<s>
In	O
cryptography	O
,	O
XTR	B-Algorithm
is	O
an	O
algorithm	O
for	O
public-key	B-Application
encryption	I-Application
.	O
</s>
<s>
XTR	B-Algorithm
stands	O
for	O
'	O
ECSTR	O
 '	O
,	O
which	O
is	O
an	O
abbreviation	O
for	O
Efficient	O
and	O
Compact	O
Subgroup	O
Trace	O
Representation	O
.	O
</s>
<s>
From	O
a	O
security	O
point	O
of	O
view	O
,	O
XTR	B-Algorithm
relies	O
on	O
the	O
difficulty	O
of	O
solving	O
Discrete	O
Logarithm	O
related	O
problems	O
in	O
the	O
full	O
multiplicative	O
group	O
of	O
a	O
finite	O
field	O
.	O
</s>
<s>
Unlike	O
many	O
cryptographic	O
protocols	O
that	O
are	O
based	O
on	O
the	O
generator	O
of	O
the	O
full	O
multiplicative	O
group	O
of	O
a	O
finite	O
field	O
,	O
XTR	B-Algorithm
uses	O
the	O
generator	O
of	O
a	O
relatively	O
small	O
subgroup	O
of	O
some	O
prime	O
order	O
of	O
a	O
subgroup	O
of	O
.	O
</s>
<s>
With	O
the	O
right	O
choice	O
of	O
,	O
computing	O
Discrete	O
Logarithms	O
in	O
the	O
group	O
,	O
generated	O
by	O
,	O
is	O
,	O
in	O
general	O
,	O
as	O
hard	O
as	O
it	O
is	O
in	O
and	O
thus	O
cryptographic	O
applications	O
of	O
XTR	B-Algorithm
use	O
arithmetics	O
while	O
achieving	O
full	O
security	O
leading	O
to	O
substantial	O
savings	O
both	O
in	O
communication	O
and	O
computational	O
overhead	O
without	O
compromising	O
security	O
.	O
</s>
<s>
Some	O
other	O
advantages	O
of	O
XTR	B-Algorithm
are	O
its	O
fast	O
key	O
generation	O
,	O
small	O
key	O
sizes	O
and	O
speed	O
.	O
</s>
<s>
XTR	B-Algorithm
uses	O
a	O
subgroup	O
,	O
commonly	O
referred	O
to	O
as	O
XTR	B-Algorithm
subgroup	O
or	O
just	O
XTR	B-Algorithm
group	O
,	O
of	O
a	O
subgroup	O
called	O
XTR	B-Algorithm
supergroup	O
,	O
of	O
the	O
multiplicative	O
group	O
of	O
a	O
finite	O
field	O
with	O
elements	O
.	O
</s>
<s>
The	O
XTR	B-Algorithm
supergroup	O
is	O
of	O
order	O
,	O
where	O
p	O
is	O
a	O
prime	O
such	O
that	O
a	O
sufficiently	O
large	O
prime	O
q	O
divides	O
.	O
</s>
<s>
The	O
XTR	B-Algorithm
subgroup	O
has	O
now	O
order	O
q	O
and	O
is	O
,	O
as	O
a	O
subgroup	O
of	O
,	O
a	O
cyclic	O
group	O
with	O
generator	O
g	O
.	O
The	O
following	O
three	O
paragraphs	O
will	O
describe	O
how	O
elements	O
of	O
the	O
XTR	B-Algorithm
supergroup	O
can	O
be	O
represented	O
using	O
an	O
element	O
of	O
instead	O
of	O
an	O
element	O
of	O
and	O
how	O
arithmetic	O
operations	O
take	O
place	O
in	O
instead	O
of	O
in	O
.	O
</s>
<s>
The	O
cost	O
of	O
arithmetic	O
operations	O
is	O
now	O
given	O
in	O
the	O
following	O
Lemma	O
labeled	O
Lemma	O
2.21	O
in	O
"	O
An	O
overview	O
of	O
the	O
XTR	B-Algorithm
public	B-Application
key	I-Application
system	O
"	O
:	O
</s>
<s>
The	O
trace	O
in	O
XTR	B-Algorithm
is	O
always	O
considered	O
over	O
.	O
</s>
<s>
Consider	O
now	O
the	O
generator	O
of	O
the	O
XTR	B-Algorithm
subgroup	O
of	O
a	O
prime	O
order	O
.	O
</s>
<s>
Remember	O
that	O
is	O
a	O
subgroup	O
of	O
the	O
XTR	B-Algorithm
supergroup	O
of	O
order	O
,	O
so	O
.	O
</s>
<s>
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
