# Questions tagged [isogeny]

Elliptic curve isogenies are structure-preserving maps between elliptic curves which have been proposed as a foundation of post-quantum cryptosystems.

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### What basic knowledge is required to understand SIKE?

I'm interested in learning about Supersingular Isogeny based Key Encapsulation mechanism. Currently, I only know all the basic knowledge about how standard Elliptic Curve Cryptography, works, using ...
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### Isogeny-based AKE: what is the difference between Gal 1 and FTTY 1?

What is the difference between (a) the AKE scheme proposed by Galbraith (based on TS2 protocol) in https://eprint.iacr.org/2018/266.pdf (page 9-10); and (b) the AKE scheme proposed by Fujioka et al. ...
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### Characteristics of an isogeny between super-singular elliptic curves

I believe I've read this before, but I can't find it despite hours of searching on Google. I've know the common definition of isogeny in elliptic curves, as $\phi:E_1 \rightarrow E_2$ a nonconstant ...
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### SIKE as Diffie-Hellman exchange

SIKE involves a Diffie-Hellman like construction using isogenies between elliptic curves. Is it possible to express it as an instance of Diffie-Hellman? I.e. what is the underlying group if so, and ...
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### How to Reduce a Quaternion Ideal into Power Smoothness?

(TL;DR) How exactly do we reduce a quaternion ideal into another powersmooth one? Given a supersingular elliptic curve, it is known that its endomorphism ring is non-commutative. Specifically, there ...
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### Partially Repeated Roots of Classical Modular Polynomial

So I was trying to compute a normalized model of elliptic curve as described here. Consider $p$= ...
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### How to Sample from Frobenius Eigenspace?

So I was implementing the $2$-point method described here, which requires to samples two points $P_0, P_1$ in the Frobenius eigenspace initially. It uses a method called Elligator, which seems to ...
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### SIDH cryptosystem question

I'm trying to understand the SIDH cryptosystem and got confused at this point: Alice fixes base $\{P_A,Q_A\}$ so that it generates $E_0[l_A^{e_A}]$. Then she chooses secret parameters $m_A,n_A$ and ...
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### How to get an optimal strategy in computing isogenies for SIDH/SIKE

How to get a strategy $(s_1,...,s_{t-1})$ as mentioned in section 1.3.7 of SIKE spec? If possible, can anyone give me an example? And why do we need to compute all leaf point? I though we just need ...
237 views

### Quantum bit level security of Supersingular Isogeny based Diffie Hellman scheme

I'm a newbie in studying and learning how SIDH works. I have a simple doubt. So my question is in the SIKE submission present in this link (https://csrc.nist.gov/Projects/Post-Quantum-Cryptography/...
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### How to compute a smooth-degree isogeny given a generator point of its kernel?

I need to compute an isogeny from $E: y^2 = x^3 + ax + b$, given a generator point for its kernel subgroup, using Python. The point has smooth order. I need both the parameters for the curve and ...
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### Supersingular isogeny Diffie–Hellman key exchange (SIDH) Library Support AES-256?

In the version 3.0 of the SIDH library, it implements SIDH/SIKEp503 (AES-128) and SIDH/SIKEp751 (AES-192) only. Are there plans to incorporate AES-256? Nathan Aw
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### Any Supersingular Isogeny-based Diffie-Hellman (SIDH) key exchange recommended Curve Domain Parameters?

For existing ECDH, I understand that there are recommended Elliptic Curve Domain Parameters. May I know if there are such similar considerations in SIDH? Any recommended Curve Domain Parameters?
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### Can Microsoft's SIDH (Supersingular Isogeny) keypairs be reused for encryption? If not, why?

I was considering using Microsoft's SIDH implementation for post-quantum public-key encryption because of its relatively small key size. I realized however, thanks to Issue #4, that it might not be as ...
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### How can I calculate the security level provided by a supersingular Elliptic Curve?

I want to know what security level is provided by an elliptic curve used in Supersingular isogeny Diffie–Hellman key exchange (SIDH). Is there any mathematical convention to follow or by looking at ...
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### Elliptic Curve Isogenies, Frobenius endomorphism relation to characteristic equation

In Schoof's 1995 paper, Counting points on elliptic curves over finite fields, page 236, Proposition 6.1(i) states: Let $\mathbb{E}$ be an elliptic curve over $\mathbb{F}_p$. Suppose that its $j$-...
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### How does the key size in supersingular isogeny schemes relate to their security level?

I'm looking at the De Feo, Jao, and Plût 2014 paper: Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. My understanding of section 3.2 Key Exchange, is that Alice'...
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### Finding the subgroup in isogeny-based cryptography

Isogeny-based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is theorem: Elliptic curves are ...
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### Supersingular Isogeny Key Exchange broken?

Found this report detailing a quantum algorithm for computing isogenies between supersingular elliptic curves. http://cacr.uwaterloo.ca/techreports/2014/cacr2014-24.pdf with the quote "...
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### DSSP reduction to DSSI

In “Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies" by DeFeo, Jao and Plut, a reduction from the Decisional Supersingular Product (DSSP) problem to Decisional ...
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### Parameter choice Supersingular Isogeny DH

In “Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies” by DeFeo, Jao and Plut (PDF), the public parameters are defined as: Supersingular curve $E$, and bases $P, Q$ ...