Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
System
(IJPEDS)
V
ol.
11,
No.
1,
March
2020,
pp.
284
290
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v11.i1.pp284-290
r
284
V
erifiable
secur
e
computation
of
linear
fractional
pr
ogramming
using
certificate
v
alidation
Nedal
M.
Mohammed
1
,
Laman
R.
Sultan
2
,
Ahmed
A.
Hamoud
3
,
Santosh
S.
Lomte
4
1,4
Department
of
Computer
Science,
Dr
.
Babasaheb
Ambedkar
Marathw
ada
Uni
v
ersity
,
Aurang
abad,
India
2
Department
of
Po
wer
Mechanics,
Basra
T
echnical
Institute,
Southern
T
echnical
Uni
v
ersity
,
Al-Basrah,
Iraq
3
Department
of
Mathematics,
Dr
.
Babasaheb
Ambedkar
Marathw
ada
Uni
v
ersity
,
Aurang
abad,
India
1
Department
of
Computer
Science,
T
aiz
Uni
v
ersity
,
T
aiz,
Y
emen.
Article
Inf
o
Article
history:
Recei
v
ed
Mar
28,
2019
Re
vised
Jul
8,
2019
Accepted
Jul
31,
2019
K
eyw
ords:
Certificate
v
alidation
LFP
Computation
outsourcing
V
erifiable
computation
V
erifiable
secure
computation
of
LFP
ABSTRA
CT
Outsourcing
of
scientific
computations
is
attracting
increasing
attention
since
it
enables
the
customers
with
limited
computing
resource
and
storage
de
vices
to
outsource
the
sophisticated
computat
ion
w
orkloads
into
po
werful
service
pro
viders.
Ho
we
v
er
,
it
also
comes
up
with
some
security
and
pri
v
ac
y
concerns
and
challenges,
such
as
the
input
and
output
pri
v
ac
y
of
the
customers,
and
cheating
beha
viors
of
the
cloud.
Moti
v
ated
by
t
hese
issues,
this
paper
focused
on
pri
v
ac
y-preserving
Linear
Fractional
Programming
(LFP)
as
a
typical
and
practically
rele
v
ant
case
for
v
erifiable
secure
multiparty
computation.
W
e
will
in
v
estig
ate
the
secure
and
v
erifiable
schema
with
correctness
guarantees,
by
using
normal
multiparty
techniques
to
compute
the
result
of
a
computation
and
then
using
v
erifiable
techniques
only
to
v
erify
that
this
result
w
as
correct.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Nedal
M.
Mohammed,
Department
of
Computer
Science,
T
aiz
Uni
v
ersity
,
T
aiz,
Y
emen.
Email:
dr
.nedal.mohammed@gmail.com
1.
INTR
ODUCTION
The
po
werful
adv
antage
of
cloud
com
pu
t
ing
is
called
outsourcing,
where
the
customers
with
l
imited
computing
res
ource
and
storage
de
vices
can
outsource
the
sophisticated
computation
w
orkloads
into
po
werful
service
pro
viders.
Despite
the
tremendous
benefits,
there
are
man
y
challenges
and
security
concerns
because
the
cloud
serv
er
and
customer
are
not
in
the
same
trusted
domain,
to
a
v
oid
these
problems
[1-4].
First,
to
combat
the
security
concern
is
applying
encryption
techniques
to
customer’
s
sensiti
v
e
information
before
outsourcing
to
the
cloud
b
ut
still,
there
is
a
challenge
ho
w
mak
es
the
task
of
computation
o
v
er
encrypted
data
[5,6].
Second,
no
guarantee
from
the
cloud
on
the
quality
of
the
computed
data
and
results.
F
or
instance,
solving
financial
linear
programs
is
useful
for
optimizing
global
profits
confidentiality
is
important
because
the
inputs
are
sensiti
v
e
information
from
multiple
companies
b
ut
correctness
is
important
because
the
outcome
represents
financial
v
alue.
In
theory
,
correctness
and
pri
v
ac
y
can
be
achie
v
ed
by
producing
cryptographic
proofs
of
correctness
in
a
multi-party
w
ay
[7,8].
In
[9]
The
y
achie
v
ed
Correctness
by
replicating
a
computation
and
comparing
the
results
this
done
ag
ainst
uncorrelated
f
ailure.
W
ithout
assuming
uncorrelated
f
ailure
or
trusted
hardw
are
the
correctness
can
be
done
e.g.,
[10]
by
instead
producing
cryptographic
proofs
of
correctness.
Also,
pri
v
ac
y
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
285
can
be
done
when
the
computation
achie
v
ed
by
multiple
computation
parties
using
multiparty
computation
protocols
e.g.
[11,12].
In
this
paper
,
we
w
ant
to
be
sure
that
the
resul
ts
are
correct
and
with
the
multiple
mutually
distrust
ing
in
putters,
also
we
w
ant
to
guarantee
the
pri
v
ac
y
of
the
inputs.