key	I-Protocol
exchange	I-Protocol
by	O
and	O
thus	O
obtaining	O
a	O
factor	O
of	O
3	O
reduction	O
in	O
representation	O
size	O
.	O
</s>
<s>
A	O
.	O
Lenstra	O
and	O
E	O
.	O
Verheul	O
give	O
this	O
algorithm	O
in	O
their	O
paper	O
titled	O
The	O
XTR	B-Algorithm
public	B-Application
key	I-Application
system	O
in	O
.	O
</s>
<s>
To	O
achieve	O
security	O
comparable	O
to	O
1024-bit	O
RSA	B-Architecture
,	O
we	O
should	O
choose	O
about	O
1024	O
,	O
i.e.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
such	O
may	O
be	O
undesirable	O
from	O
a	O
security	O
point	O
of	O
view	O
because	O
they	O
may	O
make	O
an	O
attack	O
with	O
the	O
Discrete	O
Logarithm	O
variant	O
of	O
the	O
Number	B-Algorithm
Field	I-Algorithm
Sieve	I-Algorithm
easier	O
.	O
</s>
<s>
But	O
,	O
given	O
,	O
a	O
generator	O
of	O
the	O
XTR	B-Algorithm
(	O
sub	O
)	O
group	O
can	O
be	O
found	O
by	O
determining	O
any	O
root	O
of	O
which	O
has	O
been	O
defined	O
above	O
.	O
</s>
<s>
More	O
details	O
to	O
the	O
algorithm	O
,	O
its	O
correctness	O
,	O
runtime	O
and	O
the	O
proof	O
of	O
the	O
Lemma	O
can	O
be	O
found	O
in	O
"	O
An	O
overview	O
of	O
the	O
XTR	B-Algorithm
public	B-Application
key	I-Application
system	O
"	O
in	O
.	O
</s>
<s>
In	O
general	O
,	O
XTR	B-Algorithm
can	O
be	O
used	O
in	O
any	O
cryptosystem	O
that	O
relies	O
on	O
the	O
(	O
subgroup	O
)	O
Discrete	O
Logarithm	O
problem	O
.	O
</s>
<s>
Two	O
important	O
applications	O
of	O
XTR	B-Algorithm
are	O
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
key	O
agreement	O
and	O
the	O
ElGamal	B-Algorithm
encryption	I-Algorithm
.	O
</s>
<s>
We	O
will	O
start	O
first	O
with	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
.	O
</s>
<s>
We	O
suppose	O
that	O
both	O
Alice	O
and	O
Bob	O
have	O
access	O
to	O
the	O
XTR	B-Algorithm
public	B-Application
key	I-Application
data	O
and	O
intend	O
to	O
agree	O
on	O
a	O
shared	O
secret	O
key	O
.	O
</s>
<s>
They	O
can	O
do	O
this	O
by	O
using	O
the	O
following	O
XTR	B-Algorithm
version	O
of	O
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
key	I-Protocol
exchange	I-Protocol
:	O
</s>
<s>
For	O
the	O
ElGamal	B-Algorithm
encryption	I-Algorithm
we	O
suppose	O
now	O
that	O
Alice	O
is	O
the	O
owner	O
of	O
the	O
XTR	B-Algorithm
public	B-Application
key	I-Application
data	O
and	O
that	O
she	O
has	O
selected	O
a	O
secret	O
integer	O
,	O
computed	O
and	O
published	O
the	O
result	O
.	O
</s>
<s>
Given	O
Alice	O
's	O
XTR	B-Algorithm
public	B-Application
key	I-Application
data	O
,	O
Bob	O
can	O
encrypt	O
a	O
message	O
,	O
intended	O
for	O
Alice	O
,	O
using	O
the	O
following	O
XTR	B-Algorithm
version	O
of	O
the	O
ElGamal	B-Algorithm
encryption	I-Algorithm
:	O
</s>
<s>
Bob	O
determines	O
a	O
symmetric	B-Algorithm
encryption	I-Algorithm
key	O
based	O
on	O
.	O
</s>
<s>
Bob	O
uses	O
an	O
agreed	O
upon	O
symmetric	B-Algorithm
encryption	I-Algorithm
method	O
with	O
key	O
to	O
encrypt	O
his	O
message	O
,	O
resulting	O
in	O
the	O
encryption	O
