<s>
Supersingular	B-Algorithm
isogeny	I-Algorithm
Diffie	I-Algorithm
–	I-Algorithm
Hellman	I-Algorithm
key	I-Algorithm
exchange	I-Algorithm
(	O
SIDH	B-Algorithm
or	O
SIKE	B-Algorithm
)	O
is	O
an	O
insecure	O
proposal	O
for	O
a	O
post-quantum	O
cryptographic	O
algorithm	O
to	O
establish	O
a	O
secret	O
key	O
between	O
two	O
parties	O
over	O
an	O
untrusted	O
communications	O
channel	O
.	O
</s>
<s>
It	O
is	O
analogous	O
to	O
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
key	I-Protocol
exchange	I-Protocol
,	O
but	O
is	O
based	O
on	O
walks	O
in	O
a	O
supersingular	O
isogeny	O
graph	O
and	O
was	O
designed	O
to	O
resist	O
cryptanalytic	O
attack	O
by	O
an	O
adversary	O
in	O
possession	O
of	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
Before	O
it	O
was	O
broken	O
,	O
SIDH	B-Algorithm
boasted	O
one	O
of	O
the	O
smallest	O
key	O
sizes	O
of	O
all	O
post-quantum	O
key	O
exchanges	O
;	O
with	O
compression	O
,	O
SIDH	B-Algorithm
used	O
2688-bit	O
public	B-Application
keys	I-Application
at	O
a	O
128-bit	O
quantum	O
security	O
level	O
.	O
</s>
<s>
SIDH	B-Algorithm
also	O
distinguishes	O
itself	O
from	O
similar	O
systems	O
such	O
as	O
NTRU	O
and	O
Ring-LWE	O
by	O
supporting	O
perfect	O
forward	O
secrecy	O
,	O
a	O
property	O
that	O
prevents	O
compromised	O
long-term	O
keys	O
from	O
compromising	O
the	O
confidentiality	O
of	O
old	O
communication	O
sessions	O
.	O
</s>
<s>
These	O
properties	O
seemed	O
to	O
make	O
SIDH	B-Algorithm
a	O
natural	O
candidate	O
to	O
replace	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
(	O
DHE	O
)	O
and	O
elliptic	B-Protocol
curve	I-Protocol
Diffie	I-Protocol
–	I-Protocol
Hellman	I-Protocol
(	O
ECDHE	B-Protocol
)	O
,	O
which	O
are	O
widely	O
used	O
in	O
Internet	O
communication	O
.	O
</s>
<s>
However	O
,	O
SIDH	B-Algorithm
is	O
vulnerable	O
to	O
a	O
devastating	O
key-recovery	O
attack	O
published	O
in	O
July	O
2022	O
and	O
is	O
therefore	O
insecure	O
.	O
</s>
<s>
The	O
attack	O
does	O
not	O
require	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
For	O
certain	O
classes	O
of	O
problems	O
,	O
algorithms	O
running	O
on	O
quantum	B-Architecture
computers	I-Architecture
are	O
naturally	O
capable	O
of	O
achieving	O
lower	O
time	O
complexity	O
than	O
on	O
classical	O
computers	O
.	O
</s>
<s>
That	O
is	O
,	O
quantum	B-Device
algorithms	I-Device
can	O
solve	O
certain	O
problems	O
faster	O
than	O
the	O
most	O
efficient	O
algorithm	O
running	O
on	O
a	O
traditional	O
computer	O
.	O
</s>
<s>
For	O
example	O
,	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
factor	O
an	O
integer	O
N	O
in	O
polynomial	O
time	O
,	O
while	O
the	O
best-known	O
factoring	O
classic	O
algorithm	O
,	O
the	O
general	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
,	O
operates	O
in	O
sub-exponential	O
