An Elliptic Curve-Based Signcryption Scheme with Forward Secrecy
Ali Asghar Beheshti Shirazi
An elliptic curve-based signcryption scheme is introduced
in this paper that effectively combines the functionalities of digital
signature and encryption and decreases the computational costs and communication
overheads in comparison with the traditional signature-then-encryption
schemes. It simultaneously provides the attributes of message confidentiality,
authentication, integrity, unforgeability, non-repudiation, public verifiability
and forward secrecy of message confidentiality. Since it is based on elliptic
curves and can use any fast and secure symmetric algorithm for encrypting
messages, it has great advantages to be used for security establishment
in store-and-forward applications and when dealing with resource-constrained
The encryption and digital signature are two fundamental cryptographic mechanisms
that can provide the security of communications. Until the before decade, they
have been viewed as important but distinct building blocks of various cryptographic
systems. In the public key schemes, a traditional method is to digitally sign
a message then followed by an encryption (signature-then-encryption) that can
have two problems: Low efficiency and high cost of such summation and the case
that any arbitrary scheme cannot guarantee the security. The signcryption is
a relatively new cryptographic technique that is supposed to fulfill the functionalities
of digital signature and encryption in a single logical step. It effectively
decreases the computational costs and communication overheads in comparison
with the traditional signature-then-encryption schemes. The first signcryption
scheme was introduced by Zheng (1997) but it fails the
forward secrecy of message confidentiality (Jung et al.,
2001). Zheng and Imai (1998) also proposed an elliptic
curve-based signcryption scheme that saves 58% of computational and 40% of communication
costs when compared with the traditional elliptic curve-based signature-then-encryption
schemes. Several signcryption schemes are also proposed over the years, each
of them providing different level of security services and computational costs.
The correctness, efficiency and security are the essential attributes that any
signcryption scheme should take them into account. A signcryption scheme should
simultaneously fulfill the security attributes of an encryption and those of
a digital signature. Such properties mainly include: Confidentiality, Unforgeability,
Integrity and Non-repudiation. Some signcryption schemes provide further attributes
such as Public verifiability and Forward secrecy of message confidentiality
while the others do not provide them. The public verifiability may not be required
in some applications while forward secrecy of message confidentiality has an
increasingly significance especially when the signcryption is to be done on
poorly protected devices such as mobile phones. For the case of brevity, we
do not describe the mentioned attributes since they can be easily found in the
corresponding literature such as Tso et al. (2007).
Currently, the elliptic curve cryptography is being used in a wide variety
of applications. The elliptic curve-based systems can attain to a desired security
level with significantly smaller keys than those of required for their counterparts.
This can enhance the speed and leads to an efficient use of power, bandwidth
and storage that are the basic limitations of resource-constrained devices.
The Elliptic Curve Discrete Logarithm Problem (ECDLP) that can be a computationally
infeasible problem has an essential role in the elliptic curve-based approaches
(Hankerson et al., 2004).
In this paper, a new elliptic curve-based signcryption scheme is introduced
that simultaneously provides the attributes of message confidentiality,
authentication, integrity, unforgeability, non-repudiation, public verifiability,
and forward secrecy of message confidentiality. It is an authenticated
scheme since it deploys an implicit authenticated key establishment.
THE PROPOSED SCHEME
Throughout this paper, Alice is the sender, Bob is the recipient and
Mallory is the malicious active attacker. Our proposed signcryption scheme
is depicted in Fig. 1 where some of its deployed notations
are described in Fig. 2. It consists of four phases:
Initialization, Signcryption, Unsigncryption, and Judge Verification.
The initialization phase includes selecting the domain parameters, generating
the private/public keys, and getting a certificate for the public key
of each user. In signcryption phase, Alice signcrypts her message and
sends it to Bob. In unsigncryption phase, Bob performs the unsigncryption
to recover the signcrypted text and verify the signature. The judge verification
phase is used only when any dispute occurs in which the judge decides
whether Alice has sent the signcrypted message to Bob or not.
