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250 CHAPTER 12. RSA
two large primes. Euler’s Theorem allows us to decipher messages by raising
the ciphertext to a power modulo the same product. The deciphering power
is the multiplicative inverse of the enciphering power modulo a number easily
computed from the two original primes.
When dealing with large primes, and here large means approximately 300
digits each, exponentiation must be done carefully so that the size of the numbers
does not overwhelm the computer or calculator being used. In general,
to compute the value (a b )%n, where a, b and n could all be sizable values and
n is either prime or the product of two primes, first reduce a modulo n, then
reduce b modulo n − 1 (if n is prime) or modulo (p − 1)(q − 1) (if n = pq is a
product of primes). Next, a “binary chart” is used to simultaneously write the
remainder of b in binary and compute the powers of the remainder of a. Finally,
the necessary powers of the remainder of a are combined to produce the final
value.
The RSA system has become the world’s most widely-known Public Key
cryptosystem. The private information is the deciphering key and the two
primes, and the public information is the product of the primes and the enciphering
exponent. RSA also provides for digital signatures, a method by which
the sender can prove their identity. Although RSA is slow when compared with
popular private key systems, it pairs easily with any such system: encipher your
message with a private key system, encipher the key with RSA and then send
the enciphered key and the ciphertext as a two-part message.
Outside of poor uses of the system (e.g., a bad choice of parameters), which
are generally easy to avoid, the RSA system seems very difficult to break. Its
security seems to depend on the difficulty of factoring the product of the two
large primes. While factoring has been studied for many, many years, there no
publicly-known general method that will factor the product of two well-chosen
primes in any realistically small amount of time. So, for the time being at least,
RSA is one of the most secure ciphersystems known.
12.13 Topics and Techniques
1. What is Fermat’s Theorem How to use it Under what conditions does
it apply
2. Is raising a number to a large power modulo a prime ever the same as
raising that same number to a smaller power modulo the same prime
Explain.
3. What is “double modular arithmetic”
4. If I have a large number raised to a large number that I want to reduce
modulo a smaller prime, how do I go about making this problem more
manageable
5. What is the binary expansion of a number How to find it