This document is an informal description of our main results presented in the thesis [2]. We propose heuristic modifications to the successful use of Lenstra, Lenstra and Lovasz's LLL algorithm [4] to solving the Subset Sum (or knapsack) problem

Abstract: This paper gives the definition of a coprime sequence and the concept of the lever function, describes the five algorithms and six characteristics of the REESSE1+ public-key cryptosystem based on three new hardnesses: the modular subset product problem, the multivariate arrangement problem, and the super logarithm problem in a prime field, shows the correctness of the decryption and verification algorithms, and infers that the probability that a plaintext solution is not unique is nearly zeroth. The authors discuss the relation between the lever function and a random oracle, and analyze the security of REESSE1+ against recovering a plaintext from a ciphertext, extracting a private key from a public key or a signature, and faking a digital signature via a public key or via known signatures with a public key. On the basis of analysis, believe that the security of REESSE1+ is at least equal to the time complexity of O(2 n) at present. At last, expound the idea of optimizing REESSE1+ through binary compact sequences.

1.4 Simple XOR

Average case reductions for Subset Sum and Decoding of Linear Codes Genevi`eve Arboit Master of Science Graduate Department of Computer Science University of Toronto 1999 In a 1996 paper, R. Impagliazzo and M. Naor show two average case reductions for the Subset Sum problem (SS). We use similar ideas to obtain stronger and additional such reductions for SS. Furthermore, we use modifications of these ideas to obtain similar reductions for the Decoding of Linear Codes problem (DLC). The theorems give further evidence that the hardest case for Average case SS is when the number of integers is equal to their length. For Average case DLC, the theorems give evidence that the hardest case is when the dimension of the code is equal to the channel capacity times the length of the words. Average case SS and DLC hardness assumptions can be used to obtain one-way functions, pseudorandom generators, and secure private-key cryptography. ii Acknowledgments I dedicate this work to my parents, who ha...

This thesis presents fast and practical methods for generating randomly distributed pairs of the form (x; g x mod p) or (x; x e mod N ), using precomputation. These generation schemes are of wide applicability for speeding-up public key systems that depend on exponentiation and offer a smooth memory-speed trade-off. The steps involving exponentiation in these systems can be reduced significantly in many cases. The schemes are most suited for server applications. The thesis also presents security analyses of the schemes using standard assumptions. The methods are novel in the sense that they identify and thoroughly exploit the randomness issues related to the instances generated in these public-key schemes. The constructions use random walks on Cayley (expander) graphs over Abelian groups.

The members of the Committee appointed to examine the

In this paper, we present a new class of knapsack type PKC referred to as K(II)ΣΠPKC. In K(II)ΣΠPKC, Bob randomly constructs a very small subset of Alice’s set of public key whose order is very large, under the condition that the coding rate ρ satisfies 0.01 < ρ < 0.5. In K(II)ΣΠPKC, no secret sequence such as super-increasing sequence or shifted-odd sequence but the sequence whose component is constructed by a product of the same number of many prime numbers of the same size, is used. We show that K(II)ΣΠPKC is secure against the attacks such as LLL algorithm, Shamir’s attack etc., because a subset of Alice’s public keys is chosen entirely in a probabilistic manner at the sending end. We also show that K(II)ΣΠPKC can be used as a member of the class of common key cryptosystems because the list of the subset randomly chosen by Bob can be used as a common key between Bob and Alice, provided that the conditions given in this paper are strictly observed, without notifying Alice of his secret key through a particular secret channel. Key words Public-key cryptosystem(PKC), Knapsack-type PKC, Product-sum type PKC, LLL algorithm, PQC.