<s>
In	O
public-key	B-Application
cryptography	I-Application
,	O
Edwards-curve	B-Algorithm
Digital	I-Algorithm
Signature	I-Algorithm
Algorithm	I-Algorithm
(	O
EdDSA	B-Algorithm
)	O
is	O
a	O
digital	O
signature	O
scheme	O
using	O
a	O
variant	O
of	O
Schnorr	B-Algorithm
signature	I-Algorithm
based	O
on	O
twisted	O
Edwards	O
curves	O
.	O
</s>
<s>
The	O
reference	O
implementation	O
is	O
public-domain	B-Application
software	I-Application
.	O
</s>
<s>
The	O
following	O
is	O
a	O
simplified	O
description	O
of	O
EdDSA	B-Algorithm
,	O
ignoring	O
details	O
of	O
encoding	O
integers	O
and	O
curve	O
points	O
as	O
bit	O
strings	O
;	O
the	O
full	O
details	O
are	O
in	O
the	O
papers	O
and	O
RFC	O
.	O
</s>
<s>
An	O
EdDSA	B-Algorithm
signature	O
scheme	O
is	O
a	O
choice	O
:	O
</s>
<s>
of	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
with	O
-bit	O
outputs	O
,	O
where	O
so	O
that	O
elements	O
of	O
and	O
curve	O
points	O
in	O
can	O
be	O
represented	O
by	O
strings	O
of	O
bits	O
.	O
</s>
<s>
These	O
parameters	O
are	O
common	O
to	O
all	O
users	O
of	O
the	O
EdDSA	B-Algorithm
signature	O
scheme	O
.	O
</s>
<s>
The	O
security	O
of	O
the	O
EdDSA	B-Algorithm
signature	O
scheme	O
depends	O
critically	O
on	O
the	O
choices	O
of	O
parameters	O
,	O
except	O
for	O
the	O
arbitrary	O
choice	O
of	O
base	O
point	O
—	O
for	O
example	O
,	O
Pollard	O
's	O
rho	O
algorithm	O
for	O
logarithms	O
is	O
expected	O
to	O
take	O
approximately	O
curve	O
additions	O
before	O
it	O
can	O
compute	O
a	O
discrete	O
logarithm	O
,	O
so	O
must	O
be	O
large	O
enough	O
for	O
this	O
to	O
be	O
infeasible	O
,	O
and	O
is	O
typically	O
taken	O
to	O
exceed	O
.	O
</s>
<s>
The	O
hash	B-Algorithm
function	I-Algorithm
is	O
normally	O
modelled	O
as	O
a	O
random	B-Application
oracle	I-Application
in	O
formal	O
analyses	O
of	O
EdDSA	B-Algorithm
's	O
security	O
.	O
</s>
<s>
In	O
the	O
HashEdDSA	O
variant	O
,	O
an	O
additional	O
collision-resistant	O
hash	B-Algorithm
function	I-Algorithm
is	O
needed	O
.	O
</s>
<s>
Within	O
an	O
EdDSA	B-Algorithm
signature	O
scheme	O
,	O
</s>
<s>
An	O
EdDSA	B-Algorithm
public	B-Application
key	I-Application
is	O
a	O
curve	O
point	O
,	O
encoded	O
in	O
bits	O
.	O
</s>
<s>
An	O
EdDSA	B-Algorithm
signature	O
on	O
a	O
message	O
by	O
public	B-Application
key	I-Application
is	O
the	O
pair	O
,	O
encoded	O
in	O
bits	O
,	O
of	O
a	O
curve	O
point	O
and	O
an	O
integer	O
satisfying	O
the	O
following	O
verification	O
equation	O
.	O
</s>
<s>
An	O
EdDSA	B-Algorithm
private	B-Application
key	I-Application
is	O
a	O
-bit	O
string	O
which	O
should	O
be	O
chosen	O
uniformly	O
at	O
random	O
.	O
</s>
<s>
The	O