Initialization: Domain parameters of the proposed scheme consist of
a suitably selected elliptic curve E defined over a finite field Fq
with the Weierstrass equation of the form y2 = x3 + ax
+ b and a base point G ε E(Fq) in which q is a large prime number.
In order to make the elliptic curve non-singular, a,b ε Fq
should satisfy 4a3 + 27b2 ≠ 0 (modq) (Hankerson
et al., 2004). To guard against small subgroup attacks, the point
G should be of a prime order n, or equivalently nG = O, and we should have
(Law et al., 2003). To protect against other known
attacks on special classes of elliptic curves, n should not divide qi-1
for all 1≤i≤V(V = 20 suffices in practice), n≠ q should be satisfied
and the curve should be non-supersingular (Law et al.,
||The proposed signcryption scheme
||Explanation of deployed notations
In order to keep the intractability of ECDLP to the Pollard-rho and
Pohlig-Hellman algorithms (Stinson, 2006), n should at
least satisfy n>2160 for the common applications. NIST has also
specified the minimum bit lengths of elliptic curves domain parameters
for different required level of security (NIST, 2007).
The private keys of Alice and Bob are the randomly selected integers wA,
wB εR[1, n-1]. The corresponding public keys are
calculated as WA = wAG and WB = wBG.
Alice and Bob are uniquely identified by the unique identifiers IDA
and IDB respectively. They also get the certificates CertA
and CertB from the Certificate Authority (CA) for their public keys
WA and WB. If CA is not involved in the public key generation
that is generally the case, it is necessary for CA to verify that each entity
really possesses the corresponding private key of its claimed public key. This
can be accomplished by a zero-knowledge proof. It should also be verified that
the public keys belong to the main group. Hereafter, it is also assumed that
the participants have access to an authentic copy of the CAs public key,
in order to use it for the purpose of certificate validation. The process of
certificate validation includes (Stinson, 2006):
||Verifying the integrity and authenticity of the certificate by verifying
the CAs signature on the certificate.
||Verifying that the certificate is not expired.
||Verifying that the certificate is not revoked.
Signcryption:Alice generates the signcrypted text (R, C, s) by
following the below steps:
||Checks the validity of CertB and uses it for verifying
||Randomly selects an integer rεR[1, n-1].
||Computes R = rG = (xR, yR).
in which f = log2
is the bit length of n, .
denotes the floor and .
indicates the ceiling. If K = O, she goes back to step 2. Otherwise,
she drives the session key of encryption as k = H(xK||IDA||yK||IDB)
in which H generates the required number of bits as the secret key
of deployed symmetric encryption.
||Computes the ciphertext as C = Ek(M).
||Computes the digital signature as s = twA-r (mod n) in
which t = HMACk (M ||xR||IDA||yR||IDB)
where HMAC is the Hash Message Authentication Code.
||Sends the signcrypted text (R, C, s) to Bob.
Unsigncryption: Bob who received the signcrypted text (R, C, s),
follows the below steps to extract the plaintext and verify the signature:
||Checks the validity of CertA and uses it for verifying
and derives the session key as k = H(xK||IDA||yK||IDB).
||Decrypts the received ciphertext as M = Dk(C).
||Computes t = HMACk (M||xR||IDA||yR||IDB).
||Verifies the Alices signature by verifying sG + R = tWA.
If this condition is satisfied, Bob accepts M as the correct plaintext
of Alice. Otherwise, he rejects M.
In the security policies of certain applications, it may be specified that
Bob upon verifying the signature of Alice, sends her a confirmation message
M" in addition to a tag t" = MACk (M") in which MAC is the Message
Authentication Code. In such cases, Bob should also verify the validity of the
received R = (xR, yR), in order to thwart the invalid-curve
attack (Hankerson et al., 2004). The point R acts
as an ephemeral public key while r is the corresponding ephemeral private key.
The validity verification of such ephemeral public key mainly includes checking
the following conditions (Antipa et al., 2003):
||R ≠ O.
||xR, yR ε Fq.
||R should satisfy the defining equation of E.
