Public Key Encryption Algorithm

4.3.2 Public Key Encryption Algorithm

We have introduced the problems with symmetric encryption algorithms. But how can we overcome them? In 1976, Whitfield Diffie and Martin Hellman proposed the public key encryption algorithm in their paper “New Directions in Cryptography”. Inspired by the safe and mailbox, they imagined the symmetric encryption as the safe, and the key as its code combination. The person who knows the code can open the safe and close it as well. Then can we replace the safe with the mailbox that allows everyone to put in letters but only those who get a key to open? Through an analogical analysis, they proposed a system, in which encryption and decryption involve different keys, and the decryption key cannot be deduced from the encryption key. If possible, the encryption key can be made public, while the decryption key is kept secret. That is to say, anyone who wants to correspond to Alice can encrypt the message with the public key, but the encrypted message can only be decrypted by Alice. This is the concept of public key algorithm.

Ever since the public key encryption was proposed, many encryption methods had been proposed. There are three kinds of systems that are considered secure and valid: large integer factorization system (e.g., RSA), elliptic curve cryptosystems (e.g., ECC) and discrete logarithmic system (e.g., DL).

Now the most commonly used public key system RSA was proposed by Rivet, Shamir, and Adelman, the security of which is based on large integer factorization, which is a famous nut in mathematics and researchers have not found solution yet. RSA system is the most typical method in the public key system, and it is widely found in most products with digital signature or encrypted by the public key.

From an abstract point of view, the public key system is a trapdoor one-way function. A function is said to be one-way function if ( ) f x is easy to compute for

all x in its domain, while f 1 () y is impossible to compute for almost all y in the range even f is known. But if some additional information (trapdoor information)

is given, f 1 () y will become easier computed, and then the one- way function f is called trapdoor one-way function. The public key encryption system is

designed according to this principle; the additional information is called the key. The strength of this kind of encryption relies on the computational complexity. Now there are two kinds of popular public key systems: one is based on large integer factorization, typically RSA; the other kind is based on discrete logarithm, such as ElGamal system and elliptic curve public key system. Because the capacity of factorizing large integer is becoming stronger and stronger, there is always a requirement of the key length to ensure the security of RSA. Now the 768bit RSA is not safe enough, and usually 1024bit is recommended. However for discrete logarithm, although the security is better, it is not mature yet.

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The communication of public key encryption is outlined as follows: (1) Alice and Bob choose a public key system. (2) Bob conveys his public key to Alice. (3) Alice uses Bob’s public key to encrypt her message and sends it to Bob. (4) Bob uses his private key to decrypt the message from Alice and reads it. Since Alice’s public key is not necessarily kept secret, it can be sent via an

insecure channel, which is a successful solution to the key distribution and management problem. Furthermore, public key system can be applied to digital signature, which would be introduced later.

The public key system also has its weak points:

Slow: approximately 1/1000 to 1/100 of symmetric encryption system.

Vulnerable to chosen plain text attack. Since C fMK ( , 1 ) in which K is 1 known, if there are only n possibilities of M, the attacker can encrypt all n kinds of possible plain texts and see which one of the results is the same with C, then M is known.

The weak points of public key system make it not suitable to encrypt messages with a very small space or a large amount of data. Actually the symmetric encryption algorithm and public key encryption algorithm are usually combined to serve applicable purposes; the public key system is used to distribute the key, and then the distributed key is used for encryption. That is the so-called mixed cipher mechanism.