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This paper presents a novel low-transition linear feedback shift register (LFSR) that is based on some new observations about the output sequence of a conventional LFSR. The proposed design, called bit-swapping LFSR (BS-LFSR), is composed of an LFSR and a 2 times 1 multiplexer. When used to generate test patterns for scan-based built-in self-tests, it reduces the number of transitions that occur at the scan-chain input during scan shift operation by 50% when compared to those patterns produced by a conventional LFSR. Hence, it reduces the overall switching activity in the circuit under test during test applications. The BS-LFSR is combined with a scan-chain-ordering algorithm that orders the cells in a way that reduces the average and peak power (scan and capture) in the test cycle or while scanning out a response to a signature analyzer. These techniques have a substantial effect on average- and peak-power reductions with negligible effect on fault coverage or test application time. Experimental results on ISCAS'89 benchmark circuits show up to 65% and 55% reductions in average and peak power, respectively.