The document summarizes the RSA cryptosystem, which is an asymmetric cryptographic algorithm used for secure data transmission. It works using a public key and private key pair. The public key is used to encrypt messages, while the private key is used to decrypt them. The security of the RSA cryptosystem relies on the difficulty of factoring the product of two large prime numbers. The document outlines the mathematical steps to generate the public and private keys, including using Euler's totient function and choosing exponents that are relatively prime. It then describes how encryption and decryption are performed using modular exponentiation with the keys.

1. Gorham/Weaver
Dr. Barton
27 March 2015
RSA Cryptosystem
Throughout history many civilizations have devised secret codes in order to pass along
information from one party to another. The first people to give a name to this practice were the
Greeks when they were at war with other nations1. Cryptography, or “hidden” in Greek, was the
practice and study of techniques for safe communication in the presence of a third party. The
idea was to keep the message secret during transmission and if a third party intercepted the
message they would not be able to decipher it. Early cryptography was solely concerned with
converting messages into unreadable groups of figures to protect the message’s content during
the time the message was being carried from one place to another (New World 2007).
The process of cryptography begins with the encryption of a message to be sent to
another party. The creator of this encryption shares how to decode the message with the member
he would like to send the message too. So, Party A encrypts a message, sends it to Party B, and
Party B is able to decrypt it. This is shown below in Figure 1.5
Figure 1.
Classes of encryption techniques are designed to secure symmetric cryptographic key material
for transmission using asymmetric (public-key) algorithms. To make this more clear,
symmetric-key algorithms have the same cryptographic keys for both encryption and decryption.
This being said, the RSA Cryptosystem has two different keys for encryption and decryption

2. making it very difficult for an adversary to break the code. For example, if a bank wants to have
multiple customers, they use a public key system, which allows many people to give the bank
locks of their account. The bank itself only has one key that opens everyone’s individual locks.
It is similar to a master key. Public key systems are clumsy to use in transmitting long messages.
Instead they are often used to exchange symmetric keys, which are usually short5. This
symmetric key is then used to encrypt longer messages.
In the past, many nations used a shift when they would crypt a message. If the alphabet
is placed on a number line the letters of each message would be shifted forwards or backwards
during the encryption and the shift would be reversed during the decryption. This is an example
of a symmetric key.
Since the First World War, cryptography and the process of encryption have become very
difficult. Today, the encryption and decryption processes are heavily involved with
mathematical theory. These algorithms are hard to break and almost impossible for an
inexperienced adversary to figure out. The RSA or the Rivest, Shamir, Adleman was the first
practicable public-key cryptosystem1. This cryptosystem uses an asymmetric way to crypt and
decrypt messages. An asymmetric cryptosystem requires two separate keys; one that is public,
and the other, private. Today it is used in order to secure data transmission. In this system, the
encryption key is public while the decryption key is kept a secret. Therefore the key does not
need to be transmitted. Instead of sending how to decrypt the message all you have to send is the
cipher text itself. If the message is intercepted with a symmetric-key system, the way to decrypt
it is included with the cipher text.
The RSA Cryptosystem is an algorithm based on the practical difficulty of factoring the
product of two large prime numbers. In order to create an encryption, the user must base the

3. public key off of two large prime numbers. If the prime numbers are large enough, factoring
them would take even a computer a lengthy period of time. In my opinion, factoring a large
prime number like 44,345,523 to find 2 prime numbers is harder than multiplying two prime
numbers together in order to create a public key.
To start off, the prime numbers must be kept a secret. Since the encryption key is public,
anyone can encrypt a message. However, if the public key is large enough, only someone who
knows the prime numbers can decode the message. So now we first create a key. Then we
choose two distinct prime numbers p and q; these should be of similar digit length and take the
Euler’s Totient Function (phi function) of the two numbers.
Lemma 1: If p is a positive prime number, then Φ(p) = p – 1.
Proof: If p is prime, then since its only positive divisors are 1 and p, all of the integers 1,
2, 3, … , p – 1 are relatively prime to p.
By multiplying p and q together we receive and integer y. Now we take the Euler Totient
function of y.
Lemma 2: For p and q distinct primes, Φ(pq) = Φ(p) Φ(q) = (p - 1)(q - 1).
Proof: There are pq - 1 natural numbers smaller than pq including
p, 2p, 3p, 4p, …, (q - 1)p and q, 2q, 3q, 4q, …, (p - 1)q, all of which have a factor other
than 1 in common with pq.
Since there are (q - 1) + (p - 1) of these numbers, this leaves
(pq) – 1 – (p - 1) – (q - 1) = pq – p – q + 1 = (p - 1)(q – 1)
This shows that Φ(pq) can be applied to both p and q at the same time.
So by using this function designated Φ we can compute
Φ(y) = Φ(p) Φ(q) = (p - 1)(q - 1)

