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Trustworthy computing modules like secure coprocessors (ScP) are already in extensive use today, albeit limited predominantly to scenarios where constraints on cost is not a serious limiting factor. However, inexpensive trustworthy computers are required for many evolving application scenarios. The problem of realizing inexpensive ScPs for large-scale networks consisting of low-complexity devices have not received adequate consideration thus far. We introduce two strategies toward realizing low-cost ScPs. The first is the decrypt only when necessary (DOWN) policy, which can substantially improve the ability of low-cost ScPs to protect their secrets. The DOWN policy relies on the ability to operate with fractional parts of secrets. Taking full advantage of the DOWN policy requires consideration of the nature of computations performed with secrets and even the mechanisms employed for distribution of secrets. We discuss the feasibility of extending the DOWN policy to various asymmetric and symmetric cryptographic primitives. The second is cryptographic authentication strategies which employ only symmetric cryptographic primitives, based on novel ID-based key predistribution schemes that demand very low complexity of operations to be performed by the ScP and can take good advantage of the DOWN policy.