W
e
present
certificate
v
alidation
as
a
general
technique
for
achie
ving
v
erifiable
secure
computation
of
linear
fractional
programming.
W
e
use
of
El-Gamal
encryption
[13-15]
by
combining
the
computation
stage
and
the
v
alidation
stage
rather
than
using
e
xpensi
v
e
encryption
schemes
such
as
P
aillier’
s
cryptosystem.
The
rest
of
the
paper
is
or
g
anized
as
follo
ws:
section
2.
Sho
ws
v
erifiable
computation
schema.
In
section
3.
W
e
describes
the
system
model
of
our
proposed
Protocol
for
pri
v
ac
y-preserving
outsourcing
LFP
.
In
section
4.
W
e
pro
vide
e
xperimental
result
analysis
for
the
proposed
schema.
At
last
the
w
ork
conclusion
is
presented
in
section
5.
2.
VERIFIABLE
COMPUT
A
TION
V
erifiable
computation
has
been
studied
by
plenty
of
researchers
in
v
arious
application
s
cenarios.
The
y
researched
widely
ho
w
to
v
erify
the
correctness
of
computations
performed
by
untrusted
parties
(without
pri
v
ac
y)
[16-21].
Figure
1.
System
Structure
of
V
erifiable
Computation
Scheme.
V
erifiable
computation
schemes
are
normally
based
on
either
computation
comple
xity
theory
or
cryptographic
algorithms.
Data
and
computations
can
be
outsourced
to
another
party
in
order
to
obtain
a
processing
result
in
return.
Ho
we
v
er
,
whether
the
result
is
right
or
wrong
could
cause
a
potential
risk
for
a
data
processing
result
requester
.
F
or
outsourced
data
processing
and
computations,
v
erification
of
the
computation
results
is
a
critical
issue
to
ensure
the
trust
of
Computation-as-a-Service
[22].
3.
PR
O
T
OCOL
FOR
PRIV
A
CY
-PRESER
VING
OUTSOURCING
LINEAR
FRA
CTION
AL
PR
O-
GRAMMING
W
e
present
main
protocol
for
pri
v
ac
y-preserving
outsourcing
with
correctness
guarant
ees.
W
e
compute
a
solution
and
a
so-called
certificate
using
normal
multiparty
computation,
and
then
produce
V
erifiable
secur
e
computation
of
linear
...
(Nedal
M.
Mohammed)
Evaluation Warning : The document was created with Spire.PDF for Python.
286
r
ISSN:
2088-8694
a
proof
that
the
solution
is
v
alid
with
respect
to
the
certificate
using
the
El-Gamal-based
proofs
[23].
3.1.
Functions
of
certificates
and
v
alidating
T
o
ef
ficiently
v
alidate
a
computation
result,
we
use
certificates.
In
comple
xity
theory
,
a
certificate
is
a
proof
that
a
v
alue
lies
in
a
certain
set
that
can
be
v
erified
in
polynomial
time.
Let
S
1
;
S
2
be
sets
and
Y
S
1
.
A
polynomial
time
computable
predicate
[
'
S
1
S
2
]
is
called
a
v
alidating
function
for
Y
if
Y
=
f
w
2
S
1
j9
c
2
S
2
:
'
(
w
;
c
)
g
:
If
'
(
w
;
c
)
w
2
Y
:
In
our
case,
a
computation
is
gi
v
en
by
a
computation
function
'
(
y
;
a;
r
)
;
and
a
v
alidating
function
'
(
y
;
a;
r
)
:
Here,
on
input
x
,
function
f
computes
function
output
r
and
certificate
a
;
v
alidating
function
'
checks
that
r
is
a
v
alid
output
with
respect
to
x
and
a
.
W
e
require
that
if
(
a;
r
)
=
f
(
y
)
,
then
'
(
y
;
a;
r
)
,
b
ut
we
do
not
demand
the
con
v
erse:
the
outcome
of
the
computation
might
not
be
unique,
and
might
merely
check
that
some
correct
solution
w
as
found,
not
that
it
w
as
produced
according
to
algorithm
f
.
(F
or
instance,
a
square
root
finder
may
return
the
positi
v
e
square
root
while
ne
g
ati
v
e
square
root
is
also
v
alid.)
In
our
case
study
,
we
use
that
the
optimality
of
a
solution
to
a
LFP
can
be
ef
ficiently
v
alidated
using
a
certificate.
3.2.
The
v
erifiable
multiparty
computation
pr
otocol
by
certificate
v
alidation
W
e
present
V
erifiable
Multiparty
computation
protocol
by
certificate
v
alidation
(V
erMPC)
protocol
to
compute
(
a;
r
)
=
f
(
x
)
,
and
pro
v
e
this
resul
t
X
i
is
correct.
