What is the length of the cryptographic key used in the Data Encryption Standard DES

Today, the RSA algorithm forms the security backbone of a large number of well-known secu- rity infrastructures produced by companies like Microsoft, Nokia, and Cisco. The RSA algorithm depends upon the computational difficulty inherent in factoring large prime numbers. Each user of the cryptosystem generates a pair of public and private keys using the algorithm described in the following steps: 1. Choose two large prime numbers approximately 200 digits each, labeled p and q. 2. Compute the product of those two numbers, n = p q. 3. Select a number, e, that satisfies the following two requirements: a. e is less than n.

b. e and n – 1q – 1 are relatively prime—that is, the two numbers have no common fac-

tors other than 1. 4. Find a number, d, such that ed – 1 mod p – 1q – 1 = 0. 5. Distribute e and n as the public key to all cryptosystem users. Keep d secret as the private key. If Alice wants to send an encrypted message to Bob, she generates the ciphertext C from the plaintext P using the following formula where e is Bob’s public key and n is the product of p and q created during the key generation process: C = Pe mod n When Bob receives the message, he performs the following calculation to retrieve the plain- text message: P = Cd mod n El Gamal In Chapter 9, you learned how the Diffie-Hellman algorithm uses large integers and modular arithmetic to facilitate the secure exchange of secret keys over insecure communications chan- nels. In 1985, Dr. T. El Gamal published an article describing how the mathematical principles behind the Diffie-Hellman key exchange algorithm could be extended to support an entire pub- lic key cryptosystem used for the encryption and decryption of messages. Merkle-Hellman Knapsack Another early asymmetric algorithm, the Merkle-Hellman Knapsack algorithm, was developed the year after RSA was publicized. Like RSA, it’s also based upon the difficulty of performing factoring operations, but it relies upon a component of set theory known as superincreasing sets rather than on large prime numbers. Merkle-Hellman was proven ineffective when it was broken in 1984.