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19 Encryption Decryption Questions and Answers:
1 :: What is Public-Key Cryptography?
Traditional cryptography is based on the sender and receiver of a message knowing and using the same secret key: the sender uses the secret key to encrypt the message, and the receiver uses the same secret key to decrypt the message. This method is known as secret-key or symmetric cryptography. The main problem is getting the sender and receiver to agree on the secret key without anyone else finding out. If they are in separate physical locations, they must trust a courier, or a phone system, or some other transmission medium to prevent the disclosure of the secret key being communicated. Anyone who overhears or intercepts the key in transit can later read, modify, and forge all messages encrypted or authenticated using that key. The generation, transmission and storage of keys is called key management; all cryptosystems must deal with key management issues. Because all keys in a secret-key cryptosystem must remain secret, secret-key cryptography often has difficulty providing secure key management, especially in open systems with a large number of users.
2 :: What are the Advantages and Disadvantages of Public-Key Cryptography Compared with Secret-Key Cryptography?
The primary advantage of public-key cryptography is increased security and convenience: private keys never need to transmitted or revealed to anyone. In a secret-key system, by contrast, the secret keys must be transmitted (either manually or through a communication channel), and there may be a chance that an enemy can discover the secret keys during their transmission.
Another major advantage of public-key systems is that they can provide a method for digital signatures. Authentication via secret-key systems requires the sharing of some secret and sometimes requires trust of a third party as well. As a result, a sender can repudiate a previously authenticated message by claiming that the shared secret was somehow compromised by one of the parties sharing the secret. For example, the Kerberos secret-key authentication system involves a central database that keeps copies of the secret keys of all users; an attack on the database would allow widespread forgery. Public-key authentication, on the other hand, prevents this type of repudiation; each user has sole responsibility for protecting his or her private key. This property of public-key authentication is often called non-repudiation.
Another major advantage of public-key systems is that they can provide a method for digital signatures. Authentication via secret-key systems requires the sharing of some secret and sometimes requires trust of a third party as well. As a result, a sender can repudiate a previously authenticated message by claiming that the shared secret was somehow compromised by one of the parties sharing the secret. For example, the Kerberos secret-key authentication system involves a central database that keeps copies of the secret keys of all users; an attack on the database would allow widespread forgery. Public-key authentication, on the other hand, prevents this type of repudiation; each user has sole responsibility for protecting his or her private key. This property of public-key authentication is often called non-repudiation.
3 :: Do Digital Signatures Help Detect Altered Documents and Transmission Errors?
A digital signature is superior to a handwritten signature in that it attests to the contents of a message as well as to the identity of the signer. As long as a secure hash function is used, there is no way to take someone's signature from one document and attach it to another, or to alter a signed message in any way. The slightest change in a signed document will cause the digital signature verification process to fail. Thus, public-key authentication allows people to check the integrity of signed documents. If a signature verification fails, however, it will generally difficult to determine whether there was an attempted forgery or simply a transmission error.
4 :: What is a One-Way Function?
A one-way function is a mathematical function that is significantly easier to perform in one direction (the forward direction) than in the opposite direction (the inverse direction). It might be possible, for example, to compute the function in seconds but to compute its inverse could take months or years. A trap-door one-way function is a one-way function where the inverse direction is easy given a certain piece of information (the trap door), but difficult otherwise.
5 :: What is the Significance of One-Way Functions for Cryptography?
Public-key cryptosystems are based on (presumed) trap-door one-way functions. The public key gives information about the particular instance of the function; the private key gives information about the trap door. Whoever knows the trap door can perform the function easily in both directions, but anyone lacking the trap door can perform the function only in the forward direction. The forward direction is used for encryption and signature verification; the inverse direction is used for decryption and signature generation.
In almost all public-key systems, the size of the key corresponds to the size of the inputs to the one-way function; the larger the key, the greater the difference between the efforts necessary to compute the function in the forward and inverse directions (for someone lacking the trap door). For a digital signature to be secure for years, for example, it is necessary to use a trap-door one-way function with inputs large enough that someone without the trap door would need many years to compute the inverse function.
In almost all public-key systems, the size of the key corresponds to the size of the inputs to the one-way function; the larger the key, the greater the difference between the efforts necessary to compute the function in the forward and inverse directions (for someone lacking the trap door). For a digital signature to be secure for years, for example, it is necessary to use a trap-door one-way function with inputs large enough that someone without the trap door would need many years to compute the inverse function.
6 :: How is RSA used for Encryption in Practice?
RSA is combined with a secret-key cryptosystem, such as DES, to encrypt a message by means of an RSA digital envelope.