4. Now we choose an integer x so that 1< x <Φ(y) and the gcd(x, Φ(y)) = 1
In other words, x and Φ(y) are coprime.
Lemma 3: If p is prime and k is a positive integer, then gcd(n, pk) = 1 if and only if p does
not divide n.
Proof: First of all if p|n, then gcd(n, pk) ≠ 1 because p also divides pk.
Conversely, if gcd(n, pk) ≠ 1 then n and pk are not relatively prime, i.e., they share a
common factor greater than 1.
But the only factors of pk are powers of p and so p|n.
When choosing x, x must be coprime and thus the gcd(x, Φ(y)) = 1
Lemma 4: If p is prime and k is a positive integer, then Φ(pk) = pk – pk-1.
Proof: There are precisely pk-1 integers between 1 and pk that are divisible by p, namely
{p, 2p, 3p, …, pk-1p}.
So of the pk that are positive and less than or equal to pk, pk – pk-1 of them do not have a factor in
common with pk. This shows (pk) = pk – pk-1.
Lemma 5: For n > 2, Φ(n) is an even integer.
Proof: Divide this problem into two cases:
Case 1: n is a power of 2, i.e., n = 2k (k > 1). Then
Φ(n) = Φ(2k) = 2k – 2k-1 = 2k-1(2 – 1) = 2k-1, which is even.
Case 2: n is not a power of 2. In this case n is divisible by an odd prime p, so n = pkm (k
≥ 1) and gcd(pk, m) = 1.
So, Φ(n) = Φ(pkm) = (pk – pk-1) Φ(m) = pk-1(p-1) Φ(m)
Which is even (because 2|(p-1)). So now x is released as the public key exponent.Now we must
determine d (private key).

5. d ≡ x -1(mod Φ(y))
Thus, we solve for d given that dx ≡ 1 (mod Φ(y)). And thus d is kept as the private key
exponent.
Euler’s Theorem shows us that for some prime number n, if n and a are coprime then n
and a have no common factors, n|(aΦ(n) – 1) and so, aΦ(n) ≡ 1(mod n). So far now we have the
public key which consists of the modulus y and the public exponent x. The private key however
consists of the modulus y and the private exponent d. Thus the variables a, b, Φ(y) must be kept
secret because they are used to calculate d.
The next step is to create the encryption. Therefore, give out your public key (y,x) and
keep the d value a secret. Now we wish to send out a message called TAP. Make TAP an
integer by using ASCII notation designated by b, such that T is 20, A is 01, and P is 16. Next
you produce the cipher text c corresponding to c ≡ bx (mod y). This is done through modular
exponentiation and next we send the c value to the recipient.
The decryption is done by recovering the b value from c by using the private key as
exponent d. This is done by computing:
b ≡ cd mod y.
This is a very simplified version of the RSA cryptosystem. The only problem with this
system is that a padding scheme is used in order to scramble the ciphertext before the algorithm
begins. This simplified version has a modular arithmetic system that is not 1 to 1, meaning that
you could used several different mods in order to obtain your message b. However, if a proper
padding scheme is used, the RSA system is a very effective code when securing data during
transmission. Your everyday credit card use and internet passwords are all, for the most part,
protected by this well built algorithm.

6. Euler’s Totient Function
If n is a positive integer with gcd(a, n) = 1, then aΦ(n) ≡ 1(mod n).
Proof: If n =1, we wish to show that a0 ≡ 1(mod 1) which is obviously true. So assume
that n > 1 and let S = {a1, a2, …, aΦ(n)} be the set of positive integers less than n that are
relatively prime to n.
Since gcd(a, n) = 1, and we know that aa1, aa2, …, aaΦ(n) are congruent to a1, a2, …, aΦ(n)
in some order.
aa1 ≡ a’1(mod n),
aa2 ≡ a’2(mod n),
…
aaΦ(n) ≡ a’ Φ(n)(mod n).

7. Work Cited
1http://www.faqs.org/espionage/Cou-De/Cryptology-History.html
2http://searchsecurity.techtarget.com/definition/cryptology
3http://mathworld.wolfram.com/RSAEncryption.html
4https://crypto.stanford.edu/~dabo/papers/RSA-survey.pdf
5http://www.laits.utexas.edu/~anorman/BUS.FOR/course.mat/SSim/history.html