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assumptions.yml
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assumptions.yml
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cssi:
name:
short: CSSI
long: Computational Supersingular Isogeny Problem
aliases:
- short: SSI-T
long: Supersingular Isogeny with Torsion
attacks:
- mitm
- tani
- vow
- castryck-decru-maino-martindale
- robert
reduces_to:
ssendring:
references:
JDF11: 'https://doi.org/10.1007/978-3-642-25405-5_2'
Pet17: 'https://eprint.iacr.org/2017/571'
QK+20: 'https://eprint.iacr.org/2020/633.pdf'
comment: >-
The problem of computing an isogeny of degree A between two random
supersingular curves, given its action on a torsion subgroup of
order B ≈ A, coprime A. First introduced in [JDF11] to claim
security of SIDH, where A and B are powers of small primes, and
then generalized in several works to cover general A and B.
variants:
special-curve:
name:
long: CSSI with special starting curve
attacks:
- castryck-decru
comment: >-
Variant of CSSI where one of the two curves is "special", in
the sense that its endomorphism ring is known and contains
endomorphisms of "small" norm.
sidh:
name:
short: SIDH
long: The SIDH problem
aliases:
- short: SSCDH
long: Supersingular Computational Diffie-Hellman Problem
reduces_to:
cssi>special-curve:
references:
JDF11: 'https://doi.org/10.1007/978-3-642-25405-5_2'
comment: >-
The analogue of CDH for SIDH.
ssddh:
name:
short: SSDDH
long: Supersingular Decision Diffie-Hellman Problem
reduces_to:
sidh:
references:
JDF11: 'https://doi.org/10.1007/978-3-642-25405-5_2'
comment: >-
The analogue of DDH for SIDH.
bsidh:
name:
short: B-SIDH
long: The B-SIDH problem
reduces_to:
cssi>special-curve:
references:
Cos19: 'https://eprint.iacr.org/2019/1145'
comment: >-
The analogue of the SIDH problem for B-SIDH.
osidh:
name:
long: OSIDH
attacks:
- dartois-de-feo
references:
CK20: https://eprint.iacr.org/2020/985.pdf
comment: >-
A generalisation of CSIDH in which the orientation
and class group action is hidden in intermediate
data, protecting against Kuperberg's subexponential
quantum attack
ssisopath:
name:
long: The supersingular isogeny path problem
attacks:
- delfs-galbraith
- biasse-jao-sankar
reduces_to:
ssendring:
references:
CGL06: 'https://eprint.iacr.org/2006/021'
Wes21: 'https://eprint.iacr.org/2021/919'
comment: >-
Given two random supersingular curves, the problem of finding an
isogeny walk between them. In some versions of the problem, one of
the two curves is fixed. First considered in [CGL06]. Proven
equivalent to the Endomorphism ring problem in [Wes21].
variants:
short:
name:
long: The short supersingular isogeny path problem
references:
GP+16: 'https://eprint.iacr.org/2016/859'
attacks:
- mitm
comment: >-
The variant of the isogeny path problem where the distance
between the two curves in some isogeny graph is bounded away
from the average distance of two random curves. [GP+16] proves
that this problem heuristically reduces to the problem of
finding paths of generic length.
ssendring:
name:
long: The supersingular endomorphism ring problem
reduces_to:
ssisopath:
references:
Koh96: 'http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf'
Cer04: 'https://arxiv.org/pdf/math/0404538'
Wes21: 'https://eprint.iacr.org/2021/919'
comment: >-
Given a random supersingular curve, the problem of computing its
endomorphism ring. First algorithms proposed in [Koh96]. Proven
equivalent to the Isogeny path problem in [Wes21].
ssendringfp:
name:
long: The supersingular endomorphism ring problem for curves over GF(p)
reduces_to:
ssendring:
vector>supersingular:
references:
Wes21: 'https://eprint.iacr.org/2021/1583'
comment: >-
Same as the supersingular endomorphism ring problem, but for
random curves over a prime field $\mathbb{F}_p$.
