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The Information Technology Laboratory (ITL) at the National Institute of Standards and Technology (NIST) promotes the U.S. economy and public welfare by providing technical leadership for the nation's measurement and standards infrastructure. ITL develops tests, test methods, reference data, proof of concept implementations, and technical analysis to advance the development and productive use of information technology. ITL's responsibilities include the development of technical, physical, administrative, and management standards and guidelines for the cost-effective security and privacy of sensitive unclassified information in Federal computer systems. This Special Publication 800-series reports on ITL's research, guidance, and outreach efforts in computer security and its collaborative activities with industry, government, and academic organizations.

In 1997, the National Institute of Standards and Technology (NIST) initiated a process to select a symmetric-key encryption algorithm to be used to protect sensitive (unclassified) Federal information in furtherance of NIST’s statutory responsibilities. In 1998, NIST announced the acceptance of 15 candidate algorithms and requested the assistance of the cryptographic research community in analyzing the candidates. This analysis included an initial examination of the security and efficiency characteristics for each algorithm. NIST reviewed the results of this preliminary research and selected MARS, RC™, Rijndael, Serpent and Twofish as finalists. Having reviewed further public analysis of the finalists, NIST has decided to propose Rijndael as the Advanced Encryption Standard (AES). The research results and rationale for this selection are documented in this report.

In 1997, the National Institute of Standards and Technology (NIST) initiated a process to select a symmetric-key encryption algorithm to be used to protect sensitive (unclassified) Federal information in furtherance of NIST’s statutory responsibilities. In 1998, NIST announced the acceptance of 15 candidate algorithms and requested the assistance of the cryptographic research community in analyzing the candidates. This analysis included an initial examination of the security and efficiency characteristics for each algorithm. NIST has reviewed the results of this research and selected five algorithms (MARS, RC6™, Rijndael, Serpent and Twofish) as finalists. The research results and rationale for the selection of the finalists are documented in this report. The five finalists will be the subject of further study before the selection of one or more of these algorithms for inclusion in the Advanced Encryption Standard.

(NIST) has a unique responsibility for computer systems technology within the Federal Government. NIST's Computer Systems Laboratory (CSL) develops standards and guidelines, provides technical assistance, and conducts research for computers and related telecommunications systems to achieve more effective utilization of Federal information technology resources. CSL's responsibilities include development of technical, management, physical, and administrative standards and guidelines for the cost-effective security and privacy of sensitive unclassified information processed in Federal computers. CSL assists agencies in developing security plans and in improving computer security awareness training. This Special Publication 800 series reports CSL research and guidelines to Federal agencies as well as to organizations in industry, government, and academia.

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We study here an approximation scheme suitable for knapsack and related problems, such as alx I + a2x 2 + ... + anX n = N Xl,X2,...,x n nonnegative integers Previously algorithms have been developed to solve the above and similar problems for two variables. The approximation scheme developed herein extends at least to three variables.It makes critical use of Farey fractions. The essence of the strategy is to obtain the solution of a related optimization problem recursively in terms of the solutions of subproblems resulting from approximation.We are thus able to provide a complete solution to the decision problem for the above for n=3. The resulting algorithm has given an excellent empirical performance for moderately large inputs. For example if input size is restricted to be on the order of 2"'30, the number of subproblems mentioned earlier averages about 30 for randomlygenerated input. This leaves open the possibility of polynomial-time average-case behavior. The worst-case time is in doubt. This calls into question a conjecture of Hirschberg and Wong implying that the above is solvable in polynomial time for fixed n.Our algorithm is well-suited for parallel processing, and shows that the above is solvable in linear time for n=3 with massive parallelism.Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1986ACM-0-89791 -177-6/86/0002/0510 $00.75 510