time	O
.	O
</s>
<s>
This	O
is	O
significant	O
to	O
public	B-Application
key	I-Application
cryptography	I-Application
because	O
the	O
security	O
of	O
RSA	B-Architecture
is	O
dependent	O
on	O
the	O
infeasibility	O
of	O
factoring	O
integers	O
,	O
the	O
integer	O
factorization	O
problem	O
.	O
</s>
<s>
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
also	O
efficiently	O
solve	O
the	O
discrete	O
logarithm	O
problem	O
,	O
which	O
is	O
the	O
basis	O
for	O
the	O
security	O
of	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
,	O
elliptic	B-Protocol
curve	I-Protocol
Diffie	I-Protocol
–	I-Protocol
Hellman	I-Protocol
,	O
elliptic	O
curve	O
DSA	O
,	O
Curve25519	O
,	O
ed25519	O
,	O
and	O
ElGamal	B-Algorithm
.	O
</s>
<s>
Although	O
quantum	B-Architecture
computers	I-Architecture
are	O
currently	O
in	O
their	O
infancy	O
,	O
the	O
ongoing	O
development	O
of	O
quantum	B-Architecture
computers	I-Architecture
and	O
their	O
theoretical	O
ability	O
to	O
compromise	O
modern	O
cryptographic	O
protocols	O
(	O
such	O
as	O
TLS/SSL	B-Protocol
)	O
has	O
prompted	O
the	O
development	O
of	O
post-quantum	O
cryptography	O
.	O
</s>
<s>
SIDH	B-Algorithm
was	O
created	O
in	O
2011	O
by	O
De	O
Feo	O
,	O
Jao	O
,	O
and	O
Plut	O
.	O
</s>
<s>
SIDH	B-Algorithm
provides	O
perfect	O
forward	O
secrecy	O
and	O
thus	O
does	O
not	O
rely	O
on	O
the	O
security	O
of	O
long-term	O
private	B-Application
keys	I-Application
.	O
</s>
<s>
Forward	O
secrecy	O
improves	O
the	O
long-term	O
security	O
of	O
encrypted	O
communications	O
,	O
helps	O
defend	O
against	O
mass	O
surveillance	O
,	O
and	O
reduces	O
the	O
impact	O
of	O
vulnerabilities	O
like	O
Heartbleed	B-Error_Name
.	O
</s>
<s>
The	O
supersingular	O
isogeny	O
Diffie-Hellman	B-Protocol
protocol	O
(	O
SIDH	B-Algorithm
)	O
works	O
with	O
the	O
graph	O
whose	O
vertices	O
are	O
(	O
isomorphism	O
classes	O
of	O
)	O
supersingular	O
elliptic	O
curves	O
and	O
whose	O
edges	O
are	O
isogenies	O
between	O
those	O
curves	O
.	O
</s>
<s>
The	O
setup	O
for	O
SIDH	B-Algorithm
is	O
a	O
prime	O
of	O
the	O
form	O
,	O
for	O
different	O
(	O
small	O
)	O
primes	O
and	O
,	O
(	O
large	O
)	O
exponents	O
and	O
,	O
and	O
small	O
cofactor	O
,	O
together	O
with	O
a	O
supersingular	O
elliptic	O
curve	O
defined	O
over	O
.	O
</s>
<s>
The	O
most	O
straightforward	O
way	O
to	O
attack	O
SIDH	B-Algorithm
is	O
to	O
solve	O
the	O
problem	O
of	O
finding	O
an	O
isogeny	O
between	O
two	O
supersingular	O
elliptic	O
curves	O
with	O
the	O
same	O
number	O
of	O
points	O
.	O
</s>
<s>
At	O
the	O
time	O
of	O
the	O
original	O
publication	O
due	O
to	O
De	O
Feo	O
,	O
Jao	O
and	O
Plût	O
,	O
the	O
best	O
attack	O
known	O
against	O
SIDH	B-Algorithm
was	O
based	O
on	O
solving	O
the	O
related	O
claw	O
finding	O
problem	O
,	O
which	O
led	O
to	O
a	O
