Data sharing is the main way to give full play to the value
of data itself. For small amount of data, the generator directly
sends the data to the requester peer-to-peer. With the rapid
growth of user data scale, the cost of local storage, data
security and data services grow exponentially. Cloud storage
provides a cost-efficient way for data delegation, and can
deploy stable data services. However, data security turns out
to be critical since most delegated data are in plaintext. The
build-in access control and domain separation methods are
major security strategies rather than data encryption. In recent
years, several serious data leaking accidents are caused by
improper security setting in the cloud, explaining that all
sensitive data should be in fine-grained, ciphertext manner,
and the data sharing via cloud storage should be authenticated
by a safe key, which cannot be used to decrypt the original
data. Blaze et al. [1] announced the idea of proxy re-
encryption, where a proxy can convert a confidential message
encrypted by using one partyβs public key into another one
that can be decrypted by using another partyβs private key.
The Umbral project [2] propose a threshold-based proxy re-
encryption scheme following a key encapsulation mechanism
(KEM) approach. The data owner can delegate decryption
rights to any receiver for any ciphertext in intended grained,
through a re-encryption process performed by a set of N semi-
honest proxies. Under the threshold of (t, N), where at least t
out of N of these proxies participate by performing re-
encryption algorithm, and then the receiver is able to combine
these independent segments and decrypt the original message
using its private key.
In PRE schemes, the directionality and the transitivity are
critical characters for applications. In a bi-directional PRE
scheme, the re-encryption phase is reversible, which means
the proxy can use the same re-encryption key to re-encrypt
each ciphertext both from owner to receiver and receiver to
owner with zero knowledge to data. In this case, both the
owner and receiver must combine their secret keys to produce
the re-encryption key. The bi-directionality seems to bridge a
peer-to-peer data sharing relationship. On the other hand, a
unidirectional PRE means the proxy delegation is one-way.
The ciphertext can be re-encrypted from owner to receiver,
but not the reverse. Thus, the construction of re-encryption
key only requires the ownerβs secret key. As for the
transitivity character, it represents the times for a ciphertext
to be re-encrypted. Ciphertext that can be re-encrypted only
once refers to single-hop PRE scheme, while multi-hop PRE
scheme enables the unlimited-time transitivity of ciphertext
re-encryption. The permutation of types for PRE characters
allows a various of applications such as e-mail forwarding,
authorization transfer, and data distribution. However, most
PRE schemes require a centralized proxy to perform data
storage and re-encryption, which means the proxy turns out
to be the keypath of the system and has to handle all the
delegations.
We improve the system soundness by implementing a
decentralized proxy network, where each participant
possesses a key-share of re-encryption key, and can execute
the re-encrypt task separately. Meanwhile, our proposal also
involves multi-hop PRE to release data owner from re-
encryption key generation upon an authorized request, which
frequently calls the ownerβs secret key and has to be executed
locally at owner side. The distributed PRE scheme adopts
secret sharing and multi-hop re-encryption to provide reliable
and full delegation service. Moreover, this proposal achieves
a reshare function to process the continuing node variations
in decentralized environment.
The prototype of safe PRE scheme is introduced by
Mambo and Okamoto [3] by using the partial decryptions,
without offering any extra security benefits for delegatorβs
secret key. Their proposal aggregated the decrypt and re-
encrypt into an atomic execution by taking the re-encryption
key as input, during which the cipher-state original data is
never revealed. Later, Blaze, Bleumer and Strauss proposed
the ciphertext conversion algorithm [2] on the ElGamal
This work was supported by a grant from the Core Technology Research
and Development Program of Lin-Gang.
A Multi-hop and Distributed Proxy Re-encryption Scheme with
Dynamic Re-sharing
1JUNTAO CAO, 1XIN PEI, 2XIAOCHUAN WU
1The Idol Group, Shanghai, CHINA
2School of Computer Science, Fudan University, Shanghai, CHINA
Abstract: In order to achieve delegated data sharing, a reliable proxy is required for both data storage and execution of the
delegated authorization. The PRE scheme is a representative technique for delegating data sharing, which involves a single
proxy to transform the encryption by reencrypting algorithm with an auth-key, without knowing any knowledge about the
plaintext. However, most PRE schemes are performed in a centralized environment, which means the system will crash
upon the proxy is off-work. In this paper, we optimize the PRE scheme from two aspects. Firstly, the proxy acting as the
key path is decentralized in a thresholdbased network, which will provide continuous PRE service when any t out of N
nodes work. Moreover, considering the flexible entry and exit mechanism of the decentralized nodes, this proposal presents
a re-share algorithm to ensure N live nodes. Secondly, we adopt the multi-hop re-encryption strategy for transitivity of
ciphertext, so that the data owner is released from re-encryption key generation task upon user requests, and the authorized
delegatees are able to retransform the encryption to designated users by using its own secret key.
