Multilevel Threshold Secret and Function Sharing based on the Chinese Remainder Theorem

A recent work of Harn and Fuyou presents the first multilevel (disjunctive) threshold secret sharing scheme based on the Chinese Remainder Theorem. In this work, we first show that the proposed method is not secure and also fails to work with a certain natural setting of the threshold values on compartments. We then propose a secure scheme that works for all threshold settings. In this scheme, we employ a refined version of Asmuth-Bloom secret sharing with a special and generic Asmuth-Bloom sequence called the {\it anchor sequence}. Based on this idea, we also propose the first multilevel conjunctive threshold secret sharing scheme based on the Chinese Remainder Theorem. Lastly, we discuss how the proposed schemes can be used for multilevel threshold function sharing by employing it in a threshold RSA cryptosystem as an example

Function and secret sharing extensions for Blakley and Asmuth-Bloom secret sharing schemes

Ankara : The Department of Computer Engineering and the Institute of Engineering and Science of Bilkent University, 2009.Thesis (Master's) -- Bilkent University, 2009.Includes bibliographical references leaves 65-69.Threshold cryptography deals with situations where the authority to initiate or perform cryptographic operations is distributed amongst a group of individuals. Usually in these situations a secret sharing scheme is used to distribute shares of a highly sensitive secret, such as the private key of a bank, to the involved individuals so that only when a sufficient number of them can reconstruct the secret but smaller coalitions cannot. The secret sharing problem was introduced independently by Blakley and Shamir in 1979. They proposed two different solutions. Both secret sharing schemes (SSS) are examples of linear secret sharing. Many extensions and solutions based on these secret sharing schemes have appeared in the literature, most of them using Shamir SSS. In this thesis, we apply these ideas to Blakley secret sharing scheme. Many of the standard operations of single-user cryptography have counterparts in threshold cryptography. Function sharing deals with the problem of distribution of the computation of a function (such as decryption or signature) among several parties. The necessary values for the computation are distributed to the participants using a secret sharing scheme. Several function sharing schemes have been proposed in the literature with most of them using Shamir secret sharing as the underlying SSS. In this work, we investigate how function sharing can be achieved using linear secret sharing schemes in general and give solutions of threshold RSA signature, threshold Paillier decryption and threshold DSS signature operations. The threshold RSA scheme we propose is a generalization of Shoup’s Shamir-based scheme. It is similarly robust and provably secure under the static adversary model. In threshold cryptography the authorization of groups of people are decided simply according to their size. There are also general access structures in which any group can be designed as authorized. Multipartite access structures constitute an example of general access structures in which members of a subset are equivalent to each other and can be interchanged. Multipartite access structures can be used to represent any access structure since all access structures are multipartite. To investigate secret sharing schemes using these access structures, we used Mignotte and Asmuth-Bloom secret sharing schemes which are based on the Chinese remainder theorem (CRT). The question we tried to asnwer was whether one can find a Mignotte or Asmuth-Bloom sequence for an arbitrary access structure. For this purpose, we adapted an algorithm that appeared in the literature to generate these sequences. We also proposed a new SSS which solves the mentioned problem by generating more than one sequence.Bozkurt, İlker NadiM.S

Threshold cryptography with Chinese remainder theorem

Ankara : The Department of Computer Engineering and the Institute of Engineering and Science of Bilkent University, 2009.Thesis (Master's) -- Bilkent University, 2009.Includes bibliographical references leaves 84-91.Information security has become much more important since electronic communication is started to be used in our daily life. The content of the term information security varies according to the type and the requirements of the area. However, no matter which algorithms are used, security depends on the secrecy of a key which is supposed to be only known by the agents in the first place. The requirement of the key being secret brings several problems. Storing a secret key on only one person, server or database reduces the security of the system to the security and credibility of that agent. Besides, not having a backup of the key introduces the problem of losing the key if a software/hardware failure occurs. On the other hand, if the key is held by more than one agent an adversary with a desire for the key has more flexibility of choosing the target. Hence the security is reduced to the security of the least secure or least credible of these agents. Secret sharing schemes are introduced to solve the problems above. The main idea of these schemes is to share the secret among the agents such that only predefined coalitions can come together and reveal the secret, while no other coalition can obtain any information about the secret. Thus, the keys used in the areas requiring vital secrecy like large-scale finance applications and commandcontrol mechanisms of nuclear systems, can be stored by using secret sharing schemes. Threshold cryptography deals with a particular type of secret sharing schemes. In threshold cryptography related secret sharing schemes, if the size of a coalition exceeds a bound t, it can reveal the key. And, smaller coalitions can reveal no information about the key. Actually, the first secret sharing scheme in the literature is the threshold scheme of Shamir where he considered the secret as the constant of a polynomial of degree t − 1, and distributed the points on the polynomial to the group of users. Thus, a coalition of size t can recover the polynomial and reveal the key but a smaller coalition can not. This scheme is widely accepted by the researchers and used in several applications. Shamir’s secret sharing scheme is not the only one in the literature. For example, almost concurrently, Blakley proposed another secret sharing scheme depending on planar geometry and Asmuth and Bloom proposed a scheme depending on the Chinese Remainder Theorem. Although these schemes satisfy the necessary and sufficient conditions for the security, they have not been considered for the applications requiring a secret sharing scheme. Secret sharing schemes constituted a building block in several other applications other than the ones mentioned above. These applications simply contain a standard problem in the literature, the function sharing problem. In a function sharing scheme, each user has its own secret as an input to a function and the scheme computes the outcome of the function without revealing the secrets. In the literature, encryption or signature functions of the public key algorithms like RSA, ElGamal and Paillier can be given as an example to the functions shared by using a secret sharing scheme. Even new generation applications like electronic voting require a function sharing scheme. As mentioned before, Shamir’s secret sharing scheme has attracted much of the attention in the literature and other schemes are not considered much. However, as this thesis shows, secret sharing schemes depending on the Chinese Remainder Theorem can be practically used in these applications. Since each application has different needs, Shamir’s secret sharing scheme is used in applications with several extensions. Basically, this thesis investigates how to adapt Chinese Remainder Theorem based secret sharing schemes to the applications in the literature. We first propose some modifications on the Asmuth-Bloom secret sharing scheme and then by using this modified scheme we designed provably secure function sharing schemes and security extensions.Kaya, KamerM.S

