Glimpses from Cryptography: Cryptograpic Hash functions

Hash function is a one way mapping, typically from an an infinite set to a finite set. Hash functions are deterministic functions. All hash functions are not suitable for cryptographic applications, hence we have a set of functions called Cryptographic Hash functions which posses certain properties which makes in suitable for cryptographic applications. For explaining the properties, Consider an infinite set A such that a is a member of A . The elements of set A are mapped to a finite set B. Let the mapping be represented by function h(). According to Pigeonhole principle, it is clear that more than one element of A is mapped to a unique element in B. Be it the case we design a computationally secure hash function with following properties.

• Collision resistance: It is computationally infeasible to find two elements such that both are having same hash value. That is to find a1,a2 such that h(a1)=h(a2)
• Pre-image Resistance: Given a hash value, it is computationally infeasible to find corresponding element from set A. Let b be element of B, then it is infeasible to find a such that b=h(a)
• Second-Preimage resistance: Given a pair (a1,b) such that h(a1)=b, it is computationally infeasible to find a2 such that h(a1)=b=h(a2)
• Pseudorandomness: For a single bit change in the input, the hash value should change significantly

Some of the applications of cryptographic hash functions are listed below

• Digital Signatures
• Password storage
• Malware analysis (using signature of malware)
• Data Integrity Check