Here, the unsigncryption will be terminated with failure if any of the
above-mentioned conditions fails
Judge verification: When Bob claims that he has received the signcrypted
text (R, C, s) from Alice and a dispute occurs, the trusted third party
(judge) wants Bob to provide (R, C, s, M, k). Bob is simply capable of
extracting M and k from the previously saved (R, C, s). The judge follows
the following steps to adjudicate on what Bob claims:
||Checks the validity of CertA and uses it for verifying
||Verifies M = Dk (C). If this is not the case, Bob is
||Computes t = HMACk(M||xR||IDA||yR||IDB).
||Verifies the signature of Alice by checking the sG+R = tWA
condition. If this condition is not satisfied, Bob is wrong. Otherwise,
Alice has sent (R, C, s) to Bob.
Several precautions should be taken into account in implementation of
the proposed scheme. It is strongly recommended to use a strong block
cipher (such as AES) for encrypting messages. The generated random numbers
and also the state of pseudo-random generators should be erased from the
memory as soon as the signature is generated. Using the predefined pairs
of (r, R) and saving them in an insecure storage media is not recommended.
The private keys should be stored in a secure storage media such as a
Hardware Security Module (HSM).
Although it is not recommended, the first step of the signcryption and unsigncryption
phases may be simplified after the first run of the protocol. Alice and Bob
may save the trusted public key of the other party for the future uses. However,
it is still necessary for Bob to check the revocation status of the Alices
certificate. The revocation status can be obtained using either of Certificate
Revocation List (CRL), Delta CRL, or Online Certificate Status Protocol (OCSP).
The OCSP (Myers et al., 1999) is the most profitable
solution for the resource-constrained devices that cannot save a too large CRL,
and whenever the timely information is necessary, e.g. in the funds transferring.
The OCSP responses are digitally signed with a private key that its corresponding
trusted public key is known to the participants. The OCSP server may query its
required information from a database server. The certificates are usually stored
in an LDAP (Lightweight Directory Access Protocol) directory (Zeilenga,
2006). Figure 3 depicts a typical configuration for the
proposed scheme when an OCSP server is queried for the revocation status, in
which OCSPA and OCSPB are the corresponding OCSP tokens
for the certificates of Alice and Bob. For the resource-constrained environments,
delegating the validity verifications to a trusted server will improve the performance.
The verification authority and hash chain can also be used for simplifying the
certificate validations (Satizabal, 2007).
The correctness of the proposed scheme can be simply verified. Alice
and Bob will reach to the same point K on the elliptic curve since:
Subsequently, both of participants calculate the same session key as
k = H(xK||IDA||yK||IDB) so
Bob correctly decrypts the signcrypted text and verifies the signature.
as the least significant half in binary representation of xR is just
a trade-off between security and efficiency. This corresponds with the NIST
specifications (NIST, 2007). Let c denotes the bit length
The smaller the c the less required number of operations will be and the efficiency
will be subsequently improved. However, a too small c may decrease the security.
For a similar case, Krawczyk has shown that choosing c so that
does not increase the security (Krawczyk, 2005). If L
denotes a point of elliptic curve and ρ is a positive integer, finding
ρL takes O(log|ρ|) group operations (Wagstaff, 2003).
As an example, the required number of operations for calculating
is decreased by a factor of
with respect to that is required for calculating xRWA
is taken as the half in binary representation of xR. Therefore,
setting c = f/2
provides the best trade-off between security and efficiency.
|| A typical configuration for the proposed scheme
ON THE SECURITY OF THE PROPOSED SCHEME
The proposed scheme provides a wide variety of security attributes as
it is shown in Table 1. The long-term private key of
Alice is involved in the session key generation so the session key has
resilience to disclosure of secret value r. Validity verification of the
static and ephemeral public keys and the certificates are carefully considered
so several kinds of attacks are thwarted. The proposed scheme gets its
security from several components:
||The security attributes of the session key establishment.
||The security attributes of the certificates.
||The security attributes of deployed block cipher, one-way hash function
||Intractability of ECDLP due to the selected domain parameters.
