A Family of Lightweight Twisted Edwards Curves for the Internet of Things

We introduce a set of four twisted Edwards curves that satisfy common security requirements and allow for fast implementations of scalar multiplication on 8, 16, and 32-bit processors. Our curves are defined by an equation of the form -x^2 + y^2 = 1 + dx^2y^2 over a prime field Fp, where d is a small non-square modulo p. The underlying prime fields are based on "pseudo-Mersenne" primes given by p = 2^k - c and have in common that p is congruent to 5 modulo 8, k is a multiple of 32 minus 1, and c is at most eight bits long. Due to these common features, our primes facilitate a parameterized implementation of the low-level arithmetic so that one and the same arithmetic function is able to process operands of different length. Each of the twisted Edwards curves we introduce in this paper is birationally equivalent to a Montgomery curve of the form -(A+2)y^2 = x^3 + Ax^2 + x where 4/(A+2) is small. Even though this contrasts with the usual practice of choosing A such that (A+2)/4 is small, we show that the Montgomery form of our curves allows for an equally efficient implementation of point doubling as Curve25519. The four curves we put forward roughly match the common security levels of 80, 96, 112 and 128 bits. In addition, their Weierstraß representations are isomorphic to curves of the form y^2 = x^3 - 3x + b so as to facilitate inter-operability with TinyECC and other legacy software

Efficient algorithms for supersingular isogeny Diffie-Hellman

We propose a new suite of algorithms that significantly improve the performance of supersingular isogeny Diffie-Hellman (SIDH) key exchange. Subsequently, we present a full-fledged implementation of SIDH that is geared towards the 128-bit quantum and 192-bit classical security levels. Our library is the first constant-time SIDH implementation and is up to 2.9 times faster than the previous best (non-constant-time) SIDH software. The high speeds in this paper are driven by compact, inversion-free point and isogeny arithmetic and fast SIDH-tailored field arithmetic: on an Intel Haswell processor, generating ephemeral public keys takes 46 million cycles for Alice and 54 million cycles for Bob, while computing the shared secret takes 44 million and 52 million cycles, respectively. The size of public keys is only 564 bytes, which is significantly smaller than most of the popular post-quantum key exchange alternatives. Ultimately, the size and speed of our software illustrates the strong potential of SIDH as a post-quantum key exchange candidate and we hope that these results encourage a wider cryptanalytic effort

How Not to Create an Isogeny-Based PAKE

Isogeny-based key establishment protocols are believed to be resistant to quantum cryptanalysis. Two such protocols---supersingular isogeny Diffie-Hellman (SIDH) and commutative supersingular isogeny Diffie-Hellman (CSIDH)---are of particular interest because of their extremely small public key sizes compared with other post-quantum candidates. Although SIDH and CSIDH allow us to achieve key establishment against passive adversaries and authenticated key establishment (using generic constructions), there has been little progress in the creation of provably-secure isogeny-based password-authenticated key establishment protocols (PAKEs). This is in stark contrast with the classical setting, where the Diffie-Hellman protocol can be tweaked in a number of straightforward ways to construct PAKEs, such as EKE, SPEKE, PAK (and variants), J-PAKE, and Dragonfly. Although SIDH and CSIDH superficially resemble Diffie-Hellman, it is often difficult or impossible to ``translate\u27\u27 these Diffie-Hellman-based protocols to the SIDH or CSIDH setting; worse still, even when the construction can be ``translated,\u27\u27 the resultant protocol may be insecure, even if the Diffie-Hellman based protocol is secure. In particular, a recent paper of Terada and Yoneyama and ProvSec 2019 purports to instantiate encrypted key exchange (EKE) over SIDH and CSIDH; however, there is a subtle problem which leads to an offline dictionary attack on the protocol, rendering it insecure. In this work we present man-in-the-middle and offline dictionary attacks on isogeny-based PAKEs from the literature, and explain why other classical constructions do not ``translate\u27\u27 securely to the isogeny-based setting

SPDH-Sign: towards Efficient, Post-quantum Group-based Signatures

In this paper, we present a new diverse class of post-quantum group-based Digital Signature Schemes (DSS). The approach is significantly different from previous examples of group-based digital signatures and adopts the framework of group action-based cryptography: we show that each finite group defines a group action relative to the semidirect product of the group by its automorphism group, and give security bounds on the resulting signature scheme in terms of the group-theoretic computational problem known as the Semidirect Discrete Logarithm Problem (SDLP). Crucially, we make progress towards being able to efficiently compute the novel group action, and give an example of a parameterised family of groups for which the group action can be computed for any parameters, thereby negating the need for expensive offline computation or inclusion of redundancy required in other schemes of this type

A Formula for Disaster : A Unified Approach to Elliptic Curve Special-Point-Based Attacks

The Refined Power Analysis, Zero-Value Point, and Exceptional Procedure attacks introduced side-channel techniques against specific cases of elliptic curve cryptography. The three attacks recover bits of a static ECDH key adaptively, collecting information on whether a certain multiple of the input point was computed. We unify and generalize these attacks in a common framework, and solve the corresponding problem for a broader class of inputs. We also introduce a version of the attack against windowed scalar multiplication methods, recovering the full scalar instead of just a part of it. Finally, we systematically analyze elliptic curve point addition formulas from the Explicit-Formulas Database, classify all non-trivial exceptional points, and find them in new formulas. These results indicate the usefulness of our tooling, which we released publicly, for unrolling formulas and finding special points, and potentially for independent future work.acceptedVersionPeer reviewe

