Logical Analysis and Verification of Cryptographic Protocols - Loria

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58 CHAPTER 3. PROTOCOLS WITH VULNERABLE HASH FUNCTIONS Outline. In Section 3.1, we introduce hash functions and show some of their properties. In Section 3.2, we introduce the collision vulnerability property. In Section 3.3, we give the model that we use during this chapter. The collision vulnerability property is symbolically formalised in Section 3.4, and the decidability results are given in Section 3.5. 3.1 Hash functions 3.1.1 Definition of hash functions Cryptographic hash functions, or simply hash functions play a fundamental role in modern cryptography. They take a message of arbitrary length as input and output a fixed-length message referred to as a hash- code, hash-result, hash-value, or simply hash. Definition 34 (hash function) Let D and R be respectively a set of messages of arbitrary length and a set of messages of fixed length. A hash function h is a function h : D → R which has, as a minimum, the following two properties: Compression h maps an input x of arbitrary finite length, to an output h(x) of fixed length. Ease of computation given h and an input x, h(x) is easy to compute; actually, h is a linear function, that is for any input x, h(x) is computable in time O(�x�). For any hash function h : D → R, it is easy to see that �D� > �R�. Applications. Hash functions have many applications in information security. A typical use of a cryptographic hash would be as follows: Alice poses a tough math problem to Bob, and claims she has solved it. Bob would like to try it himself, but would yet like to be sure that Alice is not bluffing. Therefore, Alice writes down her solution, appends a random nonce, computes its hash and tells Bob the hash value (whilst keeping the solution and nonce secret). This way, when Bob comes up with the solution himself a few days later, Alice can prove that she had the solution earlier by revealing the nonce to Bob. This is an example of a simple commitment scheme. Another important application of secure hashes is verification of message integrity. Determining whether any change has been made to a message (or a file) can be accomplished by comparing the hash of the message calculated before and after transmission (or any other event). The hash of a message can also serve as a means of reliably identifying a file.
3.1. HASH FUNCTIONS 59 A related application is password verification [125]: passwords are usually not stored in cleartext, for obvious reasons, but instead in hashed form. To authenticate a user, the password presented by the user is hashed and compared with the stored hash. Hash functions are also used in digital signature schemes [175, 139]: for both security and performance reasons, some signature schemes specify that only the hash of the message will be “signed”, and not the entire message. Hash functions can also be used in the generation of pseudorandom bits [153]. 3.1.2 Properties of hash functions Let h be an arbitrary hash function, h : D → R. We present here the three potential properties (in addition to ease of computation and compression given in Definition 34), for h [153]: Preimage resistance given a hash y ∈ R, if the correspondant input is not known, it is computationally infeasible, i.e. it takes too long (hundreds of years) to compute using the fastest of super computers, to find any input x ∈ D such that h(x) = y. This concept is related to that of one way function. Second preimage resistance given an input x ∈ D, it is computationally infeasible to find another input x ′ ∈ D such that x ′ �= x and h(x ′ ) = h(x). This property is sometimes referred to as weak collision resistance. Collision resistance it is computationally infeasible to find two distinct inputs x, x ′ ∈ D such that h(x ′ ) = h(x). This property is sometimes referred to as strong collision resistance. We say that a hash function is vulnerable to respectively preimage attacks, second preimage attacks, and collision attacks if it lacks respectively preimage resistance, second preimage resistance and collision resistance property. We say also that a hash function is respectively one-way, second-preimage resistant and collision resistant hash function if it is a hash function as per definition 34 with respectively the following properties: preimage resistance, second preimage resistance and collision resistance property. In [153], the authors showed the following relationships between properties of hash functions given above: • collision resistance implies second-preimage resistance of hash functions; • collision resistance does not guarantee preimage resistance; • preimage resistance does not guarantee second-preimage resistance.
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Logical Analysis and Verification o
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Logical Analysis and Verification of Cryptographic Protocols - Loria

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