corresponding	O
public	B-Application
key	I-Application
is	O
,	O
where	O
is	O
the	O
least	O
significant	O
bits	O
of	O
interpreted	O
as	O
an	O
integer	O
in	O
little-endian	O
.	O
</s>
<s>
is	O
SHA-512	B-Algorithm
,	O
with	O
.	O
</s>
<s>
The	O
original	O
team	O
has	O
optimized	O
Ed25519	O
for	O
the	O
x86-64	B-Device
Nehalem/Westmere	O
processor	O
family	O
.	O
</s>
<s>
Ed25519	O
is	O
intended	O
to	O
provide	O
attack	O
resistance	O
comparable	O
to	O
quality	O
128-bit	O
symmetric	B-Algorithm
ciphers	I-Algorithm
.	O
</s>
<s>
Public	B-Application
keys	I-Application
are	O
256	O
bits	O
long	O
and	O
signatures	O
are	O
512	O
bits	O
long	O
.	O
</s>
<s>
As	O
with	O
other	O
discrete-log-based	O
signature	O
schemes	O
,	O
EdDSA	B-Algorithm
uses	O
a	O
secret	O
value	O
called	O
a	O
nonce	O
unique	O
to	O
each	O
signature	O
.	O
</s>
<s>
In	O
the	O
signature	O
schemes	O
DSA	O
and	O
ECDSA	O
,	O
this	O
nonce	O
is	O
traditionally	O
generated	O
randomly	O
for	O
each	O
signature	O
—	O
and	O
if	O
the	O
random	O
number	O
generator	O
is	O
ever	O
broken	O
and	O
predictable	O
when	O
making	O
a	O
signature	O
,	O
the	O
signature	O
can	O
leak	O
the	O
private	B-Application
key	I-Application
,	O
as	O
happened	O
with	O
the	O
Sony	B-Operating_System
PlayStation	I-Operating_System
3	I-Operating_System
firmware	O
update	O
signing	O
key	O
.	O
</s>
<s>
In	O
contrast	O
,	O
EdDSA	B-Algorithm
chooses	O
the	O
nonce	O
deterministically	O
as	O
the	O
hash	B-Algorithm
of	O
a	O
part	O
of	O
the	O
private	B-Application
key	I-Application
and	O
the	O
message	O
.	O
</s>
<s>
Thus	O
,	O
once	O
a	O
private	B-Application
key	I-Application
is	O
generated	O
,	O
EdDSA	B-Algorithm
has	O
no	O
further	O
need	O
for	O
a	O
random	O
number	O
generator	O
in	O
order	O
to	O
make	O
signatures	O
,	O
and	O
there	O
is	O
no	O
danger	O
that	O
a	O
broken	O
random	O
number	O
generator	O
used	O
to	O
make	O
a	O
signature	O
will	O
reveal	O
the	O
private	B-Application
key	I-Application
.	O
</s>
<s>
Note	O
that	O
there	O
are	O
two	O
standardization	O
efforts	O
for	O
EdDSA	B-Algorithm
,	O
one	O
from	O
IETF	O
,	O
an	O
informational	O
and	O
one	O
from	O
NIST	O
as	O
part	O
of	O
FIPS	O
186-5	O
.	O
</s>
<s>
Notable	O
uses	O
of	O
Ed25519	O
include	O
OpenSSH	B-Language
,	O
GnuPG	B-Application
and	O
various	O
alternatives	O
,	O
and	O
the	O
signify	O
tool	O
by	O
OpenBSD	B-Operating_System
.	O
</s>
<s>
Usage	O
of	O
Ed25519	O
(	O
and	O
Ed448	O
)	O
in	O
the	O
SSH	O
protocol	B-Protocol
has	O
been	O
standardized	O
.	O
</s>
<s>
Ed448	O
is	O
the	O
EdDSA	B-Algorithm
signature	O
scheme	O
using	O
SHAKE256	B-Algorithm
and	O
Curve448	O
defined	O
in	O
.	O
</s>