.	O
</s>
<s>
Alice	O
determines	O
the	O
symmetric	B-Algorithm
key	I-Algorithm
based	O
on	O
.	O
</s>
<s>
Alice	O
uses	O
the	O
agreed	O
upon	O
symmetric	B-Algorithm
encryption	I-Algorithm
method	O
with	O
key	O
to	O
decrypt	O
,	O
resulting	O
in	O
the	O
original	O
message	O
.	O
</s>
<s>
The	O
here	O
described	O
encryption	O
scheme	O
is	O
based	O
on	O
a	O
common	O
hybrid	O
version	O
of	O
the	O
ElGamal	B-Algorithm
encryption	I-Algorithm
,	O
where	O
the	O
secret	O
key	O
is	O
obtained	O
by	O
an	O
asymmetric	B-Application
public	I-Application
key	I-Application
system	O
and	O
then	O
the	O
message	O
is	O
encrypted	O
with	O
a	O
symmetric	B-Algorithm
key	I-Algorithm
encryption	I-Algorithm
method	O
Alice	O
and	O
Bob	O
agreed	O
to	O
.	O
</s>
<s>
In	O
the	O
more	O
traditional	O
ElGamal	B-Algorithm
encryption	I-Algorithm
the	O
message	O
is	O
restricted	O
to	O
the	O
key	O
space	O
,	O
which	O
would	O
here	O
be	O
,	O
because	O
.	O
</s>
<s>
In	O
order	O
to	O
say	O
something	O
about	O
the	O
security	O
properties	O
of	O
the	O
above	O
explained	O
XTR	B-Algorithm
encryption	O
scheme	O
,	O
first	O
it	O
is	O
important	O
to	O
check	O
the	O
security	O
of	O
the	O
XTR	B-Algorithm
group	O
,	O
which	O
means	O
how	O
hard	O
it	O
is	O
to	O
solve	O
the	O
Discrete	O
Logarithm	O
problem	O
there	O
.	O
</s>
<s>
The	O
next	O
part	O
will	O
then	O
state	O
the	O
equivalency	O
between	O
the	O
Discrete	O
Logarithm	O
problem	O
in	O
the	O
XTR	B-Algorithm
group	O
and	O
the	O
XTR	B-Algorithm
version	O
of	O
the	O
discrete	O
logarithm	O
problem	O
,	O
using	O
only	O
the	O
traces	O
of	O
elements	O
.	O
</s>
<s>
The	O
security	O
of	O
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
protocol	I-Protocol
in	O
relies	O
on	O
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
(	O
DH	O
)	O
problem	O
of	O
computing	O
.	O
</s>
<s>
The	O
first	O
one	O
is	O
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
Decision	O
(	O
DHD	O
)	O
problem	O
to	O
determine	O
if	O
for	O
given	O
and	O
the	O
second	O
one	O
is	O
the	O
Discrete	O
Logarithm	O
(	O
DL	O
)	O
problem	O
to	O
find	O
for	O
a	O
given	O
.	O
</s>
<s>
To	O
attack	O
the	O
multiplicative	O
group	O
the	O
best	O
known	O
method	O
is	O
the	O
Discrete	O
Logarithm	O
variant	O
of	O
the	O
Number	B-Algorithm
Field	I-Algorithm
Sieve	I-Algorithm
or	O
alternatively	O
in	O
the	O
subgroup	O
one	O
can	O
use	O
one	O
of	O
several	O
methods	O
that	O
take	O
operations	O
in	O
,	O
such	O
as	O
Pollard	B-Algorithm
's	I-Algorithm
rho	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
XTR	B-Algorithm
parameters	O
are	O
now	O
chosen	O
in	O
such	O
a	O
way	O
that	O
is	O
not	O
small	O
,	O
is	O
sufficiently	O
large	O
and	O
cannot	O
be	O
embedded	O
in	O
a	O
true	O
subfield	O
of	O
,	O
since	O
and	O
is	O
a	O
divisor	O
of	O
,	O
but	O
it	O
does	O
not	O
divide	O
and	O
thus	O
cannot	O
be	O
a	O
subgroup	O
of	O
for	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
DL	O
problem	O
in	O
the	O
XTR	B-Algorithm
group	O
