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15.1. Secure Hash Functions
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Hash functions, as used in the world of cryptography, have some similarities to and differences from traditional hash functions. Hash functions in the world of computer science map a long string to a shorter one. Hash functions are used, for instance, to help construct hash tables in which data is stored into buckets for quick accesses. A hash function is used to determine which bucket the data should be placed in. As such, the goal of a traditional hash function is to evenly balance the data across all the buckets in the hash table. However, cryptographic hash functions have additional goals. A secure cryptographic hash function H takes as input some (potentially large) string M. It produces a message digest, MD = H(M), that has a few properties: 1. Efficiency: It should not take a lot of computational time or CPU cycles to compute H(M) even for a potentially large message. 2. Pre-image resistance: Given H(M), it should be computationally infeasible to determine M. M is often called the pre-image of the hash H(M), and H is sometimes referred to as a one-way function. 3. Collision resistance: It is computationally infeasible to find two distinct input messages M1 and M2 (M1 M2) for which H(M1) = H(M2).
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CHAPTER 15 s MACS AND SIGNATURES
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A traditional hash function that, say, simply adds the ASCII values of the characters in the message together does not satisfy all these properties. For instance, such a traditional hash function does not exhibit collision-resistance, as the messages AB and BA have the same hash value even though they are different messages. Also, note that checksum functions, such as CRC32, are used to provide redundancy checks against communication errors and may be efficient, but in general are not one-way or collision-resistant. Cryptographic hash functions use much more sophisticated techniques to ensure pre-image and collision resistance properties. Two real-world examples of hash functions that are commonly used in building security protocols are MD5 and SHA-1. MD5 is an acronym that stands for Message Digest 5, and it was developed by Ron Rivest, who also contributed to the development of the RSA asymmetric cipher. MD5 takes as input multiples of 512 bits. If a message is not a multiple of 512 bits, the message can be padded such that it becomes a multiple of 512 bits. MD5 produces a 128-bit message digest as output. SHA-1 is another hash function that takes 512 bits of input at a time, but its output is 160 bits instead of 128. SHA-1 was developed by collaboration between NIST and NSA. Over the past few years, there have been some attacks constructed against the collision resistance properties of MD5 and SHA-1. We briefly discuss these attacks in Section 15.4, after describing how these hash functions are used in MAC and digital signature schemes.
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15.2. Message Authentication Codes (MACs)
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MACs can be used to determine if a message originated from a principal that has possession of a secret key. In particular, Alice and Bob may share a key, k, and Alice may use that key to compute a MAC on a message, M, which we denote as t = MAC(M,k). Alice can then transmit M and t to Bob. The value t is often referred to as a tag. Let M' be the message that Bob receives and t' be the corresponding MAC that he receives. If Mallory attempts to modify either M or t, Bob can detect any potential tampering by checking whether t' = MAC(M',k). If the equality holds true, it means (with overwhelmingly high probability) that either the message and signature were not tampered with (M' = M and t' = t) or that Mallory knows k.
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