Lecture 4.1: Hash Functions: Introduction

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Lecture 4.1: Hash Functions: Introduction 2/24/2019 Lecture 4.1: Hash Functions: Introduction CS 436/636/736 Spring 2012 Nitesh Saxena

Course Administration 2/24/2019 Course Administration HW1 being graded Solution was emailed Expect HW2 by coming Monday Covers mostly Public Key Cryptography Will be due in 12-14 days 2/24/2019 Lecture 4.1: Hash Functions

Outline of Today’s lecture 2/24/2019 Outline of Today’s lecture Hash Functions Properties Birthday Paradox Generic Design 2/24/2019 Lecture 4.1: Hash Functions

Cryptographic Hash Functions 2/24/2019 Cryptographic Hash Functions Requirements of cryptographic hash functions: Can be applied to data of any length. Output is fixed length, usually very short Relatively easy to compute h(x), given x Function is deterministic Infeasible to get x, given h(x). One-wayness property Given x, infeasible to find y such that h(x) = h(y). Weak-collision resistance property Infeasible to find any pair x and y (x ≠ y) such that h(x) = h(y). Strong-collision resistance property or just collision resistance 2/24/2019 Lecture 4.1: Hash Functions

Some Applications of Hash Functions 2/24/2019 Some Applications of Hash Functions In general, can be used as a checksum for large amounts of data Password hashing Digital signatures Message authentication codes (will study later) Used also in RSA-OAEP, and many other cryptographic constructions 2/24/2019 Lecture 4.1: Hash Functions

Lecture 4.1: Hash Functions 2/24/2019 Hash Output Length How long should be the output (n bits) of a cryptographic hash function? To find collision - randomly select messages and check if hash matches any that we know. Throwing k balls in N = 2n bins. How large should k be, before probability of landing two balls in the same becomes greater than ½? Birthday paradox - a collision can be found in roughly sqrt(N) = 2(n/2) trials for an n bit hash In a group of 23 (~ sqrt(365)) people, at least two of them will have the same birthday (with a probability > ½) Hence n should be at least 160 2/24/2019 Lecture 4.1: Hash Functions

Lecture 4.1: Hash Functions 2/24/2019 Birthday Paradox Probability that hash values of k random messages are distinct is (that is, no collisions) is: 2/24/2019 Lecture 4.1: Hash Functions

Generic Hash Function – Merkle-Damgard Construction 2/24/2019 Generic Hash Function – Merkle-Damgard Construction This design for H() is collision-resistant given that h() is collision resistant Intuitively, this is because there is a avalanche effect – even if the inputs differ in just 1 bit, the outputs will be completely different IV is a known public constant

An Illustrative Example from Wikipedia 2/24/2019 An Illustrative Example from Wikipedia

Lecture 4.1: Hash Functions 2/24/2019 Further Reading Stallings Chapter 11 HAC Chapter 9 2/24/2019 Lecture 4.1: Hash Functions