Keywords: data sharing, proxy re-encryption, KMS, reshare, multi-hop, transitivity
Received: April 25, 2021. Revised: July 15, 2022. Accepted: August 12, 2022. Published: September 13, 2022.
1. Introduction
2. Related Works
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cryptosystem, which requires taking secret keys of both
parties in the construction procedure. In a semi-honest model,
the proxy takes a re-encryption key(RK) generated by the
delegatorβs private key and delegateeβs public key to re-
encrypt the ciphertext, which was originally encrypted using
the delegatorβs public key, into a new ciphertext that can be
decrypted by the delegateeβs private key. The classical PRE
procedure is shown in Figure 1. Thus, the delegator can share
its fine-grained data by simple authentication via generating
the re-encryption key, rather than to execute decryption each
time or give out its private key.
Fig. 1. The classical unidirectional PRE scheme
Above re-encryption scheme only supports single-hop type,
so that the ciphertext is not transformable after re-encryption.
In other words, if the delegatee intends to re-authenticate the
ciphertext to another, it has to run decryption with the private
key and then encrypt the original data with its public key to a
new ciphertext. Ateniese and Fu [4] proposed the first
unidirectional proxy re-encryption scheme, where the
delegator can assign a proxy to authorize requestors on data
access without revealing its private key. In 2009, Wang and
Cao[5] proposed the multi-hop re-encryption scheme,
enabling the decryption right of cipertext to be transferred
multiple times. Their proposal was then enhanced by Cai and
Liu [6] to CCA-secured. Based on these works, Luo et al. [7]
proposed a unidirectional, multi-hop, identity-based proxy
re-encryption scheme with CPA-security. Sun et al. suggested
the identity-based PRE encryption schemes with the ability
of ciphertext evolution [8,9,10]. However, the length of
ciphertext will grow linearly with the times of updating. To
solve this, Liang et al. [11] proposed a constant ciphertext
size scheme, with the properties of collusion-resistant,
bidirectional and multi-hop. To address the directionality
problem, Weng et al. proposed a conditional PRE approach
in which the proxy can re-encrypt ciphertexts for a user only
if it satisfies the necessary conditions [12]. The unidirectional,
multi-hop PRE scheme is shown in Figure 2.
To improve the reliability of secrets, the centralized proxy
is distributed to a multi-proxy network, where each sub-proxy
possesses a secret share and executes partial computation, so
that the aggregation of multi-proxy outputs can meet the
demand. Secret sharing is a fundamental primitive in
cryptography, and the original secret sharing model was
proposed by Shamir. In [13], a (k, n) threshold scheme was
proposed. A secret data S can be divided into n share pieces
π !,β¦,π "
and S can be obtained through any k or more
π #
pieces, and S cannot be obtained through any
π β1
or fewer
π #
pieces.
Fig. 2. The unidirectional, multi-hop PRE scheme
On the other hand, the nodes in proxy network should
periodically prove the possession of secret shares. In the
research of remote data checking, Ateniese et al. firstly
propose the βchallenge-responseβ mode [14]. For each
checking request, the remote server has to calculate a
corresponding response with local data as input. Sebe et al.
propose a periodically scheme based on PDP [15]. The
requestor randomly chooses several data blocks as part of the
whole data to check. This reduces the overhead of server, but
the probability of error recognition decreases from 100% to
[16] or [17], where c is number of the
selected blocks in N rounds, and is sector number in each
block if divided. is the average error probability of each
block or sector.
Before the construction of the proposed distributed PRE
scheme with multi-hop delegation, we introduce the secret
Re-share algorithm.
Fig. 3. The Re-share algorithm
The secret Re-share algorithm provides a mechanism for
reliability of the distributed network. The shares are
generated by a secret sharing scheme by taking the re-
encryption key and the threshold as input. Then, the N shares
are distributed to the corresponding nodes, who get the
rewards from the contributed storage and computation.
