<s>
In	O
cryptography	O
,	O
Learning	B-General_Concept
with	I-General_Concept
errors	I-General_Concept
(	O
LWE	O
)	O
is	O
a	O
mathematical	O
problem	O
that	O
is	O
widely	O
used	O
in	O
cryptography	O
to	O
create	O
secure	O
encryption	O
algorithms	O
.	O
</s>
<s>
The	O
LWE	O
problem	O
was	O
introduced	O
by	O
Oded	O
Regev	O
in	O
2005	O
(	O
who	O
won	O
the	O
2018	O
Gödel	O
Prize	O
for	O
this	O
work	O
)	O
,	O
it	O
is	O
a	O
generalization	O
of	O
the	O
parity	B-General_Concept
learning	I-General_Concept
problem	O
.	O
</s>
<s>
Subsequently	O
,	O
the	O
LWE	O
problem	O
has	O
been	O
used	O
as	O
a	O
hardness	O
assumption	O
to	O
create	O
public-key	B-Application
cryptosystems	I-Application
,	O
such	O
as	O
the	O
ring	O
learning	B-General_Concept
with	I-General_Concept
errors	I-General_Concept
key	O
exchange	O
by	O
Peikert	O
.	O
</s>
<s>
The	O
learning	B-General_Concept
with	I-General_Concept
errors	I-General_Concept
problem	I-General_Concept
is	O
to	O
find	O
,	O
given	O
access	O
to	O
polynomially	O
many	O
samples	O
of	O
choice	O
from	O
.	O
</s>
<s>
Regev	O
proposed	O
a	O
public-key	B-Application
cryptosystem	I-Application
based	O
on	O
the	O
hardness	O
of	O
the	O
LWE	O
problem	O
.	O
</s>
<s>
Private	B-Application
key	I-Application
:	O
Private	B-Application
key	I-Application
is	O
an	O
chosen	O
uniformly	O
at	O
random	O
.	O
</s>
<s>
Public	B-Application
key	I-Application
:	O
Choose	O
vectors	O
uniformly	O
and	O
independently	O
.	O
</s>
