This paper presents two Byzantine fault-tolerant state machine replication (BFT) algorithms that are minimal in several senses. First, they require only 2 f +1 replicas, instead of the usual 3 f +1. Second, the trusted service in which this reduction of replicas is based is arguably minimal: it provides an interface with a single function and is composed only by a counter and a signature generation primitive. Third, in nice executions the two algorithms run in the minimum number of communication steps for non-speculative and speculative algorithms, respectively 4 and 3 steps. The paper is also the first to present BFT algorithms with 2 f +1 replicas that require a trusted service implementable using commercial off-the-shelf trusted hardware: this service can be implemented with the Trusted Platform Module currently available as a chip in the mainboard of many commodity PCs