On the Classification of Weierstrass Elliptic Curves over Zn\mathbb{Z}_n

The development of secure cryptographic protocols and the subsequent attack mechanisms have been placed in the literature with the utmost curiosity. While sophisticated quantum attacks bring a concern to the classical cryptographic protocols present in the applications used in everyday life, the necessity of developing post-quantum protocols is felt primarily. In post-quantum cryptography, elliptic curve-base protocols are exciting to the researchers. While the comprehensive study of elliptic curves over finite fields is well known, the extended study over finite rings is still missing. In this work, we generalize the study of Weierstrass elliptic curves over finite ring $\mathbb{Z}_n$ through classification. Several expressions to compute critical factors in studying elliptic curves are conferred. An all-around computational classification on the Weierstrass elliptic curves over $\mathbb{Z}_n$ for rigorous understanding is also attached to this work.Comment: 12 pages, 2 figures, draf

An Efficient Secure Message Transmission in Mobile Ad Hoc Networks using Enhanced Homomorphic Encryption Scheme

In MANETs the nodes are capable of roaming independently. The node with inadequate physical protection can be easily captured, compromised and hijacked. Due to this huge dependency's on the nodes, there are more security problems. Therefore the nodes in the network must be prepared to work in a mode that trusts no peer. In this paper we look at the current scheme to transmit the data in MANETs. We then propose a new scheme for secure transmission of message in MANETs as Alternative scheme for DF2019;s new Ph and DF2019;s additive and multiplicative PH. Here we also provide the computational cost of the homomorphic encryption schemes. We also provide the implementation issues of our new scheme in MANETs. For the entire message to be recoverd by the attacker, the attacker needs to compromise atleast g nodes, one node from each group g and know the encryption keys to decrypt the message. The success rate of our proposed new scheme is 100% if there are more number of active paths in each group of the network

Improving the Efficiency of Homomorphic Encryption Schemes

In this dissertation, we explore different approaches to practical homomorphic encryption schemes. For partial homomorphic encryption schemes, we observe that the versatility is the main bottleneck. To solve this problem, we propose general approaches to improve versatility of them by either extending the range of supported circuits or extending the message space. These general approaches can be applied to a wide range of partial HE schemes and greatly increase the number of applications that they support. For fully homomorphic encryption schemes, the slow running speed and the large ciphertext are the main challenges. Therefore, we propose efficient implementations as well as methods to compress the ciphertext. In detail, the Gentry Halevi FHE scheme and the LTV FHE scheme are implemented and the resulting performance shows significant improvement over previous works. For ciphertext compression, the concept of scheme conversion is proposed. Given a scheme converter, we can convert between schemes with compact ciphertext for communication and homomorphic schemes for computation

The Theory and Applications of Homomorphic Cryptography

Homomorphic cryptography provides a third party with the ability to perform simple computations on encrypted data without revealing any information about the data itself. Typically, a third party can calculate one of the encrypted sum or the encrypted product of two encrypted messages. This is possible due to the fact that the encryption function is a group homomorphism, and thus preserves group operations. This makes homomorphic cryptosystems useful in a wide variety of privacy preserving protocols. A comprehensive survey of known homomorphic cryptosystems is provided, including formal definitions, security assumptions, and outlines of security proofs for each cryptosystem presented. Threshold variants of several homomorphic cryptosystems are also considered, with the first construction of a threshold Boneh-Goh-Nissim cryptosystem given, along with a complete proof of security under the threshold semantic security game of Fouque, Poupard, and Stern. This approach is based on Shoup's approach to threshold RSA signatures, which has been previously applied to the Paillier and Damg\aa rd-Jurik cryptosystems. The question of whether or not this approach is suitable for other homomorphic cryptosystems is investigated, with results suggesting that a different approach is required when decryption requires a reduction modulo a secret value. The wide variety of protocols utilizing homomorphic cryptography makes it difficult to provide a comprehensive survey, and while an overview of applications is given, it is limited in scope and intended to provide an introduction to the various ways in which homomorphic cryptography is used beyond simple addition or multiplication of encrypted messages. In the case of strong conditional oblivious tranfser, a new protocol implementing the greater than predicate is presented, utilizing some special properties of the Boneh-Goh-Nissim cryptosystem to achieve security against a malicious receiver

Tree-Structured Composition of Homomorphic Encryption: How to Weaken Underlying Assumptions

