Hi, :)
  I want to know why the RSAParameters.Exponent is always set to byte
{1,0,1},
what's the meaning of this field?
thanks very much ^_^

Hi,

3 bytes {1,0,1} in RSAParameters.Exponent is a standard recommended public
exponent = 2^16 + 1 = 65537 = 0x10001. It is recommended because it gives
good performance for operations involving public key (due to low hamming
weight). 3 as public exponent gives better performance, but 3 exponent could
not be used to encrypt the same message to more than two different
participants (i.e. public exponent 3, but different modulus) otherwise
message could be easily recovered. With 0x10001 it could be a threat only if
you send the same message encrypted with more than 65536 different public
keys that uses 0x10001 as public exponent - which is much higher threshold
and isn't usually a practical threat. 3 and 2^16+1 as public exponent have
best performance of encryption and signature verification, least amount of
messages that encrypts to themselves (message convolution), guarantees that
private exponent will be big and not be vulnerable to Weiner attack; +++.



RSA function (also called RSA trapdoor permutation) is defined following
way:

n is product of two distinct prime numbers p and q (i.e. n = p*q);

phi(n) is (p-1)*(q-1);

public exponent e is a number relative prime to phi(n) (i.e. gcd(e,phi(n)) =
1);

private exponent d is chosen such that e*d = 1 mod phi(n) [*]

then for any positive m less than n following is true:

(m^e)^d = m mod n



Therefore encryption is simple exponentiation:

cipher = m^e mod n

and decryption is another exponentiation:

cipher^d mod n = (m^e)^d mod n = m



RSA trapdoor permutation is a basis of RSA encryption. RSA encryption is
actually more that just RSA trapdoor permutation and additionally defines a
set of rules (like padding and such) about how to apply RSA trapdoor
permutation to encrypt some plain text in secure way.



If you are interested to find out about internal details of cryptography -
you should read some books. Since you are newbie in crypto - entry level
books, like Bruce Schneier's Applied Cryptography, and/or Neil Ferguson's
and Bruce Schneier's Practical Cryptography would be a good beginning for
you.



-Valery.

http://www.harper.no/valery



[*] note that private exponent only requires to satisfy e*d=1 mod
lcm(p-1,q-1), but since last requires calculation of least common
multiplier, e*d = 1 mod phi(n) is usually used instead.


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