However, there are probabilities that several nodes exit or
crash. In order to guarantee the retrievability of secret, a
heartbeat mechanism of share possession proof is established,
so that N live nodes are kept for most of the time and the Re-
share algorithm will execute when any node fails. Moreover,
the Re-share algorithm can generate any number of shares
unless the amount of live nodes is less then t as defined in the
1(1 )
c
r
--
1(1 )
cs
r
Γ
--
s
r
3. The Construction of Our Proposal
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threshold. Figure 3 illustrates a situation where a node
possessing the share E exits from the network under threshold
(3, 5). Then, the network executes the Re-share algorithm to
calculate a new share F on the polynomial via Lagrange
interpolation. Let
πΉ$
be a random number on
β€%
,
πΉ&=
π¦'π'
(
π₯
)
+π¦(π(
(
π₯
)
+π¦)π)
(
π₯
), where
π'
(
π₯
)
=*
!+$"
$#+$"β
*
!+$$
$#+$$,
π(
(
π₯
)
=*
!+$#
$"+$#β
*
!+$$
$"+$$
,
π)
(
π₯
)
=*
!+$#
$$+$#β
*
!+$"
$$+$"
, and the new
share is
πΉ = (πΉ$,πΉ&)
.
By adopting the KEM methods, the asymmetric encryption
only encrypts the data key, which is used to encrypt the
original data symmetrically. We use βcapsuleβ to term the
encryption of the data key to differ from the ciphertext of the
original data. The ability to decrypt a capsule means an
authentication to access the corresponding original data. Our
proposal consists of four pair of functions, the key
management pair Key Generation and ReKey-Generation, the
enc/dec pair Encrypt and Decrypt, the re-enc/retrieve pair re-
encrypt and Retrieve, the updating pair Resharing and
Transmit.
Firstly, we setup the public parameters and define the hash
families.
πππ‘π’π(π)
: Determine a cyclic group
πΎ
of prime order p,
according to the security parameter
8π
. Let
π,π β πΎ
be
generators. Let
π»:πΎ β β€%
, be hash functions that behave as
random oracles. Let
KDF:
πΎ β
{
0,1
}
,8
be a random oracle key
derivation function, where
π
is the fixed length according to
the security parameter
π
. The global public parameters are
represented by the tuple:
ππππππ =
(
πΎ,π,π,π»,KDF
) (1)
πΎππ¦πΊππ
(
8
) : Sample
π β β€%
uniformly at random as
private key, compute
π-
and output the keypair (
ππ,π π
)
=
(π-,π)
.
In this model, we assume A as the data owner or delegator,
B as the requestor or delegatee.
πΈππππ¦ππ‘(ππ')
: On input the public key
ππ'
, the encrypt
algorithm first samples random
π,π’ β β€%
and computes
πΈ =
π.
and
π = π/
. Computes the value
π = π’ + π βπ»0(πΈ,π)
.
The derived data key, used to encrypt the original data
symmetrically, is computed as
π·πΎ =KDF(ππ'
.1/)
. The
tuple
(πΈ,π,π)
is called capsule and allows to derive again
the data key DK. To examine the validity of the capsule, we
can check if the following equation holds:
8π28?= π βπΈ3%45678
(2)
Then, the
πΈππππ¦ππ‘
function runs a ChaCha20 to encrypt
the original data with DK into ciphertext Cipher. Finally, the
encrypt algorithm outputs
(πΆππβππ,ππππ π’ππ)
.
π
ππππ¦πΊππ(π π',ππ(,π‘,π)
: Firstly, pick a random
π‘ β1
degree polynomial in (3), where
π
is the secret generated in
πππ¦πΊππ
, and
π!~π9
are randomly chosen.
π
(
π₯
)
= π +π!π₯ +π0π₯0+β―+π9+!π₯9+!
(3)
We can calculate
π!= π
(
π₯!
)
,β¦,π"= π
(
π₯"
), and then we
generated n share group data:
(π₯!,π!),β¦,(π₯",π")
, and S can
be calculated by any t group data of
(π₯#,π#)
. Note that the
picked prime p should be bigger than both S and n, and data
range is in [0, p), and the values of
(π₯!,π!),β¦,(π₯",π")
are
computed modulo p. If the secret S is long, it can be break
into shorter blocks of m bits. Then, input Aβ secret key
π π'=
π
and Bβ public key
ππ(= π:
, a threshold
(π‘,π)
, the
π
ππππ¦πΊππ
algorithm computes
π
fragments of the re-
encryption key between A and B as follows:
(1) Sample a
π'β β€%
at random, and compute
π
'= π.#
(2) Compute
π = π»0
(
π
',ππ(,(ππ().#
), where
π
is the
result of a non-interactive Diffie-Hellman (DH) key
exchange between Bβs keypair and the ephemeral key
pair
8(π',π
')
.
