We survey the computational assumptions of various cryptographic schemes, and discuss the security threat posed by Shor's quantum algorithm.;One-way functions form the basis of public-key cryptography. Although we have candidate hard problems that are believed to be one-way, none has been proven to be so. Therefore the security of the corresponding cryptographic schemes depends on the intractability assumptions of these problems. Two major species of such problems, factoring and discrete logarithm, are widely believed to be intractable, and serve as the basis of many popular schemes. However, these two problems turned out to be polynomial-time solvable on a hypothetical quantum computer using Shor's algorithm. This is the most worrisome long-term threat to current public-key cryptosystems.;In the thesis we provide a review of existing cryptosystems, with a focus on their underlying computational assumptions and the security. Other than factoring and discrete logarithm, schemes have been proposed based on error-correcting codes, subset-sum and subset-product problems, lattice, polynomials, combinatorial group theory, number fields, etc. Many are to be furtherly evaluated in future research.
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