Digital Signature and Message Authentication

Digital Signature (DS): 

Before the advent of computer communication, handwritten signature were commonly used as a promising tool for validating proposals. 
With the rise of computer communication, handwritten signature were gradually replaced by digital signatures. 
Thus, the interplay between encryption and signature methods was enabled by their digitalization and the introduction of computational complexity as a basic for security. 
Loosely speaking, a scheme for unforgeable signature requires that
1. each user can efficiently generate their own signature on documents of their choice, 
2. each user can efficiently verify whether a given string is valid signature of another user on a specific document, and
3. no one can efficiently produce the signature of another user on documents that the users did not sign. 

Message Authentication (MA):

First, consider a scenario where an adversary is monitoring the channel and may alter the message sent transmitted through it. The communicating parties to authenticate their messages so that the intended recipient can distinguish an original message from a modified one. Loosely speaking, a scheme for message authentication requires that
1. each communicating party can efficiently generate an authentication tag for any message of their choice. 
2. each communicating party can efficiently verify whether a string is a valid authentication tag for a given message, and
3. no external adversary (i.e., a party other than the communicating parties) can efficiently produce authentication tags for messages not sent by the communicating parties. 

Relationship between DS and MA:
MA does not require third parties (who may be dishonest) to verify the validity of authentication tags produced by the designated users, whereas in the setting of DS schemes, it is required that such third parties be able to verify the validity of signatures produced by other users. 
Thus, DS provide a solution to MA problem. On the other hand, MA scheme does not necessarily constitute a digital signature scheme. 

Hence, within the scope of cryptography, DS are generally preferred over MA, as DS schemes have a broader range of applications. 

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