(3) Compute
π· = π»(ππ',ππ(,ππ(
-)
(4) Initialize set
ππΉππππππ‘ = β
and repeat
π
times:
a. Sample random
π¦
,
ππ β β€%
b. Compute
π $= π»(ππ,π·)
and
π = π&
.
c. Compute
ππ = π(π $)
d. Compute
π!= π.;
.
e. Compute
π§!= π»(π,ππ,ππ',ππ(,π!,π
')
Fig. 4 The System architecture of the multi-hop PRE in distributed environment
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Volume 19, 2022
f. Compute
π§0= π¦ β π βπ§!
g. Obtain
ππΉπππ
as the tuple
(ππ,ππ,π
',π!,π§!,π§0)
.
h. Add into
ππΉππππππ‘
as
ππΉππππππ‘ βͺ{ππΉπππ}
.
(5) Finally, output the set
ππΉππππππ‘
.
This means Aβ grants its own permissions which are in
form of encryptions under
ππ'
to B.
π
ππΈππππ¦ππ‘(ππΉπππ,ππππ π’ππ)
:" For" each" node," on
receiving a
ππΉπππ = (ππ,ππ,π
',π!,π!,π0)
and the
ππππ π’ππ = (πΈ,π,π )
, it first checks the validity of the capsule
and outputs
β₯
if the check fails. Otherwise, it runs the re-
encrypt algorithm to compute
πΈ!= πΈ.;
and
π!= π.;
, and
outputs the capsule fragment
πππππ = (πΈ!,π!,ππ,π
')
. Due
to that the threshold requires t shares to retrieve the secret, the
π
ππΈππππ¦ππ‘
algorithm should appoint at least t nodes to
calculate t pieces of
πππππ
s.
π
ππ‘ππππ£π(π π(,ππ',{πππππ#}#<!
9)
: Taking Aβs public key
ππ'
, Bβs public key
ππ(
and corresponding t
ππππππ
, being
each of them
ππΉπππ#= (πΈ!6#,π!,ππ#,π')
as input, the
π
ππ‘ππππ£π
algorithm runs as following to aggregate them
into
8ππππ π’ππβ²
, which can be decrypted by
π π(
.
(1) Compute
π· = π»(ππ',ππ'
:,ππ()
, where
ππ'
:=
ππ(
-
by adopting DH protocol.
(2) Let
π = {π $6#}#<!
9
, where
π $6# = π»(ππ#,π·)
. For all
π $6# β π
, compute:
π#6= =
h
π $6>
π $6> βπ $6#
9
><!6>?#
(3) Then, compute the values in
πππππ#
:
πΈ@=
hi
πΈ!6#
j
A&,(
9
#<!
, πβ² =
h
(π!6#)A&,(
9
#<!
8
(4) Compute
π = π»
i
π
',π
':,ππ(
j, as the result of a non-
interactive DH key exchange between Bβs keypair and the
ephemeral key pair
(π',π
')
. The value
π
'
is the same for all
the
ππππππ
that are produced by re-encryptions using a
ππΉπππ
in the set of re-encryption key fragments
ππΉππππππ‘
.
(5) Finally, output the symmetric key
π·πΎ =KDF(
(
πΈ@β
π@
)
B)
π·ππππ¦ππ‘(πΆππβππ,π·πΎ)
: Decrypts the
πΆππβππ
using the
CHACHA20 decryption function with key DK, which results
in message M if decryption is correct, and β₯ otherwise.
The entire distributed PRE procedure is ended with the
user decryption with its own secret key. However, this system
might confront two reliability issues. One relates to the shares
in the distributed PRE network, as discussed above, nodes
could exit or crash any time, so that we build the Reshare
algorithm to guarantee enough valid shares live. The other is
the delegatee point, acting as the key-path role in this system,
transforming the data access right to specified users
according to the ownerβs delegation. Therefore, the chosen
PRE algorithm must be multi-hop, otherwise, the owner has
to generate the re-encryption key upon each user request or
reveal its secret key to the delegatee, which is quite unsafe.
Besides, the delegatee proxy should have online standbys to
ensure continuous service, which is also achieve by the Key
Transform algorithm.
π
ππ βπππ({ππΉπππ#}#<!