Reusable Multi-Stage Multi-Secret Sharing Schemes Based on CRT

Three secret sharing schemes that use the Mignotte’ssequence and two secret sharing schemes that use the Asmuth-Bloom sequence are proposed in this paper. All these five secret sharing schemes are based on Chinese Remainder Theorem (CRT) [8]. The first scheme that uses the Mignotte’s sequence is a single secret scheme; the second one is an extension of the first one to Multi-secret sharing scheme. The third scheme is again for the case of multi-secrets but it is an improvement over the second scheme in the sense that it reduces the number of publicvalues. The first scheme that uses the Asmuth-Bloom sequence is designed for the case of a single secret and the second one is an extension of the first scheme to the case of multi-secrets. Novelty of the proposed schemes is that the shares of the participants are reusable i.e. same shares are applicable even with a new secret. Also only one share needs to be kept by each participant even for the muslti-secret sharing scheme. Further, the schemes are capable of verifying the honesty of the participants including the dealer. Correctness of the proposed schemes is discussed and show that the proposed schemes are computationally secure

Constructing Ideal Secret Sharing Schemes based on Chinese Remainder Theorem

Since $(t,n)$-threshold secret sharing (SS) was initially proposed by Shamir and Blakley separately in 1979, it has been widely used in many aspects. Later on, Asmuth and Bloom presented a $(t,n)$-threshold SS scheme based on the Chinese Remainder Theorem(CRT) for integers in 1983. However, compared with the most popular Shamir\u27s $(t,n)$-threshold SS scheme, existing CRT based schemes have a lower information rate, moreover, they are harder to construct. To overcome these shortcomings of the CRT based scheme, 1) we first propose a generalized $(t,n)$-threshold SS scheme based on the CRT for the polynomial ring over a finite field. We show that our scheme is ideal, i.e., it is perfect in security and has the information rate 1. By comparison, we show that our scheme has a better information rate and is easier to construct compared with existing threshold SS schemes based on the CRT for integers. 2) We show that Shamir\u27s scheme, which is based on the Lagrange interpolation polynomial, is a special case of our scheme. Therefore, we establish the connection among threshold schemes based on the Lagrange interpolation, schemes based on the CRT for integers and our scheme. 3) As a natural extension of our threshold scheme, we present a weighted threshold SS scheme based on the CRT for polynomial rings, which inherits the above advantages of our threshold scheme over existing weighted schemes based on the CRT for integers

International Symposium on Mathematics, Quantum Theory, and Cryptography

This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan. The international symposium MQC addresses the mathematics and quantum theory underlying secure modeling of the post quantum cryptography including e.g. mathematical study of the light-matter interaction models as well as quantum computing. The security of the most widely used RSA cryptosystem is based on the difficulty of factoring large integers. However, in 1994 Shor proposed a quantum polynomial time algorithm for factoring integers, and the RSA cryptosystem is no longer secure in the quantum computing model. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. This book is suitable for postgraduate students in mathematics and computer science, as well as for experts in industry working on post-quantum cryptography

International Symposium on Mathematics, Quantum Theory, and Cryptography

This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan. The international symposium MQC addresses the mathematics and quantum theory underlying secure modeling of the post quantum cryptography including e.g. mathematical study of the light-matter interaction models as well as quantum computing. The security of the most widely used RSA cryptosystem is based on the difficulty of factoring large integers. However, in 1994 Shor proposed a quantum polynomial time algorithm for factoring integers, and the RSA cryptosystem is no longer secure in the quantum computing model. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. This book is suitable for postgraduate students in mathematics and computer science, as well as for experts in industry working on post-quantum cryptography

Optics in Our Time

Optics, Lasers, Photonics, Optical Devices; Quantum Optics; Popular Science in Physics; History and Philosophical Foundations of Physic

Accountants\u27 index. Twenty-first supplement, a bibliography of accounting literature, January 1972-December 1972 (Inclusive)

https://egrove.olemiss.edu/aicpa_accind/1023/thumbnail.jp