Suppose Alice wishes to send an encrypted message to Bob. She first encrypts the message with DES, using a randomly chosen DES key. Then she looks up Bob's public key and uses it to encrypt the DES key. The DES-encrypted message and the RSA-encrypted DES key together form the RSA digital envelope and are sent to Bob. Upon receiving the digital envelope, Bob decrypts the DES key with his private key, then uses the DES key to decrypt to message itself. This combines the high speed of DES with the key-management convenience of RSA.
Suppose Alice wishes to send an encrypted message to Bob. She first encrypts the message with DES, using a randomly chosen DES key. Then she looks up Bob's public key and uses it to encrypt the DES key. The DES-encrypted message and the RSA-encrypted DES key together form the RSA digital envelope and are sent to Bob. Upon receiving the digital envelope, Bob decrypts the DES key with his private key, then uses the DES key to decrypt to message itself. This combines the high speed of DES with the key-management convenience of RSA.
7 :: What is the ElGamal Cryptosystem?
The ElGamal system is a public-key cryptosystem based on the discrete logarithm problem. It consists of both encryption and signature algorithms. The encryption algorithm is similar in nature to the Diffie-Hellman key agreement protocol.
8 :: What are Elliptic Curves?
Elliptic curves are mathematical constructions from number theory and algebraic geometry, which in recent years have found numerous applications in cryptography.
An elliptic curve can be defined over any field (e.g., real, rational, complex). However, elliptic curves used in cryptography are mainly defined over finite fields. An elliptic curve consists of elements (x, y) satisfying the equation
y2 = x3 + ax + b
together with a single element denoted O called the "point at infinity," which can be visualized as the point at the top and bottom of every vertical line. Addition of two points on a elliptic curve is defined according to a set of simple rules (e.g., point p1 plus point p2 is equal to point -p3 in Figure 2). The addition operation in an elliptic curve is the counterpart to modular multiplication in common public-key cryptosystems, and multiple addition is the counterpart to modular exponentiation.
An elliptic curve can be defined over any field (e.g., real, rational, complex). However, elliptic curves used in cryptography are mainly defined over finite fields. An elliptic curve consists of elements (x, y) satisfying the equation
y2 = x3 + ax + b
together with a single element denoted O called the "point at infinity," which can be visualized as the point at the top and bottom of every vertical line. Addition of two points on a elliptic curve is defined according to a set of simple rules (e.g., point p1 plus point p2 is equal to point -p3 in Figure 2). The addition operation in an elliptic curve is the counterpart to modular multiplication in common public-key cryptosystems, and multiple addition is the counterpart to modular exponentiation.
9 :: What are Elliptic Curve Cryptosystems?
Elliptic curve cryptosystems are analogs of public-key cryptosystems such as RSA and ElGamal, in which modular multiplication is replaced by the elliptic curve addition operation.
The curves used in elliptic curve analogs of discrete logarithm cryptosystems are normally of the form
y2 = x3 + ax + b (mod p),
where p is prime. The problem tapped by the discrete logarithm analogs in elliptic curves is the elliptic curve logarithm problem, defined as follows: given a point G on an elliptic curve with order r (number of points on the curve) and another point Y on the curve, find a unique x (0 x r - 1) such that Y = xG, i.e., Y is the xth multiple of G.
The curves used in elliptic curve analogs of discrete logarithm cryptosystems are normally of the form
y2 = x3 + ax + b (mod p),
where p is prime. The problem tapped by the discrete logarithm analogs in elliptic curves is the elliptic curve logarithm problem, defined as follows: given a point G on an elliptic curve with order r (number of points on the curve) and another point Y on the curve, find a unique x (0 x r - 1) such that Y = xG, i.e., Y is the xth multiple of G.
10 :: What are Knapsack Cryptosystems?
The Merkle-Hellman knapsack cryptosystem is a public-key cryptosystem that was first published in 1978. It is commonly referred to as the knapsack cryptosystem. It is based on the subset sum problem in combinatorics. The problem involves selecting a number of objects with given weights from a large set such that the sum of the weights is equal to a pre-specified weight. This is considered to be a difficult problem to solve in general, but certain special cases of the problem are relatively easy to solve, which serve as the "trapdoor" of the system. The-single iteration knapsack cryptosystem introduced in 1978 was broken by Shamir. Merkle then published the multiple-iteration knapsack problem which was broken by Brickell [Bri85]. Merkle offered a $100 reward for anybody able to crack the single iteration knapsack and a $1000 reward for anybody able to crack the multiple iteration cipher from his own pocket. When they were cracked, he promptly paid up.