<s>
In	O
cryptographic	B-Protocol
protocols	I-Protocol
,	O
a	O
key	B-Algorithm
encapsulation	I-Algorithm
mechanism	I-Algorithm
(	O
KEM	O
)	O
is	O
used	O
to	O
secure	O
symmetric	B-Algorithm
key	I-Algorithm
material	O
for	O
transmission	O
using	O
asymmetric	B-Application
(	O
public-key	B-Application
)	O
algorithms	O
.	O
</s>
<s>
In	O
practice	O
,	O
public	B-Application
key	I-Application
systems	O
are	O
clumsy	O
to	O
use	O
in	O
transmitting	O
long	O
messages	O
.	O
</s>
<s>
Instead	O
they	O
are	O
often	O
used	O
to	O
exchange	O
symmetric	B-Algorithm
keys	I-Algorithm
,	O
which	O
are	O
relatively	O
short	O
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
key	I-Algorithm
is	O
then	O
used	O
to	O
encrypt	O
the	O
longer	O
message	O
.	O
</s>
<s>
The	O
traditional	O
approach	O
to	O
sending	O
a	O
symmetric	B-Algorithm
key	I-Algorithm
with	O
public	B-Application
key	I-Application
systems	O
is	O
to	O
first	O
generate	O
a	O
random	O
symmetric	B-Algorithm
key	I-Algorithm
and	O
then	O
encrypt	O
it	O
using	O
the	O
chosen	O
public	B-Application
key	I-Application
algorithm	I-Application
.	O
</s>
<s>
The	O
recipient	O
then	O
decrypts	O
the	O
public	B-Application
key	I-Application
message	O
to	O
recover	O
the	O
symmetric	B-Algorithm
key	I-Algorithm
.	O
</s>
<s>
As	O
the	O
symmetric	B-Algorithm
key	I-Algorithm
is	O
generally	O
short	O
,	O
padding	O
is	O
required	O
for	O
full	O
security	O
and	O
proofs	O
of	O
security	O
for	O
padding	O
schemes	O
are	O
often	O
less	O
than	O
complete	O
.	O
</s>
<s>
KEMs	O
simplify	O
the	O
process	O
by	O
generating	O
a	O
random	O
element	O
in	O
the	O
finite	O
group	O
underlying	O
the	O
public	B-Application
key	I-Application
system	O
and	O
deriving	O
the	O
symmetric	B-Algorithm
key	I-Algorithm
by	O
hashing	O
that	O
element	O
,	O
eliminating	O
the	O
need	O
for	O
padding	O
.	O
</s>
<s>
Using	O
the	O
same	O
notation	O
employed	O
in	O
the	O
RSA	B-Architecture
system	O
article	O
,	O
say	O
Alice	O
has	O
transmitted	O
her	O
public	B-Application
key	I-Application
to	O
Bob	O
,	O
while	O
keeping	O
her	O
private	B-Application
key	I-Application
secret	O
,	O
as	O
usual	O
.	O
</s>
<s>
Bob	O
then	O
wishes	O
to	O
send	O
symmetric	B-Algorithm
key	I-Algorithm
M	O
to	O
Alice	O
.	O
</s>
<s>
Note	O
that	O
the	O
public	B-Application
key	I-Application
is	O
typically	O
2048-bits	O
or	O
even	O
longer	O
,	O
thus	O
much	O
larger	O
than	O
typical	O
symmetric	B-Algorithm
keys	I-Algorithm
.	O
</s>
<s>
To	O
avoid	O
such	O
potential	O
weakness	O
,	O
Bob	O
first	O
turns	O
M	O
into	O
a	O
larger	O
integer	O
by	O
using	O
an	O
agreed-upon	O
reversible	O
protocol	O
known	O
as	O
a	O
padding	O
scheme	O
,	O
such	O
as	O
OAEP	B-Algorithm
.	O
</s>
<s>
Alice	O
can	O
recover	O
from	O
by	O
using	O
her	O
private	B-Application
key	I-Application
exponent	O
by	O
the	O
following	O
computation	O
:	O
</s>
<s>
Instead	O
of	O
generating	O
a	O
random	O
symmetric	B-Algorithm
key	I-Algorithm
M	O
,	O
Bob	O
first	O
generates	O
a	O
random	O
with	O
.	O
</s>
<s>
He	O
derives	O
his	O
symmetric	B-Algorithm
key	I-Algorithm
M	O
by	O
,	O
where	O
is	O
a	O
key	B-Algorithm
derivation	I-Algorithm
function	I-Algorithm
,	O
such	O
as	O
a	O
cryptographic	B-Algorithm
hash	I-Algorithm
.	O
</s>
<s>
Alice	O
then	O
recovers	O
from	O
by	O
using	O
her	O
private	B-Application
key	I-Application
exponent	O
by	O
the	O
same	O
method	O
as	O
above	O
:	O
</s>
<s>
Given	O
,	O
she	O
can	O
recover	O
the	O
symmetric	B-Algorithm
key	I-Algorithm
M	O
by	O
.	O
</s>
<s>
Note	O
that	O
while	O
M	O
can	O
be	O
calculated	O
from	O
in	O
the	O
KEM	O
approach	O
,	O
the	O
reverse	O
is	O
not	O
possible	O
,	O
assuming	O
the	O
key	B-Algorithm
derivation	I-Algorithm
function	I-Algorithm
is	O
one-way	O
.	O
</s>
<s>
Thus	O
,	O
if	O
key	B-Algorithm
encapsulations	I-Algorithm
for	O
several	O
recipients	O
need	O
to	O
be	O
computed	O
,	O
independent	O
values	O
should	O
be	O
used	O
.	O
</s>
<s>
Similar	O
techniques	O
are	O
available	O
for	O
Diffie-Hellman	B-Protocol
key	I-Protocol
exchange	I-Protocol
and	O
other	O
public	B-Application
key	I-Application
methods	O
.	O
</s>
