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Cryptographic Hash Function Cryptographic Hash Function Stefan Lucks Bauhaus-Universität Weimar Summer 2019 –1– Stefan Lucks Hash Fun. (2019) Before taking this course, you must pass an introduction to Cryptography, first. This could, e.g., be “Introduction to Modern Cryptography” (Bachelor/Master), or some crypto course from your previous Bachelors’ program, . I You must not do both the “Introduction to Modern Cryptography” and this course, in the same semester. I You are not admitted to “Introduction to Modern Cryptography”, but you can do this this course, if you did study cryptography before. I You must not do this course, but you are admitted to “Introduction to Modern Cryptography”, if you did not study cryptography before. I Feel free to submit a copy of your transcripts of records from previous studies to my secretary, if you are not sure what counts as introduction to cryptography. I’ll immediately check to where you are admitted. Before the exam, we’ll require such a copy, anyway. –2– Stefan Lucks Hash Fun. (2019) General Information Some Information for You I I’ll switch between using the beamer and the blackboard. I When I am using the blackboard, you should make notes. I Information on the WWW: I Slides I Homework problems, and reading tasks I Code-example I Links to download further information and tools. –3– Stefan Lucks Hash Fun. (2019) General Information Motivation I Cryptographic hash functions are denoted the I “Workhorses”, I “Swiss army knife”, and I “Duct tape” of cryptography. I Since 2004: Many hash functions used in practice under attack. I 2007–2012: Competition for a new hash standard (Our Submission Skein has been one of five finalists) I We will also deal with Password-Hashing. Passwords are “everywhere”, to identify humans to systems, to unlock cryptographic keys, . I March 2014 – June 2015: Password Hash Competition (PHC) (Special recognition for our Submission Catena submission) –4– Stefan Lucks Hash Fun. (2019) General Information Learning Goals Learning goals: I Methods from theoretical and practical cryptography I Cryptanalysis (i.e., attacks) I Security proofs I Understanding of the state of current research in an important sub-field of cryptography. –5– Stefan Lucks Hash Fun. (2019) General Information 1: Introduction meat mincing machine (“Fleischwolf”) –6– Stefan Lucks Hash Fun. (2019) 1: Introduction What is / What Does a Hash Function? I A HF H maps a potentially long message M to small (n-bit) outputs H(M). I Consider H(M) as a “fingerprint” to identify M. I Collisions are inputs x 6= y with h(x) = h(y). Collisions are bad. If it is feasible to find a collision, the hash function is insecure. I Beyond security, you want a hash function to be fast. (Except when you are doing password hashing . ) Example: c ∗ c h :(f0; 1g ) ! f0; 1g , h(x1;:::; xn) = x1 ⊕ · · · ⊕ xn: –7– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.1: Hash Functions for Hash Tables I Hash table: data structure to implement an associative array I Example: search in a phone book I Advantages: fast access (O(1) if all goes well), small storage I Disadvantages: collisions slow down the access –8– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.1: Hash Tables Division Method I The table has p entries from 0 to p − 1 (p prime). I Write message x as a string of n m-bit values: x = (x1;:::; xn) m with xi 2 f0; 1g for 1 ≤ i ≤ n I Compute h1,..., hn: h1 = x1 h2 = (2 ∗ h1 + x2) mod p . hi = (2 ∗ hi−1 + xi) mod p and set finally h(x) = h(x1;:::; xn) = hn: –9– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.1: Hash Tables 1.2: Cryptographic Hash Functions Example use cases I Discover unauthorized manipulations of the message When first initialized, Open Source Tripwire scans the file sy- stem as directed by the administrator and stores information on each file scanned in a database. At a later date the same files are scanned and the results compared against the sto- red values in the database. Changes are reported to the user. Cryptographic hashes are employed to detect changes in a file without storing the entire contents of the file in the database. I Or in the context of digital signatures (! next slide) I Or for “Commitments” (! blackboard). –10– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.2: Cryptographic HF Digital Signatures (Hash-then-Sign) I Sign message of any length: I Typically in two steps: Hash 1. Apply hash function H to compute a short n-bit fingerprint of message 2. Apply signature scheme for n-bit messages. Hash –11– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.2: Cryptographic HF Classical security requirements Consider a hash function h : X!Y: Collision Resistance: Infeasible to find a pair x 6= y with x; y 2 X and h(x) = h(y) Second-preimage Resistance: Given a random x 2R X , infeasible to find y 2 X with h(x) = h(y). (Pure) Preimage Resistance: Definition 1: Given a random z 2R Y, infeasible to find x 2 X with h(x) = z Definition 2: Given z = h(x), where x 2R X is randomly chosen; 0 0 infeasible to find any x 2R X with h(x ) = z, regardless of x = x0 or not. Also important (a few slides ahead): “Behave like a random oracle” –12– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.2: Cryptographic HF Example Attacking the cryptographically weak division-method hash function Consider h, based on the division method with prime p = 7: 1. Find x with h(x) = 1. 2. Given x = (00110011001100), find h(x). 3. Find any x0 6= x with h(x) = h(x0). 4. Find x with h(x) = 0. 5. Find x with h(x) = 5. 6. Homework: Repeat steps 1–5 for p = 2857. –13– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.2: Cryptographic HF Relationship between security definitions and example attacks See blackboard for details Relationship: I Collision-resistant ) second-preimage-resistant. I But not (second-preimage-resistant ) collision resistant). Generic: Let h : f0; 1g∗ ! f0; 1gn be an n-bit HF. I Preimages and second preimages in time 2n. I Collisions in time 2n=2 (!) (note that the attack given here also requires memory of 2n=2; later we will see that an attack in roughly the same time with O(1) units of memory is possible) –14– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.2: Cryptographic HF 1.3: Random Oracles The Problem: I Collision, second and pure preimage resistance sometimes insufficient I The exact properties a hash function should have are difficult to model/describe . But we would need to define them to prove the security of complex cryptosystems using a hash function and other primitives. An apparent Solution: Assume the hash function behaves like an “ideal hash function”. –15– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.3: Random Oracles Random oracle = Ideal hash function . which we do not have in practice I For all X 2 f0; 1g∗: choose random H(X) := Y 2 f0; 1gn. I In other words: I If X 6= X 0, then H(X) and H(X 0) are two independent uniformly distributed random variables. I If X = X 0, then H(X) is a uniformly distributed random variable and H(X 0) = H(X). I We cannot implement such a function. (Why not?) I But we can perform “lazy sampling”: I Define List of pairs (X; H(X)), initially empty I Enter input X I If no pair (X; ∗) in the List then Choose H(X) 2 f0; 1gn at random Insert (X; H(X)) into the list I (∗ Given X, there is exactly one H(X) with (X; H(X)) on the list ∗) I Return H(X) –16– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.3: Random Oracles Example: Digital Signatures (“Full Domain Hash”) “Using RSA to sign the hash of a message“ ∗ Given: message M, RSA-key (N; e; d) and H : f0; 1g ! ZN M −! X H(M) −! Y X d mod N −! Y Sign(M) Verify(M; Y ) X H(M) X Y e mod N return Y X d mod N X 0 H(M) if X = X 0 then return true else return false –17– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.3: Random Oracles Formal Background RSA Problem: ∗ e Given N; e and a random Y 2 ZN , find X with X mod N = Y . Chosen Message Attack (against a signature scheme): For i 2 f1;:::; qg loop 1. Choose Mi d 2. Receive the signature Yi (here: Yi = H(Mi ) mod N). The adversary knows the public key and the query bound q is polynomial in the security parameter. Existential Forgery: A pair (M; Y ) 2= f(Mi ; Yi ) j 1 ≤ i ≤ qg with Verify(M; Y )=true. –18– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.3: Random Oracles Main Result Theorem 1 (Bellare-Rogaway 1993) In the random oracle model, Full Domain Hash is secure. Namely, any efficient adversary able to provide an existential forgery in a chosen message attack can efficiently solve the RSA problem. –19– Stefan Lucks Hash Fun. (2019) 1: Introduction 1.3: Random Oracles Un-Instantiability I What happens, when we replace the random oracle by a “real” hash function (hopefully a “secure” one)? I Hope: The complex cryptosystem remains secure. I Problem: Theorem 2 (Canetti-Goldreich-Halevi, 1998) Assume a secure signature scheme, either in the random oracle model or even in the standard model. Then one can define another signature scheme, which is I Provably secure in the random oracle model, but I Completely insecure in the standard model.
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