# vb.net code 39 reader The Key Distribution Problem and Public-Key Cryptography in Software Draw ANSI/AIM Code 39 in Software The Key Distribution Problem and Public-Key Cryptography

The Key Distribution Problem and Public-Key Cryptography
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Figure 4-13 In a popular 1,024-bit RSA key, the modulus is 1,024 bits, built by multiplying two 512-bit primes
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You may wonder why the modulus has to be the product of two primes. Why can t the modulus itself be a prime number The reason is that for a prime number p, (p) is (p 1). Because your modulus is public, if the modulus were p, a prime number, any attacker would be able to find (p); it s simple subtraction. Armed with (p), an attacker can easily find d. Incidentally, Satomi has a couple of brute force opportunities. First, she could try to find d by trying every value it could possibly be. Fortunately, d is a number as big as the modulus. For a 1,024-bit RSA private key, d is 1,024 bits long (maybe a bit or two smaller). No, brute force on d is not an option. A second possibility is to find p or q. Satomi could get a number b (call it b for brute force candidate) and then compute n b (n divided by b). If that doesn t work (b does not divide n evenly; there is a remainder), she tries another b. She keeps trying until she finds a b that works (one that divides n evenly). That b will be one of the factors of n. And the answer to n b is the other factor. Satomi would then have p and q. But the factors of n are half the size of the modulus (see Technical Note: MultiPrime RSA ). For a 1,024-bit RSA key, p and q are 512 bits each. So Satomi would be trying a brute force attack on a 512-bit number, and
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Technical Note: MultiPrime RSA
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Faster performance is always a goal of programmers, so anything that would speed up the RSA algorithm would be welcome. The first speed improvement came in 1982 from Belgian researchers JeanJacques Quisquater and C. Couvreur. They showed that it s possible to make private key operations (opening a digital envelope) faster if you keep the p and q around, by using what is known as the Chinese Remainder Theorem (CRT). This theorem dates to the fourth century and originated in, as the name implies, China. It s a result of research into how to count columns and columns of soldiers more quickly. Remember that an RSA private key is made up of the two numbers n and d, where n is built by multiplying two primes, p and q. When you have your d, you throw away p and q. According to the theorem, if you don t throw away your p and q, and if, while generating your key pair, you make a few other calculations and save a few more values, the private key operations you perform can run almost three times faster. The fundamental reason is that p and q are smaller than n (there s more to it than that, but at its foundation, that is the reason). Because p and q must be kept private, this technique will not help public key operations. But, as you ll see in the section Performance, RSA public key operations are already rather fast. Recently, people have been looking into using three or more primes to make up n. Here s why. When you multiply two numbers, if you add the sizes of those two numbers you get the size of the result. For example, if you multiply a 512-bit number by a 512-bit number, you get a 1,024-bit number because 512 512 1,024 (it could end up being 1,023 bits, but let s not quibble). Actually, you could multiply a 612-bit number by a 412bit number to get a 1,024-bit result, but for security reasons, it s better to have the numbers the same size or very close. Virtually all programs that generate RSA key pairs find two 512-bit primes and multiply them to make n. If you want a 1,024-bit number as a result of multiplying three smaller numbers, how big should they be One possibility is 341, 341, and 342 bits. If p and q are each 512 bits, and if private key operations are faster because they are smaller than n (which is
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