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1) Why is the RSA
cryptosystem so successful?
The success of the RSA Cipher is based on the fact that it is a secure
"public-key-cryptosystem". Such systems - also called
"asymmetric cryptosystems" - allow users to publish their public
key without revealing their private key.
2) Why are public
key systems so important?
Because they solve the key distribution problem which arises for the
symmetric cryptosystems (i.e. Linear Cipher or one time pad) where each
correspondent has to share a different secret key with each other
correspondents. Similarly, they do too. The table shows the rapidly
increasing number of keys dependent on the number of correspondents:
| #corresp.: |
2 |
3 |
4 |
5 |
6 |
10 |
100 |
1000 |
n |
| #keys: |
1 |
3 |
6 |
10 |
15 |
45 |
4950 |
499500 |
n*(n-1)/2 |
Thus, 100 Internet users require a total of 4950 keys when using any
symmetric cryptosystem. However, only 100 key pairs are needed for symmetric
cryptosystems. The 100 public encoding keys can be looked up at a central
register (similar to a telephone book), whereas each of the 100 private
decoding keys is secretly sheltered by its owner. In this way, the two
correspondents don't have to meet to define a secret key prior to their
correspondence - an enormous advantage i.e. for today's email
communication.
Exercise 1:
On this
page we generate various moduli m. If both primes p and q are less than 20,
how many moduli between 77 and 400 can be formed? Verify your
answer by generating all possible keys and en- and decode
"safe" below for such keys. Only
77, 91, 119, 133, 143, 187, 209, 221, 247 and 323 are possible moduli.
Exercise 2:
Every RSA user
has his own key because he uses a unique (his) modulus m. You may wonder:
Considering the great number of people using the RSA cryptosystem, are there
enough different RSA keys? Answer: Yes. Remember that the modulus
shall be a 200-digit number which allows many more moduli than there
will ever be RSA users. A review questions: Why do these huge moduli prevent eavesdroppers to
figure out the decoding
key d although e and m are publicly known? Given
that m is at least a 200 digit number, no computer in the world is able to
find the factors p and q.
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