# High-degree compression functions on alternative models of elliptic curves and their applications

@article{Wronski2021HighdegreeCF, title={High-degree compression functions on alternative models of elliptic curves and their applications}, author={Michal Wro'nski and Tomasz Kijko and Robert Drylo}, journal={Fundam. Informaticae}, year={2021}, volume={184}, pages={107-139} }

This paper presents method for obtaining high-degree compression functions using natural symmetries in a given model of an elliptic curve. Such symmetries may be found using symmetry of involution [–1] and symmetry of translation morphism τT = P + T, where T is the n-torsion point which naturally belongs to the E(𝕂) for a given elliptic curve model. We will study alternative models of elliptic curves with points of order 2 and 4, and specifically Huff’s curves and the Hessian family of…

## One Citation

### Application of Velusqrt algorithm to Huff's and general Huff's curves

- Computer ScienceIACR Cryptol. ePrint Arch.
- 2021

It is showed how to compute odd-degree isogeny on Huff's and general Huff’s curves using Velusqrt algorithm and x-line arithmetic for different compression functions.

## References

SHOWING 1-10 OF 62 REFERENCES

### Efficient Arithmetic on Hessian Curves

- Computer Science, MathematicsPublic Key Cryptography
- 2010

A generalized form for Hessian curves is considered and it is shown to be equivalent to the family of elliptic curves with a torsion subgroup isomorphic to ℤ/3ℤ.

### Efficient arithmetic on elliptic curves using a mixed Edwards-Montgomery representation

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2008

The observation that switching between the Edwards model and the Montgomery model is quasi cost-free is used to speed up Montgomery’s algorithm, reducing the complexity of a doubling step from 2M + 2S to 1S for suitably chosen curve parameters.

### Symmetrized Summation Polynomials: Using Small Order Torsion Points to Speed Up Elliptic Curve Index Calculus

- Computer Science, MathematicsEUROCRYPT
- 2014

The goal is to use the existence of small order torsion points on E to define new summation polynomials whose symmetrized expressions are much more compact and easier to compute, and to give new timings for the computation of relations with degree 5 extension fields.

### Efficient Arithmetic on Elliptic Curves in Characteristic 2

- Mathematics, Computer ScienceINDOCRYPT
- 2012

An algorithm for scalar multiplication with point recovery is developed which computes the multiple of a point $P$ with $4M + 4S + 2m_c + m-t$ per bit where $m_t$ is multiplication by a constant that depends on $P$.

### Efficient Montgomery-like formulas for general Huff's and Huff's elliptic curves and their applications to the isogeny-based cryptography

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2020

Formulas for doubling and differential addition after compression are provided, which for Huff’s curves are as efficient as Montgomery”s formulas for Montgomery“s curves.

### Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

- Computer Science, MathematicsLecture Notes in Computer Science
- 2009

This work discusses the construction of Authentication/Secrecy Codes, performance analysis of M-PSK Signal Constellations in Riemannian Varieties, and fast Decomposition of Polynomials with Known Galois Group.

### Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies

- Computer Science, MathematicsJ. Math. Cryptol.
- 2011

A new zero-knowledge identification scheme and detailed security proofs for the protocols, and a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data are presented.

### Huff's Model for Elliptic Curves

- Mathematics, Computer ScienceANTS
- 2010

This paper revisits a model for elliptic curves over ℚ introduced by Huff in 1948 to study a diophantine problem and finds that every elliptic curve over such a field and containing a copy of ℤ/4ℤ ×℧/2ℚ is birationally equivalent to a Huff curve over the original field.

### Twisted Hessian Curves

- Mathematics, Computer ScienceLATINCRYPT
- 2015

This paper presents new speed records for arithmetic on a large family of elliptic curves with cofactor 3: specifically, 8.77Mper bit for 256-bit variable-base single-scalar multiplication when curve…