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The REESSE1 Publickey Cryptosystem
 Computer Engineering & Science (Chinese
, 2003
"... 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+ publickey cryptosystem based on three new hardnesses: the modular subset product problem, the multivariate arrangement problem ..."
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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+ publickey 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.
Construction of New Class of KnapsackType Public Key Cryptosystem
 K(II)ΣPKC”, IEICE Technical Report, ISEC
"... In this paper, we present a new class of knapsack type PKC referred to as K(III)ΣPKC. In a sharp contrast with the conventional knapsack type PKC’s, in our proposed scheme, K(I II)ΣPKC, no conventional secret sequence but the natural binary number with noise is used. We show that the coding rate, a ..."
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In this paper, we present a new class of knapsack type PKC referred to as K(III)ΣPKC. In a sharp contrast with the conventional knapsack type PKC’s, in our proposed scheme, K(I II)ΣPKC, no conventional secret sequence but the natural binary number with noise is used. We show that the coding rate, a more conservative measure for the security on knapsack PKC, can be made approximately 1.0.
Multidimensional Subset Sum Problem
, 1997
"... 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 ..."
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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 Construction of New Classes of Knapsack Type Public Key Cryptosystem Using Uniform Secret Sequence, K(II)ΣΠPKC, Constructed Based on Maximum Length Code
"... 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 seque ..."
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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 superincreasing sequence or shiftedodd 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 Publickey cryptosystem(PKC), Knapsacktype PKC, Productsum type PKC, LLL algorithm, PQC.
Construction of A New Class of ProductSum Type Public Key Cryptosystem
 K(IV)ΣPKC and K(I)ΣPKC”, IEICE Tech. Report, ISEC
"... The author recently proposed a new class of knapsack type PKC referred to as K(II)ΣΠPKC [1]. In K(II)ΣΠPKC with old algorithm DA[I], 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 < 0.2 ..."
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The author recently proposed a new class of knapsack type PKC referred to as K(II)ΣΠPKC [1]. In K(II)ΣΠPKC with old algorithm DA[I], 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 < 0.2. In K(II)ΣΠPKC, no secret sequence such as superincreasing sequence or shiftedodd sequence but the sequence whose components are constructed by a product of the same number of many prime numbers of the same size, is used. In this paper we present a new algorithm, DA(II) for decoding K(II)ΣΠPKC. We show that with new decoding algorithm, DA(II), K(II)ΣΠPKC yields a higher coding rate and a smaller size of public key compared with K(II)ΣΠPKC using old decoding algorithm, DA(I). We further present a generalized version of K(II)ΣΠPKC, referred to as K(V)ΣΠPKC. We finally present a new decoding algorithm DA(III) and show that, in K(V)ΣΠPKC with DA(III), the relation, rF ∼ = 0, ρ ∼ = 2 3 holds, where rF is the factor ratio that will be defined in this paper. We show that K(V)ΣΠPKC yields a higher security compared with K(II)ΣΠPKC.
Average case reductions for Subset Sum and Decoding of Linear Codes
, 1999
"... 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 idea ..."
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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 oneway functions, pseudorandom generators, and secure privatekey cryptography. ii Acknowledgments I dedicate this work to my parents, who ha...
A PreComputation Scheme for Speeding Up PublicKey Cryptosystems
, 1998
"... 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 speedingup public key systems that depend on exponentiation and offer a smooth me ..."
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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 speedingup public key systems that depend on exponentiation and offer a smooth memoryspeed tradeoff. 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 publickey schemes. The constructions use random walks on Cayley (expander) graphs over Abelian groups.
ALGEBRAIC NUMBER RINGS Abstract
, 2010
"... The members of the Committee appointed to examine the ..."