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The Key Distribution Problem and Public-Key Cryptography
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Figure 4-23 The man-in-themiddle attack
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Key Sizes
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The bigger the key, the greater the level of security and the slower any public-key algorithm will run. You want the algorithm to run as fast as possible but maintain a particular level of security. The question is, how low can you go before you jeopardize security The conventional wisdom is that a 1,024-bit RSA or DH key is equivalent in security to a 160-bit ECC key. There is a little contention on that issue, but research continues. In this book, when making comparisons, we look at 1,024-bit RSA or DH, and 160-bit ECC. With RSA, the modulus is made up of three primes; with DH, the private value is 160 bits. In April 2000, RSA Labs published a paper that analyzed how long it would take to break the RSA algorithm at various key sizes if an attacker had $10 million to throw at the problem. Table 4-1 summarizes the research; the symmetric key and ECC key columns are there for comparison. With ECC, you could probably get the same results with smaller key
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Table 4-1
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Symmetric Key (Size in Bits) 56 80 96 128 ECC Key (Size in Bits) 112 160 192 256 RSA Key (Size in Bits) 430 760 1,020 1,620
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Time to Break Keys of Various Sizes with $10 Million to Spend
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Time to Break Less than 5 minutes 600 months 3 million years 10 years
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Number of Machines 105 4,300 114 0.16
Amount of Memory Trivial 4GB 170GB 120TB
sizes. However, the assumption in the report is that the public key algorithm should use a key size at least twice as long as the symmetric key (regardless of performance) for security reasons. The table says that with $10 million, an attacker could buy 105 specially made computers to crack a 56-bit symmetric key, a 112-bit ECC key, or a 430-bit RSA key in a few minutes. Actually, that $10 million would probably buy more than 105 machines, but 105 is all it would take. With the same amount of money, at the next key level the attacker could buy 4,300 machines specially built to solve the problem; at the next key level, 114, and at the next level, 0.16. Why does the money buy fewer machines as the key size increases The reason is that the amount of required memory increases. The base computer is the same, but to break bigger keys, the attacker needs more memory (120 terabytes, or about 120 trillion bytes, in the case of a 1,620-bit RSA key), and buying memory would eat up the budget. In fact, the attacker will probably need more than $10 million to break a 1,620-bit RSA key because that amount of money would only buy 0.16, or about 1/6, of a machine.
Performance
If no algorithm wins on security, you might think that you should choose the fastest one. But there is no simple answer there. Comparing the per-
The Key Distribution Problem and Public-Key Cryptography
formance of the public key operations (initiating the contact, or creating the digital envelope) shows that RSA is significantly faster than ECC, which in turn is faster than DH. For the private key operations (receiving the contact or opening the digital envelope), ECC is somewhat faster than DH, and both are faster than RSA. For many machines, though, the difference in performance is negligible. The two times might be 0.5 milliseconds and 9 milliseconds. Even though one algorithm may be 18 times faster, there s no discernible difference between times that are that fast. But if the processor performing the action is a slow device, such as a smart card, a Palm device, or other handheld device, the difference might be 0.5 seconds versus 9 seconds. Or maybe one of the correspondents is a server that must make many connections, maybe several per second. Then the comparison might be 111 per second versus 2,000 per second. Another factor with ECC is whether you use acceleration tables to speed the private key operations. If you do, you must store extra values in addition to your key. Those extra values amount to about 20,000 bytes. If the device is a server, that s no problem but will a smart card or handheld device have that kind of storage space So the most suitable algorithm depends on which is more important public-key or private-key operations in your application. Table 4-2 lists estimates from RSA Security Engineering on the relative performance of the two algorithms. The baseline is an RSA public-key operation, which is 1 unit. As shown in the table, if a particular computer can create an RSA digital envelope in 1 millisecond, it would take that same computer 13 milliseconds to open it. Or it would take that same computer 18 milliseconds to initiate an ECDH exchange and 2 milliseconds to receive one using acceleration tables.
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