Cryptographic primitives based on infinite families of progressively weaker assumptions have been proposed by Hofheinz-Kiltz and by Shacham (the n-Linear assumptions) and by Escala et al. (the Matrix Diffie-Hellman assumptions). All of these assumptions are extensions of the decisional Diffie-Hellman (DDH) assumption. In contrast, in this paper, we construct (additive) homomorphic encryption (HE) schemes based on a new infinite family of assumptions extending the decisional Composite Residuosity (DCR) assumption. This is the first result on a primitive based on an infinite family of progressively weaker assumptions not originating from the DDH assumption. Our assumptions are indexed by rooted trees, and provides a completely different structure compared to the previous extensions of the DDH assumption. Our construction of a HE scheme is generic; based on a tree structure, we recursively combine copies of building-block HE schemes associated to each leaf of the tree (e.g., the Paillier cryptosystem, for our DCR-based result mentioned above). Our construction for depth-one trees utilizes the share-then-encrypt multiple encryption paradigm, modified appropriately to ensure security of the resulting HE schemes. We prove several separations between the CPA security of our HE schemes based on different trees; for example, the existence of an adversary capable of breaking all schemes based on depth-one trees, does not imply an adversary against our scheme based on a depth-two tree (within a computational model analogous to the generic group model). Moreover, based on our results, we give an example which reveals a type of non-monotonicity for security of generic constructions of cryptographic schemes and their building-block primitives; if the building-block primitives for a scheme are replaced with other ones secure under stronger assumptions, it may happen that the resulting scheme becomes secure under a weaker assumption than the original

Performance Optimization of Network Protocols for IEEE 802.11s-based Smart Grid Communications

The transformation of the legacy electric grid to Smart Grid (SG) poses numerous challenges in the design and development of an efficient SG communications network. While there has been an increasing interest in identifying the SG communications network and possible SG applications, specific research challenges at the network protocol have not been elaborated yet. This dissertation revisited each layer of a TCP/IP protocol stack which basically was designed for a wired network and optimized their performance in IEEE 802.11s-based Advanced Metering Infrastructure (AMI) communications network against the following challenges: security and privacy, AMI data explosion, periodic simultaneous data reporting scheduling, poor Transport Control Protocol (TCP) performance, Address Resolution Protocol (ARP) broadcast, and network interoperability. To address these challenges, layered and/or cross-layered protocol improvements were proposed for each layer of TCP/IP protocol stack. At the application layer, a tree-based periodic time schedule and a time division multiple access-based scheduling were proposed to reduce high contention when smart meters simultaneously send their reading. Homomorphic encryption performance was investigated to handle AMI data explosion while providing security and privacy. At the transport layer, a tree-based fixed Retransmission Timeout (RTO) setting and a path-error aware RTO that exploits rich information of IEEE 802.11s data-link layer path selection were proposed to address higher delay due to TCP mechanisms. At the network layer, ARP requests create broadcast storm problems in IEEE 802.11s due to the use of MAC addresses for routing. A secure piggybacking-based ARP was proposed to eliminate this issue. The tunneling mechanisms in the LTE network cause a downlink traffic problem to IEEE 802.11s. For the network interoperability, at the network layer of EPC network, a novel UE access list was proposed to address this issue. At the data-link layer, to handle QoS mismatch between IEEE 802.11s and LTE network, Dual Queues approach was proposed for the Enhanced Distributed Channel Access. The effectiveness of all proposed approaches was validated through extensive simulation experiments using a network simulator. The simulation results showed that the proposed approaches outperformed the traditional TCP/IP protocols in terms of end to end delay, packet delivery ratio, throughput, and collection time

Trapdooring Discrete Logarithms on Elliptic Curves over Rings

This paper introduces three new probabilistic encryption schemes using elliptic curves over rings. The cryptosystems are based on three specific trapdoor mechanisms allowing the recipient to recover discrete logarithms on di#erent types of curves. The first scheme is an embodiment of Naccache and Stern's cryptosystem and realizes a discrete log encryption as originally wanted in [23] by Vanstone and Zuccherato

New foundations for efficient authentication, commutative cryptography, and private disjointness testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 105-115).This dissertation presents new constructions and security definitions related to three areas: authentication, cascadable and commutative crytpography, and private set operations. Existing works relevant to each of these areas fall into one of two categories: efficient solutions lacking formal proofs of security or provably-secure, but highly inefficient solutions. This work will bridge this gap by presenting new constructions and definitions that are both practical and provably-secure. The first contribution in the area of efficient authentication is a provably-secure authentication protocol named HB+. The HB+ protocol is efficient enough to be implemented on extremely low-cost devices, or even by a patient human with a coin to flip. The security of HB+ is based on the hardness of a long-standing learning problem that is closely related to coding theory. HB+ is the first authentication protocol that is both practical for low-cost devices, like radio frequency identification (RFID) tags, and provably secure against active adversaries. The second contribution of this work is a new framework for defining and proving the security of cascadable cryptosystems, specifically commutative cryptosystems.(cont.) This new framework addresses a gap in existing security definitions that fail to handle cryptosystems where ciphertexts produced by cascadable encryption and decryption perations may contain some message-independent history. Several cryptosystems, including a new, practical commutative cryptosystem, are proven secure under this new framework. Finally, a new and efficient private disjointness testing construction named HW is offered. Unlike previous constructions, HW is secure in the face of malicious parties, but without the need for random oracles or expensive zero-knowledge protocols. HW is as efficient as previous constructions and may be implemented using standard software libraries. The security of HW is based on a novel use of subgroup assumptions. These assumptions may prove useful in solving many other private set operation problems.by Stephen A. Weis.Ph.D