vector:
name:
long: Vectorization
aliases:
- short: GAIP
long: Group Action Inverse Problem
attacks:
- kuperberg
- galbraith
reduces_to:
parallelization:
quantum: yes
references:
BY01: 'https://doi.org/10.1007/3-540-38424-3_7'
Cou06: 'https://eprint.iacr.org/2006/291'
Sto10: 'https://dx.doi.org/10.3934/amc.2010.4.215'
GP+18: 'https://eprint.iacr.org/2018/1199'
comment: >-
Given two random elements $x_0$, $x_1$ in a set acted upon transitively
by a group $G$, find an element of G that maps $x_0$ to $x_1$. By random
self-reducibility, this is equivalent to the problem where $x_0$ is
fixed.
variants:
ordinary:
name:
long: Vectorization (ordinary)
comment: >-
The variant where the set is a set of ordinary curves with
endomorphism ring isomorphic to an order $\mathcal{O}$, and the set is the
class group of $\mathcal{O}$.
supersingular:
name:
long: Vectorization (supersingular)
references:
CL+18: 'https://eprint.iacr.org/2018/383'
CPV19: 'https://eprint.iacr.org/2019/1202'
Wes21: 'https://eprint.iacr.org/2021/1583'
reduces_to:
ssendringfp:
comment: >-
The variant where the set is a set of supersingular curves
defined over a prime field $\mathbb{F}_p$, and the set is the
class group ($\textrm{Cl}(\sqrt{-p})$. The reduction to the Endomorphism ring
problem over $\mathbb{F}_p$ was done in steps in [CPV19] and
[Wes21].
oriented:
name:
long: Vectorization (oriented)
references:
CK20: 'https://eprint.iacr.org/2020/985'
Wes21: 'https://eprint.iacr.org/2021/1583'
attacks:
- dartois-de-feo
reduces_to:
ssendringfp:
comment: >-
The variant where the set is a set of supersingular curves
with an *orientation* (an explicit embedding of a quadratic
imaginary order into the endomorphism ring of the curve) and
the group is the class group of the orientation. Shown to be
equivalent to $\mathcal{O}$-EndRing in [Wes21].
parallelization:
name:
long: Parallelization
aliases:
- short: GA-CDH
long: Group Action Computational Diffie-Hellman
references:
Cou06: 'https://eprint.iacr.org/2006/291'
Sto10: 'https://dx.doi.org/10.3934/amc.2010.4.215'
GP+18: 'https://eprint.iacr.org/2018/1199'
reduces_to:
vector:
comment: >-
The anaologue of CDH for group actions.
variants:
ordinary:
name:
long: Parallelization (ordinary)
supersingular:
name:
long: Parallelization (supersingular)
references:
CL+18: 'https://eprint.iacr.org/2018/383'
oriented:
name:
long: Parallelization (oriented)
aliases:
long: O-DiffieHellman
references:
CK20: 'https://eprint.iacr.org/2020/985'
Wes21: 'https://eprint.iacr.org/2021/1583'
comment: >-
Shown to be equivalent to $\mathcal{O}$-EndRing in [Wes21]
gaddh:
name:
short: GA-DDH
long: Group Action Decisional Diffie-Hellman
reduces_to:
parallelization:
references:
Sto10: 'https://dx.doi.org/10.3934/amc.2010.4.215'
AD+20: 'https://eprint.iacr.org/2020/1188'
CSV20: 'https://eprint.iacr.org/2020/151'
comment:
The anaologue of DDH for group actions.
variants:
ordinary:
name:
long: GA-DDH (ordinary)
supersingular:
name:
long: GA-DDH (supersingular)
references:
CL+18: 'https://eprint.iacr.org/2018/383'
oriented:
name:
long: GA-DDH (oriented)
references:
CK20: 'https://eprint.iacr.org/2020/985'
CH+22: 'https://eprint.iacr.org/2022/345'
dssp:
name:
short: DSSP
long: Decisional Supersingular Product
reduces_to:
ssisopath>short:
references:
DJP14: 'https://eprint.iacr.org/2011/506'
DD+21: "https://eprint.iacr.org/2021/1023"
comment: >-
Given two supersingular isogenies $\phi: E_0 \to E_1$ and
$\phi': E_2 \to E_3$ of the
same degree $A$, and given an integer $B$ coprime to $A$, decide
whether there exist isogenies $\psi: E_0 \to E_2$ and
$\psi': E_1 \to E_3$ of degree
$B$ such that $\phi,\phi',\psi,\psi'$ form a commutative diagram.