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The majority of current attacks on reduced-round variants of block ciphers seeks to maximize the number of rounds that can be broken, using less data than the entire codebook and less time than exhaustive key search. In this paper, we pursue a different approach, restricting the data available to the adversary to a few plaintext/ciphertext pairs. We argue that consideration of such attacks (which received little attention in recent years) improves our understanding of the security of block ciphers and of other cryptographic primitives based on block ciphers. In particular, these attacks can be leveraged to more complex attacks, either on the block cipher itself or on other primitives (e.g., stream ciphers, MACs, or hash functions) that use a small number of rounds of the block cipher as one of their components. As a case study, we consider the Advanced Encryption Standard (AES)-the most widely used block cipher. The AES round function is used in many cryptographic primitives, such as the hash functions Lane, SHAvite-3, and Vortex or the message authentication codes ALPHA-MAC, Pelican, and Marvin. We present attacks on up to four rounds of AES that require at most three known/chosen plaintexts. We then apply these attacks to cryptanalyze an AES-based stream cipher (which follows the leak extraction methodology), and to mount the best known plaintext attack on six-round AES.