Duncan S Wong Department of Computer Science City University of Hong Kong Joint work with Guomin Yang Chik How Tan and Qiong Huang 1 2 What is PKE with Equality Test Is it related to ID: 129926
Download Presentation The PPT/PDF document "Probabilistic Public Key Encryption with..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Probabilistic Public Key Encryption with Equality Test
Duncan S. WongDepartment of Computer ScienceCity University of Hong Kong
Joint work with Guomin Yang, Chik How Tan and Qiong Huang
1Slide2
2
What is PKE with Equality Test?
Is it related to PKE with Keyword Search or Deterministic PKE?Applications
Our
constructionWhat security level can it achieve?Impossibility of achieving IND-ATK (e.g. IND-CPA or IND-CCA1/2)Extension: a non-pairing variantW-IND-CCA2
OutlineSlide3
3
What is
PKE with Equality Test (PKE-ET)?
Enc
M
1
pk
1
C
1
Enc
M
2
pk
2
C
2
M
1
=?
M
2
Test
C
1
C
2
1
iff
M
1
=
M
2Slide4
4
What is PKE with Equality Test (PKE-ET)?
1. Perfect Consistency2. Soundness
For every
M
in plaintext space PtSp(k), Pr[ Test(C1, C2
) = 1 ] = 1
if (
pk
1
,
sk
1
)
G(1
k), (pk2,
sk2) G(1
k), C1 E
(pk1, M) and
C2 E(
pk2, M).
For any PPT A
, Pr[ Test(C
1, C2) = 1
M1 M2
M1 M2
] (k)where (
C1, C2, sk1
, sk2) A(1
k), M1
D(sk1,
C1), M2
D(sk2
, C2).Slide5
5
Is PKE-ET related to PKE with Keyword Search?
PKE with Keyword Search (PKES)w : keywordC
= Enc(
pk
, w)TW = Trapdoor(sk, w)Test(pk,
C
,
T
W
) = 1
iff
C is an encryption of w under
pkEquality Test
Test(pk, C1
, TW) = 1 & Test(pk,
C2, TW) = 1
Both C1 and
C2 are encryptions of the same w.
LimitationsA tag
TW can only be generated if sk
is known.Test: only
applicable to ciphertexts generated under the same pk.Slide6
6
Is PKE-ET related to Deterministic PKE?
Deterministic Public Key Encryption (DPKE)S = Enc(pk, M
)
M
= Dec(sk, C)Equality TestGiven C1 = Enc(pk
,
M
1
) &
C
2
= Enc(
pk
, M2)
C1 = C2
M1 = M2.
LimitationOnly applicable to
ciphertexts generated under the same pk.Slide7
7
Applications of PKE-ETOutsourced Database, data are stored in encrypted form.
Searchable Encryption: anyone is able to search keywords of encrypted messages even if they are generated under different public keys.E.g. building a search engine capable of searching encrypted messages provided by different vendors
Partitioning Encrypted Data
: DBMS or the public is able to categorize or obtain statistical information on messages without any help from the encrypted message owners.
E.g. partitioning encrypted files based on file types such as images from videosSlide8
8
Our PKE-ET ConstructionSystem Parameters
G1, G2: cyclic groups of prime order qg
: generator of
G
1Bilinear pairing e: G1 x G1 G2
PtSp
:
G
1
\{1}
KeyGen
(1
k
)
sk =
x R
Zq*pk =
y = gx
Enc(pk,
m)r
R Zq
*Ciphertext C := (
U, V, W) where U = g
r, V
= mr,
W = H(U, V, y
r) m
||rDec(sk,
C)m||r
WH
(U, V, Ux)
Verify r
Zq* m
G
1
\{1}
U
=
g
r
V
=
m
r
If true, return
m
, else return
Test(
C
1
,
C
2
)
Given
C
1
= (
U
1
,
V
1
,
W
1
) and
C
2
= (
U
2
,
V
2
,
W
2
), if
e
(
U
1
,
V
2
) =
e
(
U
2
,
V
1
), return 1, else return 0.Slide9
9
What Security Level can our PKE-ET scheme achieve?(Impossibility of Achieving IND-ATK)
In general, PKE-ET cannot achieve IND-ATK (e.g. IND-CPA or IND-CCA1/2).
IND-ATK:
Reason why PKE-ET cannot achieve IND-ATK
: adversary knows the challenge plaintexts
x
0
and
x
1
; does not even need to resort its plaintext choosing capability.Slide10
10
What Security Level can our PKE-ET scheme achieve?
After challenge phase, the adversary knows:public key: pkchallenge plaintexts:
x
0
and x1challenge ciphertext: y
Adversary
A
2
computes
y
’ = Enc(
pk
’,
x
1
)returns Test(
y, y’)Slide11
11
What Security Level can our PKE-ET scheme achieve?
It achieves one-way under chosen
ciphertext
attack (OW-CCA2)
.OW-ATK:Slide12
12
What Security Level can our PKE-ET scheme achieve?OW-CCA2 security in the random oracle model under the CDH assumption
Proof Idea:Game 1: the original scheme
Enc(
pk
, m) : U = gr, V = mr
,
W
=
H
(
U
,
V
, yr)
m||r
Game 2: Replace H
(U*, V*, yr*) of the challenge ciphertext
with a random string
Enc(pk, m*) : U*
= gr*, V*
= mr*, W* =
R* m||r
Game 1 and Game 2 are indistinguishable under the CDH assumption.
The adversary only has a negligible probability to win in Game 2 under the CDH assumption.Slide13
13
Extension: a non-pairing variant
In the PKE-ET, pairing is used in Test only.
If we remove
Test
, the scheme is a conventional PKE.KeyGen(1k)sk = x
R
Z
q
*
pk
=
y
= g
xEnc(pk
, m)r
R Zq
*Compute U =
gr, V = m
r, W = H(U, V, y
r)m||
rC := (U, V, W
)Dec(sk, C
)m||r
WH(
U, V, Ux)Verify r
R Zq
* m G1
\{1} U = g
r V = m
rIf true, return m
, else return
Observation:
in
a non-bilinear group,
this
PKE
achieves
a higher security
level.
The PKE can be implemented using a non-bilinear group. So we have more curves to choose from during implementation.Slide14
14
Extension: a non-pairing variant
Bad News: still cannot achieve IND-ATK
A
1
chooses x0 = gr0, x1 = g
r
1
where
r
0
r1
challenge stage: b {0,1}, Enc(pk
, xb) = (U = g
r, V = xbr, W
)A2 returns 0 if
V = Ur0; otherwise, returns 1.
Good News:
can achieve something stronger than OW-CCA2
W-IND-ATK where the adversary cannot select challenge plaintexts but the adversary is given the challenge plaintexts.Slide15
15
W-IND-ATK
In the random oracle model, the PKE in a non-bilinear group is W-IND-CCA2 secure under the DDH assumption.Slide16
16
Future Work
Standard model constructionAchieving IND-CCA2 for Test-removed version
Question:
is there any application for the property that the same scheme is PKE-ET
on bilinear group while being a PKE on non-bilinear group?Slide17
17
Q&AMore details can be found in
the Proc. of CT-RSA 2010