Envisioning the Future of Cyber Security in Post-Quantum Era: A Survey on PQ Standardization, Applications, Challenges and Opportunities

The rise of quantum computers exposes vulnerabilities in current public key cryptographic protocols, necessitating the development of secure post-quantum (PQ) schemes. Hence, we conduct a comprehensive study on various PQ approaches, covering the constructional design, structural vulnerabilities, and offer security assessments, implementation evaluations, and a particular focus on side-channel attacks. We analyze global standardization processes, evaluate their metrics in relation to real-world applications, and primarily focus on standardized PQ schemes, selected additional signature competition candidates, and PQ-secure cutting-edge schemes beyond standardization. Finally, we present visions and potential future directions for a seamless transition to the PQ era

Selecting Elliptic Curves for Cryptography: An Efficiency and Security Analysis

We select a set of elliptic curves for cryptography and analyze our selection from a performance and security perspective. This analysis complements recent curve proposals that suggest (twisted) Edwards curves by also considering the Weierstrass model. Working with both Montgomery-friendly and pseudo-Mersenne primes allows us to consider more possibilities which help to improve the overall efficiency of base field arithmetic. Our Weierstrass curves are backwards compatible with current implementations of prime order NIST curves, while providing improved efficiency and stronger security properties. We choose algorithms and explicit formulas to demonstrate that our curves support constant-time, exception-free scalar multiplications, thereby offering high practical security in cryptographic applications. Our implementation shows that variable-base scalar multiplication on the new Weierstrass curves at the 128-bit security level is about 1.4 times faster than the recent implementation record on the corresponding NIST curve. For practitioners who are willing to use a different curve model and sacrifice a few bits of security, we present a collection of twisted Edwards curves with particularly efficient arithmetic that are up to 1.42, 1.26 and 1.24 times faster than the new Weierstrass curves at the 128-, 192- and 256-bit security levels, respectively. Finally, we discuss how these curves behave in a real-world protocol by considering different scalar multiplication scenarios in the transport layer security (TLS) protocol. The proposed curves and the results of the analysis are intended to contribute to the recent efforts towards recommending new elliptic curves for Internet standards

Embedded document security using sticky policies and identity based encryption

Data sharing domains have expanded over several, both trusted and insecure environments. At the same time, the data security boundaries have shrunk from internal network perimeters down to a single identity and a piece of information. Since new EU GDPR regulations, the personally identifiable information sharing requires data governance in favour of a data subject. Existing enterprise grade IRM solutions fail to follow open standards and lack of data sharing frameworks that could efficiently integrate with existing identity management and authentication infrastructures. IRM services that stood against cloud demands often offer a very limited access control functionality allowing an individual to store a document online giving a read or read-write permission to other individual identified by email address. Unfortunately, such limited information sharing controls are often introduced as the only safeguards in large enterprises, healthcare institutions and other organizations that should provide the highest possible personal data protection standards. The IRM suffers from a systems architecture vulnerability where IRM application installed on a semi-trusted client truly only guarantees none or full access enforcement. Since no single authority is contacted to verify each committed change the adversary having an advantage of possessing data-encrypting and key-encrypting keys could change and re-encrypt the amended content despite that read only access has been granted. Finally, the two evaluated IRM products, have either the algorithm security lifecycle (ASL) relatively short to protect the shared data, or the solution construct highly restrained secure key-encrypting key distribution and exposes a symmetric data-encrypting key over the network. Presented here sticky policy with identity-based encryption (SPIBE) solution was designed for secure cloud data sharing. SPIBE challenges are to deliver simple standardized construct that would easily integrate with popular OOXML-like document formats and provide simple access rights enforcement over protected content. It leverages a sticky policy construct using XACML access policy language to express access conditions across different cloud data sharing boundaries. XACML is a cloud-ready standard designed for a global multi-jurisdictional use. Unlike other raw ABAC implementations, the XACML offers a standardised schema and authorisation protocols hence it simplifies interoperability. The IBE is a cryptographic scheme protecting the shared document using an identified policy as an asymmetric key-encrypting a symmetric data-encrypting key. Unlike ciphertext-policy attribute-based access control (CP-ABE), the SPIBE policy contains not only access preferences but global document identifier and unique version identifier what makes each policy uniquely identifiable in relation to the protected document. In IBE scheme the public key-encrypting key is known and could be shared between the parties although the data-encrypting key is never sent over the network. Finally, the SPIBE as a framework should have a potential to protect data in case of new threats where ASL of a used cryptographic primitive is too short, when algorithm should be replaced with a new updated cryptographic primitive. The IBE like a cryptographic protocol could be implemented with different cryptographic primitives. The identity-based encryption over isogenous pairing groups (IBE-IPG) is a post-quantum ready construct that leverages the initial IBE Boneh-Franklin (IBE-BF) approach. Existing IBE implementations could be updated to IBE-IPG without major system amendments. Finally, by applying the one document versioning blockchain-like construct could verify changes authenticity and approve only legitimate document updates, where other IRM solutions fail to operate delivering the one single authority for non-repudiation and authenticity assurance