complexity	O
of	O
O( 	O
p1/4	O
)	O
for	O
classical	O
computers	O
and	O
O( 	O
p1/6	O
)	O
for	O
quantum	B-Architecture
computers	I-Architecture
.	O
</s>
<s>
This	O
suggested	O
that	O
SIDH	B-Algorithm
with	O
a	O
768-bit	O
prime	O
(	O
p	O
)	O
would	O
have	O
a	O
128-bit	O
security	O
level	O
.	O
</s>
<s>
A	O
more	O
intricate	O
attack	O
strategy	O
is	O
based	O
on	O
exploiting	O
the	O
auxiliary	O
elliptic-curve	O
points	O
present	O
in	O
SIDH	B-Algorithm
public	B-Application
keys	I-Application
,	O
which	O
in	O
principle	O
reveal	O
a	O
lot	O
of	O
additional	O
information	O
about	O
the	O
secret	O
isogenies	O
,	O
but	O
this	O
information	O
did	O
not	O
seem	O
computationally	O
useful	O
for	O
attackers	O
at	O
first	O
.	O
</s>
<s>
Petit	O
in	O
2017	O
first	O
demonstrated	O
a	O
technique	O
making	O
use	O
of	O
these	O
points	O
to	O
attack	O
some	O
rather	O
peculiar	O
SIDH	B-Algorithm
variants	O
.	O
</s>
<s>
Despite	O
follow-up	O
work	O
extending	O
the	O
attack	O
to	O
much	O
more	O
realistic	O
SIDH	B-Algorithm
instantiations	O
,	O
the	O
attack	O
strategy	O
still	O
failed	O
to	O
break	O
"	O
standard	O
"	O
SIDH	B-Algorithm
as	O
employed	O
by	O
the	O
NIST	O
PQC	O
submission	O
SIKE	B-Algorithm
.	O
</s>
<s>
In	O
July	O
2022	O
,	O
Castryck	O
and	O
Decru	O
published	O
an	O
efficient	O
key-recovery	O
attack	O
on	O
SIKE	B-Algorithm
that	O
exploits	O
the	O
auxiliary	O
points	O
.	O
</s>
<s>
The	O
attack	O
relies	O
on	O
gluing	O
together	O
multiple	O
of	O
the	O
elliptic	O
curves	O
appearing	O
in	O
the	O
SIDH	B-Algorithm
construction	O
,	O
giving	O
an	O
abelian	O
surface	O
(	O
more	O
generally	O
,	O
an	O
abelian	O
variety	O
)	O
,	O
and	O
computing	O
a	O
specially	O
crafted	O
isogeny	O
defined	O
by	O
the	O
given	O
auxiliary	O
points	O
on	O
that	O
higher-dimensional	O
object	O
.	O
</s>
<s>
It	O
should	O
be	O
stressed	O
that	O
the	O
attack	O
crucially	O
relies	O
on	O
the	O
auxiliary	O
points	O
given	O
in	O
SIDH	B-Algorithm
,	O
and	O
there	O
is	O
no	O
known	O
way	O
to	O
apply	O
similar	O
techniques	O
to	O
the	O
general	O
isogeny	O
problem	O
.	O
</s>
<s>
With	O
these	O
compression	O
techniques	O
,	O
SIDH	B-Algorithm
has	O
a	O
similar	O
bandwidth	O
requirement	O
to	O
traditional	O
3072-bit	O
RSA	B-Architecture
signatures	O
or	O
Diffie-Hellman	B-Protocol
key	I-Protocol
exchanges	I-Protocol
.	O
</s>
<s>
This	O
small	O
space	O
requirement	O
makes	O
SIDH	B-Algorithm
applicable	O
to	O
context	O
that	O
have	O
a	O
strict	O
space	O
requirement	O
,	O
such	O
as	O
Bitcoin	B-Protocol
or	O
Tor	B-Application
.	O
</s>
<s>
Tor	B-Application
's	O
data	O
cells	O
must	O
be	O
less	O
than	O
517	O
bytes	O
in	O
length	O
,	O
so	O
they	O
can	O
hold	O
330-byte	O
SIDH	B-Algorithm
keys	O
.	O
</s>
<s>
By	O
contrast	O
,	O
NTRUEncrypt	O
must	O
exchange	O
approximately	O
600	O
