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In this paper we present a new search method for test set embedding using an accumulator driven with an additive constant C. We formulate the problem of finding the location of a test pattern in the generated sequence in terms of a linear Diophantine equation with two variables, which is known to be solved quickly in linear time. We show that only one Diophantine equation needs to be solved per test set irrespective of its size. Next we show how to find the starting state, for a given constant C and test set T, such that the generated sequence can reproduce T with minimum length. Finally, we show that the best constant Copt (in terms of shortest test length) for the embedding of T using an accumulator of size n can be found in O(2ldrn+Fldr|T|) steps, instead of O(nldr2nldr|T|) steps of a previous approach, where F depends on the particular test set and can be significantly smaller than its worst case value of 2n-2. The value of F can also be further reduced while providing a guaranteed approximation bound of the shortest test length. Experimental results show the computational improvements.