9)
: Under the nodes live checking
mechanism, the
π
ππ βπππ
algorithm will generate a new
ππΉπππ
without updating the existing ones. When any node
crashes, the system samples an
π₯ β β€C
at random, and use
Lagrange interpolation to calculate the corresponding
π¦ =
β
π¦>π>(π₯)
;
><D
, where
π>
(
π₯
)
=
β
$+$&
$)+$&
;
#<D6#?>
on the given
polynomial. Thus, the calculated point
(π₯,π¦)
turns out to be
a new
ππΉπππ
assigned to a live node.
πππππ ππ8π‘({ππΉπππ'E(6#}#<!
9,π π(,ππ))
: To transmit an
authorization from
π΄ β π΅
to
π΅ β πΆ
requires
{ππΉπππ'E(6# }#<!
9
,
π π(
,
ππ)
as input. Compared with
the
8π
ππππ¦πΊππ
algorithm, the difference is that the
πππππ πππ‘
algorithm achieves transitivity, which enables the
delegator to authorize any delegatee without using his own
secret key
π π'
.
The proposal PRE scheme has the following properties:
Unidirectionality. The delegation from the data owner to the
delegatee is unidirectional, while the reversal requires
another setup.
Non-interactivity. The adopted Diffie-Hellman protocol
between the delegator and delegateeβs keypair makes a shared
secret, enabling the re-encryption key generation of the
scheme non-interactive.
Transitivity. The encryption can be re-delegated more than
once by a series of transform keys.
Multi-hop. The designated delegatees are allowed to be the
delegators that can re-delegate the encryption under the
ownerβs authorizations, chaining the transformations.
Collusion safety. During the re-encryption, none of the
delegatees is able to collude with the distributed proxy nodes
that hold the transform key to recover the private key of the
delegator.
In this section discuss, we evaluate the performances of re-
encryption key generation algorithm, capsule re-encryption
algorithm, data key retrieval algorithm and
ππΉπππ
reshare
algorithm. The results of all tests are the average of 20 times.
Figure 5 illustrates a complete text process on above
algorithms by a given threshold (3, 5).
Figure 6 illustrates the time cost of
ππΉππππ ππ‘
generation.
There are 8 tuples of threshold included, where any t out of N
pieces can recover the re-encryption key. The growing rate is
decided by both t and N, due to that the t factor decides the
order of the polynomial, and N determines the total shares.
Fig. 6. Re-encryption key shares generation time
Figure 7 shows the computation cost in re-encryption.
Since out proposal adopts KEM strategy, the complexity only
relates to t in the threshold, with no regards to N or the size
of original file. However, this proposal ignores the node crash
probability during the re-encryption procedure, and several
4. Experiments of Our Proposal
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Volume 19, 2022
backup nodes possessing their
ππΉπππ
can be arranged to run
re-encryption algorithm to enhance the correctness.
Fig. 7. Re-encryption time of an increasing size of files
Figure 8 shows the cost of retrieving a data key with
different thresholds. This procedure can be regarded as the
reverse execution of data key encryption. We compare 8
tuples of thresholds, similarly, the cost only relates to t
numbers of
πππππ
, corresponding to the re-encrypted
ππΉππππ .
Fig. 8. Data key retrieval time
We also test the cost of Reshare algorithm. Given a fixed
polynomial, to sample an x at random and calculate
corresponding y is quite easy. But taking any t points to
recover the polynomial by using Lagrange interpolation is
costly. For a threshold with t = 50, the calculation time
Fig. 5. A complete verification on all algorithms of this proposal by a threshold (3, 5)
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requires 4872 milli seconds. In a decentralized network
environment, the participants are required to pledge their
assets for providing stable service, so that the Reshare
algorithm also means the crash times of participated nodes.
This paper proposes a distributed proxy re-encryption
network with multi-hop delegatees for reliable encryption
transformation. The proposed scheme is unidirectional, non-
interactive, transitive, multi-hop and collusion-safe. Also, we
have implemented this scheme in a test environment, which
only involves one local node that performed multiple roles in
the scheme. Experiments show that most of the overhead of
algorithm computation is in millisecond level except for
resharing in huge threshold, while the generations of re-
encryption key and multi-hop re-encrypting as well as the
reshare algorithms are regardless of the raw data size. Next,
we will combine this proposal with a DAG structured
blockchain to provide decentralized KMS service.
This work was supported by a grant from the Core Tech-
nology Research and Development Program of Lin-Gang.
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5. Conclutions
Acknowledgement
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WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS
DOI: 10.37394/23209.2022.19.18
Juntao Cao, Xin Pei, Xiaochuan Wu
E-ISSN: 2224-3402
185
Volume 19, 2022