W
e
use
passi
v
ely
secure
multiparty
computation
protocols
based
on
(
t;
n
)
Shamir
sharing
with
n
=
2
t
+
1
.
In
these
protocols,
the
input
parties
encrypt
and
announce
their
inputs,
then
mak
es
a
proof
of
kno
wledge
of
the
corresponding
plainte
xt
then
broadcast
for
this
encryption
and
proof.
Ne
xt,
the
parties
pro
vide
the
plainte
xt
and
randomness
of
the
encryption
to
the
tw
o
computation
parties
who
will
later
pro
v
e
the
result
is
correct.
The
tw
o
computation
p
a
rties
check
if
the
pro
vided
sharing
of
the
input
is
consistent
with
the
encryptions
that
were
broadcast
for
pre
v
enting
corrupted
input
parties
learns
information
about
both
their
encrypted
and
their
secret
shared
inputs,
this
done
by
encrypting
their
shares
of
the
inputs
then
using
the
homomorphic
property
of
the
cryptosystem
for
checking
correctness.
Then,
the
actual
computation
tak
es
place
in
the
third
computation
party
.
The
tw
o
parties
holding
additi
v
e
shares
of
the
input
Shamir
-share
them
between
all
three
computation
parties,
then
the
computation
is
performed
between
the
three
partie
s.
These
tw
o
of
the
computation
parties
produce
the
encrypted
result
and
pro
v
e
its
correctness
[24].
The
computation
pa
rties
send
their
additi
v
e
shares
of
the
result
and
the
randomness
of
their
encryption
shares
results
to
the
resulting
party
(the
encryptions
of
the
certificate
and
proof
of
correctness)
[13,25-27].
The
result
party
checks
the
proofs
of
kno
wledge
pro
vided
by
the
in
putters
computes
the
encrypted
results
from
its
shares
and
use
V
erify
algorithm
to
v
erify
the
correctness.
Figure
2.
System
structure
of
v
erifiable
computation
protocol
by
certificate
v
alidation
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
1,
March
2020
:
284
–
290
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
287
3.3.
Secur
e
and
v
erifiable
linear
fractional
pr
ogramming
The
LFP
is
a
special
class
of
mathematical
optimization
e
xpressed
in
the
follo
wing
s
tandard
form
[28]:
max
Z
=
cy
+
dy
+
S.t.
Ay
b;
B
y
0
;
(1)
where
(1)
the
objecti
v
e
function
is
a
linear
fractional
function
(ratio
of
tw
o
linear
functions)
y
is
an
n
1
v
ector
of
v
ariables
which
are
to
be
determined,
c
and
d
are
n
1
column
v
ectors
of
coef
ficients,
and
set
of
constraints
are
a
system
of
linear
equalities
and
inequalities
(af
fine
constraints)
A
is
m
n
matrix
of
coef
ficients,
b
is
m
1
column
v
ector
of
coef
ficients
and
;
are
constants.
B
is
n
n
nonsingular
matrix.
F
or
ins
tance,
the
LFP
represents
the
problem
to
find
x
1
;
x
2
satisfying
max
Z
=
2
x
1
+
3
x
2
x
1
+
x
2
+
1
S.t.
x
1
+
x
2
3
x
1
+
2
x
2
3
x
1
;
x
2
0
:
T
o
find
the
optimal
solution
of
a
fractional
linear
program,
typically
an
iterati
v
e
algorithm
called
the
simple
x
algorithm
is
used
after
con
v
ert
LFP
to
LP
[29].
max
F
(
y
)
=
2
y
1
+
3
y
2
S.t.
4
y
1
+
4
y
2
3
4
y
1
+
5
y
2
3
y
1
;
y
2
0
:
A
=
4
4
4
5
;
b
=
3
3
;
c
=
2
3
:
Each
iteration
in
v
olv
es
se
v
eral
comparisons
and
a
Gaussian
elimination
step,
making
it
quite
hea
vy
for
multiparty
computation.
F
or
relati
v
ely
small
instances,
passi
v
ely
secure
linear
fractional
programming
is
feasible
[11],
b
ut
acti
v
ely
secure
MPC
much
less
so
when
including
pre-processing.
Theor
em:
W
e
pro
v
e
that
y
it
is
optimal
using
the
optimal
solution
p
of
the
so-called
dual
LP
maximise
b
p
such
that
A
p
c;
p
0
:
Pr
oof:
The
solutions
(
y
1
q
;
;
y
n
q
)
and
(
p
1
q
;
;
p
m
q
)
(
y
2
Z
n
;
p
2
Z
m
;
q
2
N
+
)
are
both
optimal
if
the
follo
wing
conditions
hold:
q
1;
p
b
=
c
y
;
A
y
q
b
;
y
0
;
A
T
p
q
c
;
p
0
:
Also,
the
simple
x
algorithm
for
finding
y
turns
out
to
also
directly
gi
v
e
p
.
