(vi) Data Encryption Standard (DES)
DES is a popular cryptographic system [2] used by governments and private companies around the world. It is based on a symmetric key algorithm and uses a shared secret key that is known only to the sender and receiver. It was designed by IBM and approved by the National Bureau of Standards (NBS4) in 1976. It is a block cipher and a message is split into 64-bit message blocks. The algorithm is employed in reverse to decrypt each cipher text block.
Today, DES is considered to be insecure for many applications as its key size (56 bits) is viewed as being too small, and the cipher has been broken in less than 24 h. This has led to it being withdrawn as a standard and replaced by the Advanced Encryption Standard (AES), which uses a larger key of 128 bits or 256 bits.
The DES algorithm uses the same secret 56-bit key for encryption and decryption. The key consists of 56 bits taken from a 64-bit key that includes 8 parity bits. The parity bits are at position 8, 16,…, 64, and so every eighth bit of the 64-bit key is discarded leaving behind only the 56-bit key.
The algorithm is then applied to each 64-bit message block and the plaintext message block is converted into a 64-bit cipher text block. An initial permutation is first applied to M to create M′, and M′is divided into a 32-bit left half L0and a 32-bit right half R0. There are then 16 iterations, with the iterations having a left half and a right half:
Li¼ Ri1
Ri¼ Li1fðRi1;KiÞ
The functionfis a function that takes a 32-bit right half and a 48-bit round key Ki (each Ki contains a different subset of the 56-bit key) and produces a 32-bit output. Finally, the pre-cipher text (R16, L16) is permuted to yield thefinal cipher text C. The functionfoperates on half a message block and involves Table 10.3.
The decryption of the cipher text is similar to the encryption and it involves running the algorithm in reverse.
DES has been implemented on a microchip. However, it has been superseded in recent years by AES due to security concerns with its small 56-bit key size.
The AES uses a key size of 128 bits or 256 bits.
message does not mean that the person has sufficient information to decrypt messages.
The public key cryptosystem is based on the Table10.4:
The advantages and disadvantages of public key cryptosystems Table10.5:
The implementation of public key cryptosystems is based ontrapdoor one-way functions. A functionf: X!Y is a trapdoor one-way function if
• fis easy to computer
• f−1is difficult to compute
• f−1 is easy to compute if a trapdoor (secret information associated with the function) becomes available.
Table 10.3 DES Encryption Step Description
1 Expansion of the 32-bit half block to 48 bits (by duplicating half of the bits) 2 The 48-bit result is combined with a 48-bit subkey of the secret key using an XOR
operation
3 The 48-bit result is broken into 8 * 6 bits and passed through 8 substitution boxes to yield 8 * 4 = 32 bits
(This is the core part of the encryption algorithm)
4 The 32-bit output is rearranged according to afixed permutation
Message M
Encryption C = Eek(M)
Decryption M= Ddk(C)
Message M Public Channel
(Insecure)
Hostile Attack (Enemy)
Decryption Key (Private) Encryption Key
(Public)
Fig. 10.7 Public key cryptosystem Table 10.4 Public key encryption system
Item Description
1 It uses the concept of a key pair (ek,dk)
2 One half of the pair can encrypt messages and the other half can decrypt messages 3 One key is private and one key is public
4 The private key is kept secret and the public key is published (but associated with trading partner)
5 The key pair is associated with exactly one trading partner
A function satisfying just the first two conditions above is termed a one-way function.
Examples of Trapdoor and One-way Functions
(i) The function f: pq!n (wherep and qare primes) is a one-way function since it is easy to compute. However, the inverse functionf−1is difficult to compute problem for largensince there is no efficient algorithm to factorize a large integer into its prime factors (integer factorization problem).
(ii) The functionfg, N:x!gx(mod N) is a one-way function since it is easy to compute. However, the inverse functionf−1is difficult to compute as there is no efficient method to determinexfrom the knowledge of gx(mod N) and g and N (the discrete logarithm problem).
(iii) The functionfk, N:x!xk(mod N) (where N =pqandpandqare primes) andkk′1 (modu(n)) is a trapdoor function. It is easy to compute but the inverse off(thekth root modulo N) is difficult to compute. However, if the trapdoork′is given then fcan easily be inverted as (xk)k′ x(mod N)
10.5.1 RSA Public Key Cryptosystem
Rivest, Shamir and Adleman proposed a practical public key cryptosystem (RSA) based on primality testing and integer factorization in the late 1970s.
The RSA algorithm wasfiled as a patent (Patent No. 4,405, 829) at the U.S. Patent Office in December 1977. The RSA public key cryptosystem is based on the following assumptions:
Table 10.5 Advantages and disadvantages of public key cryptosystems
Advantages Disadvantages
Only the private key needs to be kept secret Public keys must be authenticated The distribution of keys for encryption is convenient
as everyone publishes their public key and the private key is kept private
It is slow and uses more computer resources
It provides message authentication as it allows the use of digital signatures (which enables the recipient to verify that the message is really from the particular sender)
Security Compromise is possible (if private key compromised)
The sender encodes with the private key that is known only to sender. The receiver decodes with the public key and therefore knows that the message is from the sender
Loss of private key may be irreparable (unable to decrypt messages)
Detection of tampering (digital signatures enable the receiver to detect whether message was altered in transit)
Provides for nonrepudiation
10.5 Public Key Systems 167
• It is straightforward tofind two large prime numbers.
• The integer factorization problem is infeasible for large numbers
The algorithm is based on modnarithmetic, where nis a product of two large prime numbers.
The encryption of a plaintext messageMto produce the cipher text C is given by CMeðmodnÞ;
whereeis the public encryption key,Mis the plaintext,Cis the cipher text, andnis the product of two large primes pand q. Botheand n are made public, and eis chosen such that 1 <e</(n), where/(n) is the number of positive integers that are relatively prime ton.
The cipher textCis decrypted by
MCdðmodnÞ;
wheredis the private decryption key that is known only to the receiver, anded1 (mod/(n)) and dand /(n) are kept private.
The calculation of /(n) is easy if bothp and q are known, as it is given by /(n) = (p −1)(q −1). However, its calculation for large n is infeasible ifp and qare unknown.
ed1ðmod/ð ÞÞn
)ed¼1þk/ð Þfor somen k2Z
We discussed Euler’Theorem in Chap. 3, and this result states that ifaandnare positive integers with gcd(a,n) = 1 thena/(n)1 (modn). Therefore, M/(n)1 (modn) and Mk/(n)1 (modn). The decryption of the cipher text is given by:
CdðmodnÞ MedðmodnÞ M1þk/ð ÞnðmodnÞ M1Mk/ð ÞnðmodnÞ M:1ðmodnÞ MðmodnÞ
10.5.2 Digital Signatures
The RSA public key cryptography may also be employed to obtain digital signa- tures. SupposeAwishes to send a secure message toBas well as a digital signature.
This involves signature generation using the private key, and signature verification using the public key. The steps involved are: (Table10.6):
The National Institute of Standards and Technology (NIST) proposed an algo- rithm for digital signatures in 1991. The algorithm is known as the Digital Signature Algorithm (DSA) and later became the Digital Signature Standard (DSS).