<s>
The	O
notion	O
of	O
non-malleable	B-Algorithm
codes	I-Algorithm
was	O
introduced	O
in	O
2010	O
by	O
Dziembowski	O
,	O
Pietrzak	O
,	O
and	O
Wichs	O
,	O
for	O
relaxing	O
the	O
notion	O
of	O
error-correction	B-Error_Name
and	I-Error_Name
error-detection	I-Error_Name
.	O
</s>
<s>
Non-malleable	B-Algorithm
codes	I-Algorithm
provide	O
a	O
useful	O
and	O
meaningful	O
security	O
guarantee	O
in	O
situations	O
where	O
traditional	O
error-correction	B-Error_Name
and	I-Error_Name
error-detection	I-Error_Name
is	O
impossible	O
;	O
for	O
example	O
,	O
when	O
the	O
attacker	O
can	O
completely	O
overwrite	O
the	O
encoded	B-Algorithm
message	O
.	O
</s>
<s>
To	O
know	O
the	O
operation	O
schema	O
of	O
non-malleable	B-Algorithm
code	I-Algorithm
,	O
we	O
have	O
to	O
have	O
a	O
knowledge	O
of	O
the	O
basic	O
experiment	O
it	O
based	O
on	O
.	O
</s>
<s>
A	O
source	O
message	O
is	O
encoded	B-Algorithm
via	O
a	O
(	O
possibly	O
randomized	O
)	O
procedure	O
,	O
yielding	O
a	O
code-word	O
=	O
.	O
</s>
<s>
The	O
tampering	O
experiment	O
can	O
be	O
used	O
to	O
model	O
several	O
interesting	O
real-world	O
settings	O
,	O
such	O
as	O
data	O
transmitted	O
over	O
a	O
noisy	B-General_Concept
channel	I-General_Concept
,	O
or	O
adversarial	O
tampering	O
of	O
data	O
stored	O
in	O
the	O
memory	O
of	O
a	O
physical	O
device	O
.	O
</s>
<s>
Having	O
this	O
experimental	O
base	O
,	O
we	O
would	O
like	O
to	O
build	O
special	O
encoding/decoding	O
procedures	O
,	O
which	O
give	O
us	O
some	O
meaningful	O
guarantees	O
about	O
the	O
results	O
of	O
the	O
above	O
tampering	O
experiment	O
,	O
for	O
large	O
and	O
interesting	O
families	O
of	O
tampering	O
functions	O
.	O
</s>
<s>
One	O
very	O
natural	O
guarantee	O
,	O
called	O
error-correction	B-Error_Name
,	O
would	O
be	O
to	O
require	O
that	O
for	O
any	O
tampering	O
function	O
and	O
any	O
source-message	O
s	O
,	O
the	O
tampering	O
experiment	O
always	O
produces	O
the	O
correct	O
decoded	O
message	O
.	O
</s>
<s>
A	O
weaker	O
guarantee	O
,	O
called	O
error-detection	B-Error_Name
,	O
requires	O
that	O
the	O
tampering-experiment	O
always	O
results	O
in	O
either	O
the	O
correct	O
value	O
or	O
a	O
special	O
symbol	O
indicating	O
that	O
tampering	O
has	O
been	O
detected	O
.	O
</s>
<s>
This	O
notion	O
of	O
error-detection	B-Error_Name
is	O
a	O
weaker	O
guarantee	O
than	O
error-correction	B-Error_Name
,	O
and	O
achievable	O
for	O
larger	O
F	O
of	O
tampering	O
functions	O
.	O
</s>
<s>
A	O
non-malleable	B-Algorithm
code	I-Algorithm
ensures	O
that	O
either	O
the	O
tampering	O
experiment	O
results	O
in	O
a	O
correct	O
decoded-message	O
,	O
or	O
the	O
decoded-message	O
is	O
completely	O
independent	O
of	O
and	O
unrelated	O
to	O
the	O
source-message	O
.	O
</s>
<s>
Compared	O
to	O
error	B-Error_Name
correction	I-Error_Name
or	I-Error_Name
error	I-Error_Name
detection	I-Error_Name
,	O
the	O
"	O
right	O
"	O
formalization	O
of	O
non-malleable	B-Algorithm
codes	I-Algorithm
is	O
somewhat	O
harder	O
to	O
define	O
.	O
</s>
<s>
Let	O
be	O
a	O
random	O
variable	O
for	O
the	O
value	O
of	O
the	O
decoded-message	O
,	O
which	O
results	O
when	O
we	O
run	O
the	O
tampering	O
experiment	O
with	O
source-message	O
and	O
tampering-function	O
,	O
over	O
the	O
randomness	O
of	O
the	O
encoding	B-Algorithm
procedure	O
.	O
</s>
<s>
Intuitively	O
,	O
we	O
wish	O
to	O
say	O
that	O
the	O
distribution	O
of	O
is	O
independent	O
of	O
the	O
encoded	B-Algorithm
