Wireless Security.pdf - PDF Archive

Cryptography 291
in RSA is less simple than the encryption operations, but it reveals the security of the
algorithm. Each RSA key is derived from a couple of large prime numbers, usually
denoted as p and q . The public modulus n is the product of p and q, and this is where the
security comes in 3 —it is generally considered to be impossible to determine the prime
factors of very large numbers with any method other than brute-force. Given a sufficiently
large number, even a modern computer cannot calculate the prime factors of that number
in a reasonable amount of time, which is usually defined in terms of years of computing
power. A secure algorithm typically implies thousands or millions of years (or more) of
computing power is required for a brute force attack. The trick to RSA is that the private
key ( d ) is derived from p and q such that exponentiation with modulus n will result in
the retrieval of the plaintext message from the encrypted message (which was calculated
by applying the public exponent e to the original message). All of the mathematics
here basically leads to the result that RSA is slow, and something needs to be done if
it is going to be utilized on a lower-performance system. As we mentioned, hardware
assistance is really the only way to really speed up the algorithm, but there is a method
that utilizes a property of the modular math used by RSA—it is based on an ancient
concept called the Chinese Remainder Theorem, or CRT. CRT basically divides the RSA
operation up using the prime factors p and q used to derive the public and private keys.
Instead of calculating a full private exponent from p and q, the private key is divided
amongst several CRT factors that allow the algorithm to be divided up into smaller
operations. This doesn’t translate into much performance gain unless it is implemented on
a parallel processor system, but any gain can be useful for a relatively slow, inexpensive
embedded CPU.
Using CRT to speed up RSA is pretty much the only known software method for
optimizing RSA—it is simply a CPU-cycle-eating algorithm. For this reason, hardware
optimization is really the only choice. This was recognized early on, and in the early 1980s
there was even work to design a chip dedicated to RSA operations. Today’s PCs are now
fast enough to run RSA entirely in software at the same level of performance as the original
hardware-based solutions. However, given that a large number of modern embedded
processors possess similar resources and performance characteristics to computers in the
3 This description of the RSA algorithm is derived from “PKCS #1 v2.1: RSA Cryptography
Standard (June 14, 2002),” from RSA Security Inc. Public-Key Cryptography Standards (PKCS),
www.rsa.com.
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