4) The
encryption function is encrypt(T) = (T^E) mod PQ, where T is the plaintext (a
positive integer) and ^ indicates exponentiation. 5) The decryption function is
decrypt(C) = (C^D) mod PQ, where C is the ciphertext (a positive integer) and ^
indicates exponentiation. (Litterio http://world.std.com/~franl/crypto/rsa-guts.html)
The public key is the pair (PQ, E), and the private key is the number D. The
product PQ is the modulus. E is the public exponent, while D is the private
exponent. It is difficult to obtain the private key D from the public key (PQ,
E) (Frösen 6). If one could factor PQ into P and Q, then once could obtain the
private key D (6-7). Thus the entire security of RSA is predicted on the
assumption that factoring is difficult; an easy factoring method would break RSA
(7). Encryption and authentication take place without any sharing of private
keys: each person uses only other people’s public keys and his or her own
private key (7). Anyone can send an encrypted message or verify a signed
message, using only public keys, but only someone in possession of the correct
private key can decrypt or sign a message. RSA operations are all based on prime
numbers and a series of multiplications. It is easier to do a multiplication
than to undo it. In practical applications, it is common to choose a small
public exponent for the public key. Entire groups of users can use the same
public exponent. This makes encryption faster than decryption and verification
faster and signing. According to RSA Laboratories, when implemented entirely in
software, DES is at least a hundred times faster than RSA (3). Implemented in
hardware, it may outperform the RSA algorithm by a thousand or even ten thousand
times (3). This is primarily due to the fact that the DES S-boxes are simple
table-lookup functions, while RSA depends on large-integer arithmetic (3).
Encryption is used more now than ever before, for sole purposes to keep data
private. It has been used by military and diplomatic forces for years, and now
is used by everyday people.
Bibliography
Frösen, Janne. Practical Cryptosystems and their Strength. Proceedings of HUT
Seminar on Network Security Helsinki University of Technology, February 1996.
Gardner, Martin. Codes, Ciphers and Secret Writing. NY: Dover, 1972. Litterio,
Francis. “The Mathematical Guts of RSA Encryption.” Cryptography.
http://world.std.com/~franl/crypto/rsa-guts.html (15 Nov. 2000)
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