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Questions and Answers
Consider a cryptographic hash function $H: {0,1}^* \rightarrow {0,1}^n$ constructed using the Merkle-Damgard construction with compression function $f: {0,1}^{\ell + n} \rightarrow {0,1}^n$. Which of the following statements regarding the security properties of $H$ is most accurate, assuming an ideal compression function $f$?
Consider a cryptographic hash function $H: {0,1}^* \rightarrow {0,1}^n$ constructed using the Merkle-Damgard construction with compression function $f: {0,1}^{\ell + n} \rightarrow {0,1}^n$. Which of the following statements regarding the security properties of $H$ is most accurate, assuming an ideal compression function $f$?
- If $f$ is collision-resistant, then $H$ is always collision-resistant regardless of the padding scheme.
- If $f$ is preimage-resistant, then $H$ is always preimage-resistant, provided the padding scheme is length-preserving.
- The collision resistance of $H$ is contingent upon the collision resistance of $f$ and the properties of the padding scheme used. (correct)
- If $f$ is second preimage resistant, then $H$ is second preimage resistant irrespective of the Merkle-Damgard construction.
Given a cryptographic hash function $H$, demonstrating preimage resistance for every element in the range of $H$ necessarily implies that $H$ is both collision-resistant and second preimage resistant.
Given a cryptographic hash function $H$, demonstrating preimage resistance for every element in the range of $H$ necessarily implies that $H$ is both collision-resistant and second preimage resistant.
False (B)
Describe a scenario where the security of a Merkle-Damgard hash function, utilizing a collision-resistant compression function, is compromised due to a poorly designed padding scheme. Provide specific details of the vulnerability.
Describe a scenario where the security of a Merkle-Damgard hash function, utilizing a collision-resistant compression function, is compromised due to a poorly designed padding scheme. Provide specific details of the vulnerability.
A scenario where collision resistance is compromised involves a padding scheme that allows for trivial collision creation. For example, if the padding scheme does not include a length encoding or allows for arbitrary post-padding (adding any number of bits after mandatory padding), an attacker could create two messages that, after padding, result in different but colliding initial blocks ($m_1, m_1'$) and then append the same subsequent blocks to force a full collision.
In the Merkle-Damgard construction, the internal state variable $s_i$ is iteratively updated via the application of a ______ function, which compresses the previous state $s_{i-1}$ and the current message block $m_i$.
In the Merkle-Damgard construction, the internal state variable $s_i$ is iteratively updated via the application of a ______ function, which compresses the previous state $s_{i-1}$ and the current message block $m_i$.
Match each security property of cryptographic hash functions with its corresponding definition:
Match each security property of cryptographic hash functions with its corresponding definition:
Consider a cryptographic hash function $H$ with an output size of $t$ bits. An attacker, Mallory, attempts to find either a preimage or a second preimage for a given hash value. Given her computational constraints, which of the following strategies would be the MOST efficient for Mallory to adopt?
Consider a cryptographic hash function $H$ with an output size of $t$ bits. An attacker, Mallory, attempts to find either a preimage or a second preimage for a given hash value. Given her computational constraints, which of the following strategies would be the MOST efficient for Mallory to adopt?
A hash function that is preimage resistant and second preimage resistant is, by definition, collision resistant.
A hash function that is preimage resistant and second preimage resistant is, by definition, collision resistant.
A cryptographer designs a new hash function, claiming it offers 128-bit security against collision attacks. According to the birthday paradox, approximately how many hash computations would an adversary need to perform to have a reasonable chance of finding a collision?
A cryptographer designs a new hash function, claiming it offers 128-bit security against collision attacks. According to the birthday paradox, approximately how many hash computations would an adversary need to perform to have a reasonable chance of finding a collision?
Given a cryptographic hash function $H$ and two distinct messages, $m_1$ and $m_2$, the property that $H(m_1) = H(m_2)$ is known as a ______.
Given a cryptographic hash function $H$ and two distinct messages, $m_1$ and $m_2$, the property that $H(m_1) = H(m_2)$ is known as a ______.
Match the following security properties of cryptographic hash functions with their corresponding descriptions:
Match the following security properties of cryptographic hash functions with their corresponding descriptions:
A software development team is designing a secure system that uses a cryptographic hash function. From a security perspective, which of the following application scenarios MOST critically requires collision resistance, as opposed to just preimage or second preimage resistance?
A software development team is designing a secure system that uses a cryptographic hash function. From a security perspective, which of the following application scenarios MOST critically requires collision resistance, as opposed to just preimage or second preimage resistance?
Explain how the birthday paradox impacts the security of collision-resistant hash functions and why the output size of a hash function needs to be significantly larger than the desired security level (e.g., 256 bits for 128-bit security).
Explain how the birthday paradox impacts the security of collision-resistant hash functions and why the output size of a hash function needs to be significantly larger than the desired security level (e.g., 256 bits for 128-bit security).