may	O
be	O
assumed	O
as	O
hard	O
as	O
the	O
DL	O
problem	O
in	O
.	O
</s>
<s>
Cryptographic	O
protocols	O
that	O
are	O
based	O
on	O
Discrete	O
Logarithms	O
can	O
use	O
many	O
different	O
types	O
of	O
subgroups	O
like	O
groups	O
of	O
points	O
of	O
elliptic	O
curves	O
or	O
subgroups	O
of	O
the	O
multiplicative	O
group	O
of	O
a	O
finite	O
field	O
like	O
the	O
XTR	B-Algorithm
group	O
.	O
</s>
<s>
As	O
we	O
have	O
seen	O
above	O
the	O
XTR	B-Algorithm
versions	O
of	O
the	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
and	O
ElGamal	B-Algorithm
encryption	I-Algorithm
protocol	O
replace	O
using	O
elements	O
of	O
the	O
XTR	B-Algorithm
group	O
by	O
using	O
their	O
traces	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
security	O
of	O
the	O
XTR	B-Algorithm
versions	O
of	O
these	O
encryption	O
schemes	O
is	O
no	O
longer	O
based	O
on	O
the	O
original	O
DH	O
,	O
DHD	O
or	O
DL	O
problems	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
XTR	B-Algorithm
versions	O
of	O
those	O
problems	O
need	O
to	O
be	O
defined	O
and	O
we	O
will	O
see	O
that	O
they	O
are	O
equivalent	O
(	O
in	O
the	O
sense	O
of	O
the	O
next	O
definition	O
)	O
to	O
the	O
original	O
problems	O
.	O
</s>
<s>
We	O
define	O
the	O
XTR-DH	O
problem	O
as	O
the	O
problem	O
of	O
computing	O
given	O
and	O
and	O
we	O
write	O
.	O
</s>
<s>
The	O
XTR-DHD	O
problem	O
is	O
the	O
problem	O
of	O
determining	O
whether	O
for	O
.	O
</s>
<s>
Given	O
,	O
the	O
XTR-DL	O
problem	O
is	O
to	O
find	O
,	O
i.e.	O
</s>
<s>
After	O
introducing	O
the	O
XTR	B-Algorithm
versions	O
of	O
these	O
problems	O
the	O
next	O
theorem	O
is	O
an	O
important	O
result	O
telling	O
us	O
the	O
connection	O
between	O
the	O
XTR	B-Algorithm
and	O
the	O
non-XTR	O
problems	O
,	O
which	O
are	O
in	O
fact	O
equivalent	O
.	O
</s>
<s>
This	O
implies	O
that	O
the	O
XTR	B-Algorithm
representation	O
of	O
elements	O
with	O
their	O
traces	O
is	O
,	O
as	O
can	O
be	O
seen	O
above	O
,	O
faster	O
by	O
a	O
factor	O
of	O
3	O
than	O
the	O
usual	O
representation	O
without	O
compromising	O
security	O
.	O
</s>
<s>
The	O
XTR-DL	O
problem	O
is	O
(	O
1	O
,	O
1	O
)	O
-equivalent	O
to	O
the	O
DL	O
problem	O
in	O
.	O
</s>
<s>
The	O
XTR-DH	O
problem	O
is	O
(	O
1	O
,	O
2	O
)	O
-equivalent	O
to	O
the	O
DH	O
problem	O
in	O
.	O
</s>
<s>
The	O
XTR-DHD	O
problem	O
is	O
(	O
3	O
,	O
2	O
)	O
-equivalent	O
to	O
the	O
DHD	O
problem	O
in	O
.	O
</s>
<s>
This	O
means	O
that	O
an	O
algorithm	O
solving	O
either	O
XTR-DL	O
,	O
XTR-DH	O
or	O
XTR-DHD	O
with	O
non-negligible	O
probability	O
can	O
be	O
transformed	O
into	O
an	O
algorithm	O
solving	O
the	O
corresponding	O
non-XTR	O
problem	O
DL	O
,	O
DH	O
or	O
DHD	O
with	O
non-negligible	O
probability	O
and	O
vice	O
versa	O
.	O
</s>
<s>
implies	O
that	O
determining	O
the	O
small	O
XTR-DH	O
key	O
(	O
being	O
an	O
element	O
of	O
)	O
is	O
as	O
hard	O
as	O
determining	O
the	O
whole	O
DH	O
key	O
(	O
being	O
an	O
element	O
of	O
)	O
in	O
the	O
representation	O
group	O
.	O
</s>
