Arithmetic Considerations for Isogeny Based Cryptography

In this paper we investigate various arithmetic techniques which can be used to potentially enhance the performance in the supersingular isogeny Diffie-Hellman (SIDH) key-exchange protocol which is one of the more recent contenders in the post-quantum public-key arena. Firstly, we give a systematic overview of techniques to compute efficient arithmetic modulo $2^xp^y\pm 1$. Our overview shows that in the SIDH setting, where arithmetic over a quadratic extension field is required, the approaches based on Montgomery reduction for such primes of a special shape are to be preferred. Moreover, the outcome of our investigation reveals that there exist moduli which allow even faster implementations. Secondly, we investigate if it is beneficial to use other curve models to speed-up the elliptic curve scalar multiplication. The use of twisted Edwards curves allows one to search for efficient addition-subtraction chains for fixed scalars while this is not possible with the differential addition law when using Montgomery curves. Our preliminary results show that despite the fact that we found such efficient chains, using twisted Edwards curves does not result in faster scalar multiplication arithmetic in the setting of SIDH

Optimizations of Isogeny-based Key Exchange

Supersingular Isogeny Diffie-Hellman (SIDH) is a key exchange scheme that is believed to be quantum-resistant. It is based on the difficulty of finding a certain isogeny between given elliptic curves. Over the last nine years, optimizations have been proposed that significantly increased the performance of its implementations. Today, SIDH is a promising candidate in the US National Institute for Standards and Technology’s (NIST’s) post-quantum cryptography standardization process. This work is a self-contained introduction to the active research on SIDH from a high-level, algorithmic lens. After an introduction to elliptic curves and SIDH itself, we describe the mathematical and algorithmic building blocks of the fastest known implementations. Regarding elliptic curves, we describe which algorithms, data structures and trade-offs regard- ing elliptic curve arithmetic and isogeny computations exist and quantify their runtime cost in field operations. These findings are then tailored to the situation of SIDH. As a result, we give efficient algorithms for the performance-critical parts of the protocol

On the post-quantum future of Elliptic Curve Cryptography

This thesis is a literature study on current published quantum-resistant isogeny-based key exchange protocols. Here we cover the topic from foundations. Chapters 1 and 2 discuss classical computation models, algorithm complexity, and how these concepts support the security of modern elliptic curve cryptography methods, such as ECDH and ECDSA. Next, in Chapters 3 to 5, we present quantum computation models, and how Shor's algorithm on quantum computers presents a threat to the future security of classical asymmetric cryptography. We explore the foundations of isogeny-based cryptography, and two key exchange protocols of this kind: SIDH and CSIDH. Appendices A and B are provided for readers wanting more in-depth background explanations on the algebraic geometry of elliptic curves, and quantum mechanics respectively

Proxy Blind Signature using Hyperelliptic Curve Cryptography

Blind signature is the concept to ensure anonymity of e-coins. Untracebility and unlinkability are two main properties of real coins and should also be mimicked electronically. A user has to fulll above two properties of blind signature for permission to spend an e-coin. During the last few years, asymmetric cryptosystems based on curve based cryptographiy have become very popular, especially for embedded applications. Elliptic curves(EC) are a special case of hyperelliptic curves (HEC). HEC operand size is only a fraction of the EC operand size. HEC cryptography needs a group order of size at least 2160. In particular, for a curve of genus two eld Fq with p 280 is needeed. Therefore, the eld arithmetic has to be performed using 80-bit long operands. Which is much better than the RSA using 1024 bit key length. The hyperelliptic curve is best suited for the resource constraint environments. It uses lesser key and provides more secure transmisstion of data

Computing supersingular isogenies on Kummer surfaces

We apply Scholten\u27s construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over $GF(p^2)$ and corresponding abelian surfaces over $GF(p)$. Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2,2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography

Heuristics on pairing-friendly elliptic curves

We present a heuristic asymptotic formula as $x\to \infty$ for the number of isogeny classes of pairing-friendly elliptic curves with fixed embedding degree $k\geq 3$, with fixed discriminant, with rho-value bounded by a fixed $\rho_0$ such that $1<\rho_0<2$, and with prime subgroup order at most $x$.Comment: text substantially rewritten, tables correcte