Asymmetric ciphers are used to allow to communicate securely without using a shared secret key. A "public" and a "private"
key are derived from a pair of very large prime numbers. However, it is possible
to demonstrate the formula using small numbers.
RedTitan use RSA (named for its inventors: Ron
Rivest, Adi Shamir, and Leonard Adleman) cryptography. It is a good idea to
choose a popular algorithm because there will be a choice of authentication
mechanisms on the destination computer and the prudent user can get a
second opinion. In some jurisdictions, RSA signatures have a legal status and
may be used to validate a document in much the same way as a handwritten
signature.





Originally postulated by Cliffords Cocks at the
British GCHQ the trapdoor function that became the basis for RSA relies on the
difficulty of factoring the multiple to two very large prime numbers. The
formula is very simple.
 P is any prime number
 Q is any prime number
 E (public exponent) is chosen to be a whole number greater than
1 but less than P*Q
where E and (P  1) * (Q  1) are relatively
prime.
i.e. they have no prime factors in common.
 D (private exponent) is chosen
to be any whole number such that the formula (D * E  1) / (P  1) *
(Q  1) is a whole positive integer.
e.g. any prime exceeding
max(P, Q) (and less than (P*Q)
) would be a valid choice for D.
To encrypt a number T where T <= P * Q apply the formula
(T ^{E}) mod ( P * Q )
To decrypt a number C that
was produced from the encryption formula apply.
(C^{D}) mod (P * Q)
For example,
P = 11
Q = 3
E = 3
D = 7
To encrypt the number 5 calculate (5^{3} mod (11 *3))
== (5 * 5 * 5 mod 33) == (125 mod 33) == 26
To decrypt the number 26 calculate (26^{7} mod (11 * 3
)) == (26*26* 26 *26*26*26*26 mod 33)
==(8031810176 mod 33)== 5
The public key is the pair of numbers (P * Q, E) or (33, 3) and the private key is (D) or the
number 7. In fact, these are the smallest numbers we can choose
and the formula is very easy to calculate by hand. This example could not be
used as a very good cipher  many numbers in the range (0
to 33) encrypt to exactly the same value. Although the
modulus (P *Q) is never published as a pair of prime
numbers, it is easy to guess that 33 can only factored to
3 and 11 and with this information it is
possible to deduce the private key.
Picking the numerically larger value to be
d allows implementing code to take advantage of certain
computational improvements to be had in the encryption/decryption arithmetic if
the original values of p and q are available, as they could be for
the "private" key
RSA becomes very difficult indeed
to crack when much larger pair of prime numbers are chosen. Blocks
of 128 bytes at time can be encrypted using a 1024 bit key. In this case, two
prime 512 bit prime numbers are used to calculate the modulus. Given the modulus
from a public key, a brute force scan for a suitable pair of primes will
take more time than is left in the universe (It is possible that the
CIA know better)