The proposed scheme deploys a strong key establishment. Until now, many authenticated
key exchange protocols are introduced, each of them having their own problems
and limitations. The MQV protocols (Law et al., 2003)
are possibly the most efficient of all known authenticated Diffie-Hellman protocols
that use public-key authentication. The MQV has been widely standardized and
has been selected by the NSA to protect the classified information of USA government.
Although HMQV (Krawczyk, 2005) tries to thwart the MQVs
vulnerabilities by basically introducing an additional hash function, it also
has several vulnerabilities (Menezes, 2005; Menezes
and Ustaoglu, 2006). The elliptic curve-based session key establishment
process of the proposed scheme does not exactly correspond with that of Law
et al. (2003) and Krawczyk (2005), but it
tries to improve and match such ideas for its own case. The session key establishment
part of the proposed scheme has itself the following security attributes:
||Known session key security: Each execution of the protocol
results in a unique session key. The session key will differ for different
sessions because the ephemeral random number r is introduced in the
session key establishment process so the compromise of one session
key does not compromise the keys of the other sessions. Since the
private keys and identifiers of both participants are involved in
the session key derivation function, it will differ even if Alice
uses the same random number r for signcrypting the same message for
||Resilience to the Unknown-Key Share attack: In an UKS attack (Kaliski,
2001), two parties compute the same session key but have different views
of their peers in the key exchange. This attack is feasible when a key exchange
protocol fails to provide an authenticated binding between the session key
and identifiers of the honest entities. In the proposed scheme, validity
of certificates and also the static and ephemeral public keys are verified.
The UKS attack is thwarted because the identifiers of both Alice and Bob
are explicitly involved in the session key derivation function.
||Resilience to the Key Compromise Impersonation (KCI) attack: In
a KCI attack (Strangio, 2006), Mallory who could
obtain the private key of Alice (but does not have the private key of Bob)
tries to impersonate another honest party Bob to Alice. Resistance to the
KCI attack is an important feature since as long as Mallory cannot actively
control Alice, any session that is established by Alice remains secure.
Under intractability of the ECDLP, the KCI attack is thwarted in the proposed
scheme. An adversary that could obtain wA, should find the corresponding
r of R in order to deduce the corresponding session key that is generally
in deposit of solving the ECDLP.
||The provided attributes of different signcryption schemes
|aThe proofs are similar to the proofs of
bThe proofs are provided by Toorani and Beheshti
||Partial Forward secrecy: A system has the attribute of partial
forward secrecy if compromising the long-term private key of one entity
(or more but not all the entities) does not compromise the previously
established session keys. In the proposed scheme, even if wA
is revealed, the attacker should have the value of the corresponding
r that is generally in deposit of solving the ECDLP.
As it is indicated in Table 1, the proposed scheme
provides many attributes. Hereunder, a brief proof is given for the claimed
||Confidentiality: The proposed scheme deploys a strong block
cipher so according to the Kerckhoffs principle, the secrecy
will be entirely resided in the established session keys. The session
key establishment process has itself several security attributes.
Ultimately, an adversary has only two ways to defeat the confidentiality:
having wB, or deriving both wA and r. Deriving
the private keys and finding the corresponding r of a specific R is
in deposit of solving the ECDLP that is computationally infeasible
with the selected domain parameters. If wB is revealed,
the attacker can only decrypt those ciphertexts that are signcrypted
by WB. This is an inevitable feature of all the one-pass
protocols with implicit authenticated key exchange. However, for a
definite sender, the attacker should have both wA and the
corresponding r of the clearly sent R. Otherwise, he cannot derive
the true session key and the decryption will be computationally infeasible.