T
o
turn
the
abo
v
e
criterion
into
a
set
of
polynomial
equations,
we
define
the
certificate
to
consist
of
bit
decompositions
of
(
q
b
A
y
)
i
;
y
i
;
(
q
c
A
T
p
)
i
;
and
p
i
;
and
pro
v
e
that
each
bit
decomposition
b
0
;
b
1
;
:::
sums
up
to
the
correct
v
alue
v
(with
equation
v
=
b
0
+
2
b
1
+
)
and
contains
only
bits
(with
equations
b
i
(1
b
i
)
=
0)
:
V
erifiable
secur
e
computation
of
linear
...
(Nedal
M.
Mohammed)
Evaluation Warning : The document was created with Spire.PDF for Python.
288
r
ISSN:
2088-8694
4.
EXPERIMENT
AL
RESUL
T
The
e
xperimental
results
are
the
a
v
erage
of
multiple
trials
.
W
e
design
numerical
e
xperiments
to
e
v
aluate
the
ef
ficienc
y
of
the
mechanism.
W
e
ran
our
mechanism
on
se
v
eral
LFPs.
W
e
measured
the
time
to
solv
e
the
LFP
and
to
compute
the
certificate
(this
depends
on
the
LFP
size,
number
of
iterations
needed,
and
the
bit
length
for
internal
computations),
the
time
for
PolyPro
v
e
to
produce
a
proof,
and
for
PolyV
er
to
v
erify
it
(this
depends
on
the
LFP
size
and
bit
length
for
the
proof).
Figure
3.
Sho
ws
the
performance
numbers
of
our
e
xperiments.
T
able
1.
Performance
of
the
proposed
scheme
for
infeasible
case
Problem
No.
of
V
erify
Pro
v
e
Compute
Size
Iterations
Algorithm
Algorithm
Certificate
m=5,
n=5
4
11.450
85
110
m=20,
n=20
9
79.50
200
347
m=48,
n=70
25
300.340
986
1100
m=103,
n=150
62
1000
4806
8500
Figure
3.
T
ime
cost
for
each
phase
of
v
erifiable
secure
computation
of
LFP
using
certificate
v
alidation
F
or
the
LFPs
in
our
e
xperiments,
we
find
that
producing
proof
adds
little
o
v
erhead
to
compute
the
solution
and
that
v
eri
fying
the
proof
is
much
f
aster
than
participating
in
the
computation.
As
a
consequence,
for
the
recipient,
outsourcing
both
guarantees
correctness
and
sa
v
es
t
ime
compared
to
participating
in
the
computation.
In
general,
one
e
xpects
the
dif
ference
between
computing
the
solution
and
pro
ving
its
correctness
to
be
more
pronounced
for
lar
ger
problems:
indeed,
both
the
computation
and
the
correctness
v
erification
scale
in
the
size
of
the
LFP
,
b
ut
computation
additionally
scales
in
the
number
of
iterations
needed
to
reach
the
optimal
solution.
This
number
of
iterations
typically
gro
ws
with
the
LFP
size.
Ho
we
v
er
,
we
only
found
this
for
the
biggest
LFP
,
where
pro
ving
is
o
v
er
f
o
ur
times
f
aster
than
computing,
for
the
other
programs,
this
f
actor
w
as
around
tw
o.
An
e
xplanation
for
t
his
is
that
also
the
bit
length
of
solutions
(which
influences
pro
ving
time)
typically
gro
ws
with
the
number
of
iterations.
5.
CONCLUSION
In
this
paper
,
we
combined
passi
v
ely
secure
three-party
computation
with
El-Gamal-based
proofs.
W
e
ha
v
e
sho
wn
ho
w
to
use
certificate
v
alidation
to
obtain
correctness
guarantees
for
pri
v
ac
y-preserving
outsourcing
of
LFP
.
The
securi
ty
guarantees
of
our
model
lie
in
between
pa
ssi
v
e
security
(that
does
not
guarantee
correctness
in
case
of
acti
v
e
att
acks)
and
acti
v
e
security
(
that
also
guarantees
pri
v
ac
y
in
this
case).
F
or
LFP
,
v
erifying
results
tak
es
much
less
time
than
participating
in
an
acti
v
ely
secure
computation;
in
f
act,
it
e
v
en
tak
es
less
time
than
participating
in
a
passi
v
ely
secure
computation
without
an
y
correctness
guarantees.
Hence,
for
computations
on
inputs
of
mutually
distrusting
parties,
pri
v
ac
y-preserving
outsourcing
with
correct-
ness
guarantees
pro
vides
a
compelling
combination
of
correctness
and
pri
v
ac
y
guarantees.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
1,
March
2020
:
284
–
290
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
289
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