We outline constructions for both pseudo-random generators and one-way hash functions. These constructions are based on the exact TSP (XTSP), a special variant of the well known traveling salesperson problem. We prove that these constructions are secure if the XTSP is infeasible. Our constructions are easy to implement, appear to be fast, but require a large amount of memory.

Abstract. Quite recently, in [4], a new time-memory tradeoff algorithm was presented. The original goal of this algorithm was to count the number of points on an elliptic curve, however, the authors claimed that their approach could be applied to other problems. In this paper, we describe such an application and show a new way to attack the Permuted Kernel Problem. This new method is faster than any previously known technique but still requires exponential time. In practice, we find that attacking PKP for the original size proposed by Shamir in [6] could be done on a single PC in 125 years. 1

Abstract. The appearance of the theory of zero-knowledge, presented by Goldwasser, Micali and Rackoff in 1985, opened a way to secure identification schemes. The first application was the famous Fiat-Shamir scheme based on the problem of modular square roots extraction. In the following years, many other schemes have been proposed, some Fiat-Shamir extensions but also new discrete logarithm based schemes. Therefore, all of them were based on problems from number theory. Their main common drawback is high computational load because of arithmetical operations modulo large integers. Implementation on low-cost smart cards was made difficult and inefficient. With the Permuted Kernels Problem (PKP), Shamir proposed the first efficient scheme allowing for an implementation on such low-cost smart cards, but very few others have afterwards been suggested. In this paper, we present an efficient identification scheme based on a combinatorial N P-complete problem: the Permuted Perceptrons Problem (PPP). This problem seems hard enough to be unsolvable even with very small parameters, and some recent cryptanalysis studies confirm that position. Furthermore, it admits efficient zero-knowledge proofs of knowledge and so it is well-suited for cryptographic purposes. An actual implementation completes the optimistic opinion about efficiency and practicability on low-cost smart cards, and namely with less than 2KB of EEPROM and just 100 Bytes of RAM and 6.4 KB of communication.