An IT security architect is tasked with selecting a cryptographic hash function for ensuring data integrity in a high-throughput system. Considering the trade-offs between security and performance, which attribute of the selected hash function is LEAST relevant in this specific context, assuming adequate collision resistance?
An IT security architect is tasked with selecting a cryptographic hash function for ensuring data integrity in a high-throughput system. Considering the trade-offs between security and performance, which attribute of the selected hash function is LEAST relevant in this specific context, assuming adequate collision resistance?
Flashcards
Preimage Resistance
Preimage Resistance
Hard to find a message that produces a specific hash value.
Second Preimage Resistance
Second Preimage Resistance
Given one message, it's difficult to find a different message with the same hash value.
Collision Resistance
Collision Resistance
It should be computationally infeasible to find two distinct messages that produce the same hash value.
Padding
Padding
Extending a message by adding bits to achieve a required length for cryptographic functions.
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Merkle-Damgard Construction
Merkle-Damgard Construction
A method that processes arbitrarily long inputs using a fixed-size compression function applied iteratively.
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Cryptographic Hash Function
Cryptographic Hash Function
A function that takes arbitrary length bit strings as inputs and outputs a fixed-length bit string (digest or hash value).
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One-Way Hash Function
One-Way Hash Function
A function that is one-way, meaning it's computationally infeasible to find an input that produces a specific output.
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Collision (Hash Function)
Collision (Hash Function)
Finding two different inputs (m and m') such that H(m) = H(m').
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Birthday Paradox (in Hashing)
Birthday Paradox (in Hashing)
The principle that states after q queries to a hash function with an output size of t bits, the probability of finding a collision is approximately q^2 / 2^t.
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Queries to Find a Collision
Queries to Find a Collision
Approximately 2^(t/2) queries.
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Queries to Find pre-image
Queries to Find pre-image
Approximately 2^t
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- A cryptographic hash function (H) transforms arbitrary-length bit strings into fixed-length outputs, known as digests or hash values.
- Cryptographic hash functions differ from standard hash functions through their one-way property. It should be computationally infeasible to find any value x in the domain of H such that H(x) = z, given a string z from H's codomain.
- The one-way property is known as preimage resistance.
Preimage Resistance
- A hash function possessing the one-way property is considered preimage resistant. Given a hash function producing t-bit outputs, finding preimages requires O(2^t) time.
Second Preimage Resistance
- A cryptographic hash function should be second preimage resistant. Given a message m, it should be hard to find a different message m' such that H(m') = H(m).
- Finding a second preimage in a cryptographic hash function with t-bit outputs requires approximately 2^t queries.
Collision Resistance
- Collision resistance requires that the domain D is much larger than the codomain C when mapping elements from D to C using a function H.
- A function is collision resistant if it is infeasible to find two distinct values m and m' such that H(m) = H(m').
- A function H is collision resistant or H-CR secure if finding a collision for the function is believed to be infeasible, indicating that two elements in the domain map to the same element in the codomain.
- Constructing collision resistant hash functions is more challenging than creating one-way hash functions, related to the birthday paradox.
- To find a collision in a hash function H, compute H(m1), H(m2), H(m3) until a collision occurs.
- If a function has an output size of t bits, the probability of finding a collision after q queries to H is approximately q^2/2^t. A collision can be found at 2^(t/2) queries. 128 bits of output are needed to achieve ~2^128.
- To achieve a security level of 128 bits for a collision resistant hash function, roughly 256 bits of output are needed.
Properties of Cryptographic Hash Functions:
- Hard to find a message with a given hash value (preimage resistant).
- Given one message, hard to find another message with the same hash value (second preimage resistant).
- Hard to find two messages with the same hash value (collision resistant).
Lemmas
- Assuming a function H is preimage resistant for every element in the range of H is a weaker assumption than assuming it is either collision resistant or second preimage resistant.
- Assuming a function is second preimage resistant is a weaker assumption than assuming it is collision resistant.
Message Padding
- Input messages of l bits are padded to become a message of b bits, extending messages by adding extra bits to reach the required length.
Merkle-Damgård Construction
- This construction describes the basic Merkle-Damgård method for creating a hash function that accepts inputs of arbitrary length, using hash functions that take inputs of a fixed length.
- The construction relies on a family of compression functions, which maps (n + l)-bit strings to n-bit strings.
- The construction uses an internal state variable si, an n-bit string updated by applying the compression function.
- The input message m is padded using a padding scheme, so that the output is a multiple of l bits in length.
- The input m is divided into t blocks of length l bits (m1... mt).
Algorithm details:
- For i = 1 to t, compute si = f (si-1, mi), where f is the compression function.
- Return st.
- Collision resistance in the Merkle-Damgård construction depends on the padding scheme used.
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