what is cross-query collisions?

Cross-query collisions refer to a specific kind of vulnerability or attack scenario where a collision is found between the results of different cryptographic operations across multiple queries. This concept is often discussed in the context of cryptographic protocols or systems where multiple queries are made to an oracle (a black-box function or service), and the goal is to find two distinct inputs (from potentially different queries) that produce the same output.

In Cryptographic Context:

• Collision: In cryptography, a collision occurs when two different inputs produce the same output under a cryptographic function, such as a hash function or an HMAC (Hash-based Message Authentication Code). For instance, if H(x1) = H(x2) where x1 ≠ x2, then x1 and x2 are said to collide.
• Cross-query: The term “cross-query” implies that the collision involves inputs from separate queries or operations. For example, suppose you have a scenario where you are interacting with a cryptographic system via multiple queries (e.g., sending different inputs to a hash function or encryption system). A cross-query collision would occur if you find two distinct inputs x1 from one query and x2 from another that produce the same output.

Example:

Suppose you are interacting with an HMAC oracle. You make several queries to the oracle with different inputs and receive the corresponding HMAC outputs:

• Query 1: HMAC(Key, message1) -> h1
• Query 2: HMAC(Key, message2) -> h2
• Query 3: HMAC(Key, message3) -> h3

A cross-query collision would occur if, for example, h1 == h3, indicating that message1 and message3 have produced the same HMAC value despite being different messages.

Significance:

Cross-query collisions are significant because they can undermine the security of cryptographic systems. For example:

• In Digital Signatures: A cross-query collision in a hash function used for generating digital signatures could allow an attacker to forge a valid signature for a different message.
• In MACs (Message Authentication Codes): A cross-query collision might allow an attacker to deceive the verification mechanism, leading to a successful authentication of a tampered or unintended message.
• In Protocols: Cross-query collisions can break security protocols by allowing attackers to replay or mix messages across different sessions or operations.

Preventing Cross-query Collisions:

Monitoring and Limiting Queries: In scenarios where oracles are used, it’s crucial to limit the number of queries or monitor for patterns that might indicate a collision search is taking place.

Use Strong Cryptographic Primitives: Ensure that hash functions or MAC algorithms used are resistant to collision attacks (e.g., using SHA-256 or SHA-3).

Salting or Adding Nonces: Introducing random values (salts or nonces) that are unique to each query can make it much harder for an attacker to find collisions.