It is noteworthy that the proposed scheme does not have the feature
of past recovery so Alice cannot retrieve her previously signcrypted
texts without having the corresponding r of R. Compromising the past
recovery is a common feature of all the forward secure systems.
||Authentication: The implicit authentication is provided in
three ways: the proposed scheme is certificate-based and the certificates
are verified by both sender and recipient. An implicit authentication
is also involved in the session key establishment so only the correct
party who has the true private key can reach to the correct key agreement
and perform the unsigncryption. An authentication is also accomplished
when Bob verifies the signature by checking the sG+R = t WA
||Unforgeability: Mallory cannot forge the valid (M, R, s)
with his malicious (M, R, s). A valid forged
signature s should satisfy tWA-sG
= t WA-sG or equivalently s = s+(t-t)wA
so the knowledge of t, t and wA is necessary. To
find the true value of t = HMACk(M||xR||IDA||yR||IDB)
and t = HMACk(M||xR||IDA||yR||IDB),
Mallory should know the session key k that is in deposit of his knowledge
of wB, or both wA and r. Even if he knows wB,
he can only find the values of t and t but his knowledge of
wA is still necessary to forge the signature as s
= s+(t-t)wA. Otherwise, he cannot truly forge the
signature and his forged signature will be recognized in unsigncryption
and judge verification phases when checking the sG+R = tWA
||Non-repudiation: This attribute can be deduced from the unforgeability.
It is computationally infeasible to forge the Alices signature
without having wA. The signature is verified through the
certificate and public keys validation and when checking the sG+R
= tWA condition.
||Integrity: The hash value of message concatenated with some
variable parameters is involved in the signature generation so any
message alteration will change the signature. The integrity is guarantied
by the security attributes of HMAC and unforgeability of the signature.
Mallory should also have the valid session key to encrypt his modified
message. Otherwise, his modified message will not be correctly decrypted
by Bob. The session key is also involved in signature generation.
He should have wB, or both wA and r to find
the valid session key. He should also find a collision for the HMAC
that is computationally infeasible. Otherwise, he cannot forge a valid
signature. The integrity is implicitly verified when Bob checks the
sG+R = tWA condition.
||Public verifiability: Given (R, C, s, M, k) anybody can verify
the signature by checking the sG+R = tWA condition, without
any need for the private key of Alice or Bob. However, in some cases,
it may be desirable to keep the message confidentiality so that any
third party can verify the senders signature without any need
for knowledge of plaintext and the corresponding session key. This
can be referred to as the directly public verifiability. If necessary,
the proposed scheme can be modified so that the signature can be directly
verified by anyone who observes the transmitted pairs of (R, C, s).
Since directly public verification is not required in many applications,
it will be introduced as an optional variant in this paper.
||Forward secrecy of message confidentiality: It means that
even if the long-term private key of the sender is revealed, the adversary
is not capable of decrypting the previously signcrypted texts. The
only way to defeat the message confidentiality is to have wB,
or both wA and r. When wA is revealed, for decryption
to be possible, it is necessary to have the value of random number
r for the corresponding session. Otherwise, the ephemeral session
key cannot be recovered. Generally, the forward secrecy will be compromised
only if the attacker can solve the ECDLP that is computationally infeasible
with the selected domain parameters. As a one-pass protocol, since
there is not any session-specific input from Bob, we cannot prospect
the proposed scheme for the Perfect Forward Secrecy (PFS). However,
the proposed scheme provides the partial forward secrecy under intractability
of the ECDLP.
Here, the time complexity of the proposed scheme is evaluated. Table
2 gives a comparison between the computational costs of the proposed
scheme and those of the other schemes, in which the computational costs
of verifications and symmetric encryption are neglected. Since different
schemes provide different number of security attributes, it is necessary
to take a glimpse at the provided security attributes in Table
1 before comparing the computational costs of different schemes in
Table 2. In Table 2, the computational
costs are categorized with respect to different kinds of required operations,
each of them requiring a different number of bit operations. The total
running time of a signcryption scheme depends on the efficiency of deployed
algorithm for each of required operations (hardware-independent), and
the speed of deployed hardware (hardware-dependent).