bytes	O
to	O
achieve	O
a	O
128-bit	O
security	O
and	O
cannot	O
be	O
used	O
within	O
Tor	B-Application
without	O
increasing	O
the	O
cell	O
size	O
.	O
</s>
<s>
In	O
2014	O
,	O
researchers	O
at	O
the	O
University	O
of	O
Waterloo	O
developed	O
a	O
software	O
implementation	O
of	O
SIDH	B-Algorithm
.	O
</s>
<s>
For	O
a	O
768-bit	O
modulus	O
they	O
were	O
able	O
to	O
complete	O
the	O
key	O
exchange	O
computations	O
in	O
200	O
milliseconds	O
thus	O
demonstrating	O
that	O
the	O
SIDH	B-Algorithm
is	O
computationally	O
practical	O
.	O
</s>
<s>
In	O
2016	O
,	O
researchers	O
from	O
Microsoft	O
posted	O
software	O
for	O
the	O
SIDH	B-Algorithm
which	O
runs	O
in	O
constant	O
time	O
(	O
thus	O
protecting	O
against	O
timing	O
attacks	O
)	O
and	O
is	O
the	O
most	O
efficient	O
implementation	O
to	O
date	O
.	O
</s>
<s>
They	O
write	O
:	O
"	O
The	O
size	O
of	O
public	B-Application
keys	I-Application
is	O
only	O
564	O
bytes	O
,	O
which	O
is	O
significantly	O
smaller	O
than	O
most	O
of	O
the	O
popular	O
post-quantum	O
key	O
exchange	O
alternatives	O
.	O
</s>
<s>
Ultimately	O
,	O
the	O
size	O
and	O
speed	O
of	O
our	O
software	O
illustrates	O
the	O
strong	O
potential	O
of	O
SIDH	B-Algorithm
as	O
a	O
post-quantum	O
key	O
exchange	O
candidate	O
and	O
we	O
hope	O
that	O
these	O
results	O
encourage	O
a	O
wider	O
cryptanalytic	O
effort.	O
"	O
</s>
<s>
In	O
2016	O
,	O
researchers	O
from	O
Florida	O
Atlantic	O
University	O
developed	O
efficient	O
ARM	O
implementations	O
of	O
SIDH	B-Algorithm
and	O
provided	O
a	O
comparison	O
of	O
affine	O
and	O
projective	O
coordinates	O
.	O
</s>
<s>
In	O
2017	O
,	O
researchers	O
from	O
Florida	O
Atlantic	O
University	O
developed	O
the	O
first	O
FPGA	O
implementations	O
of	O
SIDH	B-Algorithm
.	O
</s>
<s>
While	O
several	O
steps	O
of	O
SIDH	B-Algorithm
involve	O
complex	O
isogeny	O
calculations	O
,	O
the	O
overall	O
flow	O
of	O
SIDH	B-Algorithm
for	O
parties	O
A	O
and	O
B	O
is	O
straightforward	O
for	O
those	O
familiar	O
with	O
a	O
Diffie-Hellman	B-Protocol
key	I-Protocol
exchange	I-Protocol
or	O
its	O
elliptic	O
curve	O
variant	O
.	O
</s>
<s>
Notationally	O
,	O
the	O
SIDH	B-Algorithm
key	O
exchange	O
between	O
parties	O
A	O
and	O
B	O
works	O
as	O
follows	O
:	O
</s>
<s>
A	O
predecessor	O
to	O
the	O
SIDH	B-Algorithm
was	O
published	O
in	O
2006	O
by	O
Rostovtsev	O
and	O
Stolbunov	O
.	O
</s>
<s>
They	O
created	O
the	O
first	O
Diffie-Hellman	B-Protocol
replacement	O
based	O
on	O
elliptic	O
curve	O
isogenies	O
.	O
</s>
<s>
In	O
March	O
2014	O
,	O
researchers	O
at	O
the	O
Chinese	O
State	O
Key	O
Lab	O
for	O
Integrated	O
Service	O
Networks	O
and	O
Xidian	O
University	O
extended	O
the	O
security	O
of	O
the	O
SIDH	B-Algorithm
to	O
a	O
form	O
of	O
digital	O
signature	O
with	O
strong	O
designated	O
verifier	O
.	O
</s>
