possible values is from 0 to about 1 trillion. A 56-bit key is 0 to about 72 quadrillion. The range of a 128-bit key is so large that it s easier just to say it s a 128-bit key (see Figure 2-8).
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Figure 2-8 The larger the key size, the greater the range of possible values a key can be. Each bit in each position, whether 0 or 1, is important
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Each bit of key size you add doubles the time required for a brute-force attack. If a 40-bit key takes 3 hours to break, a 41-bit key would take 6 hours, a 42-bit key, 12 hours, and so on. Why Each additional bit doubles the number of possible keys. For example, there are eight possible numbers of size 3 bits:
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000 001 010 011 100 101 110 111
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These are the numbers from zero to seven. Now add one more bit:
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Every number possible with 3 bits is possible with 4 bits, but each of those numbers is possible twice : once with the first bit not set, and again with it set. So if you add a bit, you double the number of possible keys. If you double the number of possible keys, you double the average time it takes for brute-force attack to find the right key. In short, if you want to make the attacker s job tougher, you choose a bigger key. Longer keys mean greater security. How big should a key be Over the years, RSA Laboratories has offered challenges. The first person or organization to crack a particular message wins a money prize. Some of the challenges have been tests of brute-force time. In 1997, a 40-bit key fell in 3 hours, and a 48-bit key lasted 280 hours. In 1999, the Electronic Frontier Foundation found a 56-bit key in 24 hours. In each case, a little more than 50 percent of the key space was searched before the key was found. In January 1997, a 64-bit challenge was issued. As of December 2000, it has still not been solved. In all these situations, hundreds or even thousands of computers were operating cooperatively to break the keys. In fact, with the 56-bit DES challenge that the Electronic Frontier Foundation broke in 24 hours, one of those computers was a custom-built DES cracker. This kind of computer does only one thing: check DES keys. An attacker working secretly would probably not be able to harness the power of hundreds of computers and might not possess a machine built specifically to crack a particular algorithm. That s why, for most attackers, the time it takes to break the key would almost certainly be dramatically higher. On the other hand, if the attacker were a government intelligence agency with enormous resources, the situation would be different. We can devise worst-case scenarios. Let s use as our baseline an exaggerated worst-case scenario: examining 1 percent of the key space of a 56-bit key takes 1 second, and examining 50 percent takes 1 minute (see Table 2-1). Each time that we add a bit to the key size, we double the search time. Currently, 128 bits is the most commonly used symmetric-key size. If technology advances and brute-force attackers can improve on these numbers (maybe they can reduce the 128-bit times to a few years), then we would need to use a 256-bit key. You may be thinking, Technology is always advancing, so I ll have to keep increasing key sizes again and again. Won t there come a time when I ll need a key so big it becomes too unwieldy to handle The answer is
Table 2-1
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