Cryptography Tutorial

RSA algorithm is an asymmetric cryptographic algorithm as it creates 2 different keys for the purpose of encryption and decryption. It is public key cryptography as one of the keys involved is made public. RSA stands for Ron Rivest, Adi Shamir and Leonard Adleman who first publicly described it in 1978. RSA makes use of prime numbers (arbitrary large numbers) to function. Key generation. The keys for the RSA algorithm are generated in the following way: Choose two distinct prime numbers p and q. For security purposes, the integers p and q should be chosen at random, and should be similar in magnitude but differ in length by a few digits to make factoring harder.

Randomness and the RSA Algorithm Entropy is randomness used by cryptographic systems to generate cryptographic keys. Good entropy is necessary to generate strong keys. A recent story highlights the results of using bad entropy on the RSA key generation itself. The report concludes that 'device manufacturers, website and network administrators. RSA keys are generated using carefully designed cryptographic random number generators. Under the hood these may make use of cryptographic hash functions like sha512 as one piece of the algorithm, but you would typically treat these as a black box for the purpose of key generation. – James Reinstate Monica Polk Jun 3 '17 at 20:28. GITHUB RSA Algorithm is widely used in secure data transmission. It was invented by Rivest, Shamir, and Adleman in the year 1978 and hence the name is RSA. It is an asymmetric cryptography algorithm which basically means this algorithm works on two different keys i.e. Public Key and Private Key.

Cryptography Useful Resources

Selected Reading

Public Key Cryptography

Unlike symmetric key cryptography, we do not find historical use of public-key cryptography. It is a relatively new concept.

Symmetric cryptography was well suited for organizations such as governments, military, and big financial corporations were involved in the classified communication.

With the spread of more unsecure computer networks in last few decades, a genuine need was felt to use cryptography at larger scale. The symmetric key was found to be non-practical due to challenges it faced for key management. This gave rise to the public key cryptosystems.

The process of encryption and decryption is depicted in the following illustration −

The most important properties of public key encryption scheme are −

Different keys are used for encryption and decryption. This is a property which set this scheme different than symmetric encryption scheme.
Each receiver possesses a unique decryption key, generally referred to as his private key.
Receiver needs to publish an encryption key, referred to as his public key.
Some assurance of the authenticity of a public key is needed in this scheme to avoid spoofing by adversary as the receiver. Generally, this type of cryptosystem involves trusted third party which certifies that a particular public key belongs to a specific person or entity only.
Encryption algorithm is complex enough to prohibit attacker from deducing the plaintext from the ciphertext and the encryption (public) key.
Though private and public keys are related mathematically, it is not be feasible to calculate the private key from the public key. In fact, intelligent part of any public-key cryptosystem is in designing a relationship between two keys.

There are three types of Public Key Encryption schemes. We discuss them in following sections −

RSA Cryptosystem

This cryptosystem is one the initial system. It remains most employed cryptosystem even today. The system was invented by three scholars Ron Rivest, Adi Shamir, and Len Adleman and hence, it is termed as RSA cryptosystem.

We will see two aspects of the RSA cryptosystem, firstly generation of key pair and secondly encryption-decryption algorithms.

Generation of RSA Key Pair

Each person or a party who desires to participate in communication using encryption needs to generate a pair of keys, namely public key and private key. The process followed in the generation of keys is described below −

Generate the RSA modulus (n)
- Select two large primes, p and q.
- Calculate n=p*q. For strong unbreakable encryption, let n be a large number, typically a minimum of 512 bits.
Find Derived Number (e)
- Number e must be greater than 1 and less than (p − 1)(q − 1).
- There must be no common factor for e and (p − 1)(q − 1) except for 1. In other words two numbers e and (p – 1)(q – 1) are coprime.
Form the public key
- The pair of numbers (n, e) form the RSA public key and is made public.
- Interestingly, though n is part of the public key, difficulty in factorizing a large prime number ensures that attacker cannot find in finite time the two primes (p & q) used to obtain n. This is strength of RSA.
Generate the private key
- Private Key d is calculated from p, q, and e. For given n and e, there is unique number d.
- Number d is the inverse of e modulo (p - 1)(q – 1). This means that d is the number less than (p - 1)(q - 1) such that when multiplied by e, it is equal to 1 modulo (p - 1)(q - 1).
- This relationship is written mathematically as follows −

The Extended Euclidean Algorithm takes p, q, and e as input and gives d as output.

Example

An example of generating RSA Key pair is given below. (For ease of understanding, the primes p & q taken here are small values. Practically, these values are very high).

Let two primes be p = 7 and q = 13. Thus, modulus n = pq = 7 x 13 = 91.
Select e = 5, which is a valid choice since there is no number that is common factor of 5 and (p − 1)(q − 1) = 6 × 12 = 72, except for 1.
The pair of numbers (n, e) = (91, 5) forms the public key and can be made available to anyone whom we wish to be able to send us encrypted messages.
Input p = 7, q = 13, and e = 5 to the Extended Euclidean Algorithm. The output will be d = 29.
Check that the d calculated is correct by computing −

Hence, public key is (91, 5) and private keys is (91, 29).

Encryption and Decryption

Once the key pair has been generated, the process of encryption and decryption are relatively straightforward and computationally easy.

Interestingly, RSA does not directly operate on strings of bits as in case of symmetric key encryption. It operates on numbers modulo n. Hence, it is necessary to represent the plaintext as a series of numbers less than n.

RSA Encryption

Suppose the sender wish to send some text message to someone whose public key is (n, e).
The sender then represents the plaintext as a series of numbers less than n.
To encrypt the first plaintext P, which is a number modulo n. The encryption process is simple mathematical step as −

In other words, the ciphertext C is equal to the plaintext P multiplied by itself e times and then reduced modulo n. This means that C is also a number less than n.
Returning to our Key Generation example with plaintext P = 10, we get ciphertext C −

RSA Decryption

The decryption process for RSA is also very straightforward. Suppose that the receiver of public-key pair (n, e) has received a ciphertext C.
Receiver raises C to the power of his private key d. The result modulo n will be the plaintext P.

Total war rome 2 factions. Returning again to our numerical example, the ciphertext C = 82 would get decrypted to number 10 u

Key Generation Using Rsa Algorithm