message	O
.	O
</s>
<s>
Notice	O
that	O
non-malleability	O
is	O
a	O
weaker	O
guarantee	O
than	O
error	B-Error_Name
correction/detection	I-Error_Name
;	O
the	O
latter	O
ensure	O
that	O
any	O
change	O
in	O
the	O
code-word	O
can	O
be	O
corrected	O
or	O
at	O
least	O
detected	O
by	O
the	O
decoding	O
procedure	O
,	O
whereas	O
the	O
former	O
does	O
allow	O
the	O
message	O
to	O
be	O
modified	O
,	O
but	O
only	O
to	O
an	O
unrelated	O
value	O
.	O
</s>
<s>
However	O
,	O
when	O
studying	O
error	B-Error_Name
correction/detection	I-Error_Name
we	O
usually	O
restrict	O
ourselves	O
to	O
limited	O
forms	O
of	O
tampering	O
which	O
preserve	O
some	O
notion	O
of	O
distance	O
(	O
e.g.	O
,	O
usually	O
hamming	O
distance	O
)	O
between	O
the	O
original	O
and	O
tampered	O
code-word	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
already	O
impossible	O
to	O
achieve	O
error	B-Error_Name
correction/detection	I-Error_Name
for	O
the	O
simple	O
family	O
of	O
functions	O
which	O
,	O
for	O
every	O
constant	O
,	O
includes	O
a	O
"	O
constant	O
"	O
function	O
that	O
maps	O
all	O
inputs	O
to	O
.	O
</s>
<s>
The	O
prior	O
works	O
on	O
non-malleable	B-Algorithm
codes	I-Algorithm
show	O
that	O
one	O
can	O
construct	O
non-malleable	B-Algorithm
codes	I-Algorithm
for	O
highly	O
complex	O
tampering	O
function	O
families	O
for	O
which	O
error	B-Error_Name
correction/detection	I-Error_Name
can	O
not	O
be	O
achievable	O
.	O
</s>
<s>
Therefore	O
,	O
as	O
we	O
have	O
mentioned	O
,	O
error-correction	B-Error_Name
and	I-Error_Name
error-detection	I-Error_Name
cannot	O
be	O
achieved	O
w.r.t.	O
</s>
<s>
Nevertheless	O
,	O
the	O
following	O
can	O
show	O
an	O
efficient	O
non-malleable	B-Algorithm
code	I-Algorithm
for	O
this	O
powerful	O
family	O
.	O
</s>
<s>
Unfortunately	O
,	O
random	O
coding	O
schemes	O
cannot	O
be	O
efficiently	O
represented	O
,	O
nor	O
is	O
the	O
encoding/decoding	O
function	O
likely	O
to	O
be	O
efficient	O
.	O
</s>
<s>
Moreover	O
,	O
this	O
result	O
also	O
highlights	O
the	O
difference	O
between	O
"	O
error-correction/detection	O
"	O
and	O
"	O
non-malleability	O
"	O
since	O
a	O
result	O
of	O
this	O
form	O
could	O
not	O
be	O
true	O
for	O
the	O
former	O
notions	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
does	O
tell	O
us	O
that	O
non-malleable	B-Algorithm
codes	I-Algorithm
exist	O
with	O
respect	O
to	O
all	O
efficient	O
functions	O
,	O
but	O
this	O
is	O
misleading	O
as	O
we	O
know	O
that	O
efficient	O
non-malleable	B-Algorithm
codes	I-Algorithm
(	O
and	O
ultimately	O
we	O
are	O
only	O
interested	O
in	O
such	O
)	O
cannot	O
be	O
non-malleable	O
w.r.t.	O
</s>
<s>
very	O
general	O
classes	O
of	O
functions	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
.	O
</s>
<s>
By	O
using	O
non-malleable	B-Algorithm
code	I-Algorithm
for	O
this	O
purpose	O
we	O
have	O
the	O
conclusion	O
:	O
Let	O
be	O
any	O
coding	O
scheme	O
which	O
is	O
non-malleable	O
w.r.t	O
,	O
then	O
can	O
also	O
be	O
tamper-simulate	O
w.r.t.	O
</s>
<s>
For	O
every	O
family	O
with	O
,	O
there	O
exist	O
non-malleable	B-Algorithm
codes	I-Algorithm
against	O
with	O
rate	O
arbitrarily	O
close	O
to	O
1	O
−	O
(	O
this	O
is	O
achieved	O
w.h.p.	O
</s>
<s>
For	O
families	O
of	O
size	O
against	O
which	O
there	O
is	O
no	O
non-malleable	B-Algorithm
code	I-Algorithm
of	O
rate	O
1	O
−	O
(	O
in	O
fact	O
this	O
is	O
the	O
case	O
w.h.p	O
for	O
a	O
random	O
family	O
of	O
this	O
size	O
)	O
.	O
</s>