Let ζ = log2n+1
denotes the bit-length of the modulus n. Since n is an odd prime number,
the resulted calculations in the modular arithmetic have at most a bit-
length of ζ. Adding and subtracting two ζ-bit integers each takes
O(ζ) bit operations (Rosen, 1988) so we have:
Multiplying two ζ-bit integers, and dividing a 2ζ-bit dividend by
a ζ-bit divisor takes O(ζ2) bit operations using the conventional
method (Rosen, 1988). Thus,
However, one can multiply and divide faster than what is specified by the conventional
methods, e.g. a divide-and-conquer algorithm due to Karatsuba and Ofman reduces
the complexity of multiplication to
(Hankerson et al., 2004). There is also a fast algorithm
for multiplying two ζ-bit integers in O(ζ log ζ log log ζ)
bit operations but it is not so useful unless ζ is very large (Wagstaff,
2003). However, for having a fair comparison between the computational costs
of different schemes, it is reasonable to just consider what is required by
the conventional methods since such results are the results of the worst situation
and anyone can decrease the required number of bit operations by deploying faster
algorithms. For the modular inverse calculation using the conventional method,
we have (Rosen, 1988):
The running time of the modular exponentiation using the conventional methods
will be of (Rosen, 1988):
||Computational costs of different signcryption schemes
The HMAC executes in approximately the same time as its embedded hash function
for long messages but its running time depends on the kind of its embedded hash
function. Total number of operations for the HMAC-SHA1 calculations (Elkeelany
et al., 2002) is as:
THMAC-SHAI = 32+1110x(2+nk)
While for the case of HMAC-MD5, it is:
THMAC-MD5 = 32+744x(2+nk)
is the number of input blocks to the embedded hash function where N is the
bit-length of the total message, and k is the bit-length of extra-appended inner
form of the key (Elkeelany et al., 2002).
The computations in elliptic curves depend on many factors including
the deployed coordinates. The computational cost of an Elliptic Curve
Point Addition (ECPA) in an Affine coordinate is as:
While for the Jacobian projective coordinate, we have:
The Elliptic Curve Point Multiplication (ECPM) is a basic and time-consuming
operation in ECC. The execution times of the EC-based schemes are typically
dominated by the point multiplication. It typically takes O(log|k|) group operations
to compute kP when k ≠ 0 (Wagstaff, 2003). Several methods
for the scalar multiplication are proposed in the literature. However, selecting
the suitable algorithm is complicated by platform characteristics, coordinate
selection, memory and other constraints, security considerations, and interoperability
requirements (Hankerson et al., 2004). For elliptic
curve P-192 defined over GF(p = 2192-264-1) that provides
the same level of security as that of RSA with a 1024-bits modulus (NIST,
2000), the total required number of field operations for a point multiplication
for an unknown point using the Window NAF method on
Jacobian-Chudnovsky coordinate is calculated in (Hankerson
et al., 2004) as:
However, for a fixed point P, using offline pre-computations and Comb 2-table
method in Jacobian-Affine coordinate, it can be calculated as (Hankerson
et al., 2004):
that is explicitly less than what is specified in (11). For other curves,
coordinates and algorithms, the required number of operations will differ
from what is specified in (11) and (12). The computational costs of the
proposed scheme and those of the other schemes can be fairly compared
using Table 2 and expressions (2-11). Figure
4 depicts the total required number of bit operations for executing
signcryption and unsigncryption in the signcryption schemes that are described
in Table 2, in which the computational costs of verifications
and symmetric encryption are neglected. As it is apparent from Fig.
4, the computational costs of our proposed scheme in unsigncryption
phase is slightly more than the other depicted EC-based schemes, as it
provides the most feasible security attributes, but it has a great computational
advantage over its exponentiation-based counterparts.
||Total Number of bit operations for different signcryption
Here, some improvements are introduced to the proposed scheme. They are
considered as optional improvements since they may not be required in
some applications and systems.
Delegating verifications to a trusted server: When the participants
are dealing with processing and memory constraints, delegating the validity
verifications to a trusted server will offer significant advantages. The required
time of sending a certificate to a validation server and subsequently, receiving
and authenticating the response can be considerably less than the required time
of performing the certificate path discovery and validation by a resource- constrained
device. The performance of the proposed scheme can be greatly enhanced by delegating
the validation processes to a Delegated Validation (DV) server, as it is depicted
in Fig. 5. The DV server can be independent of the communication
network. It accomplishes the certificate validation via the Delegated Path Validation
(DPV) protocol but its duties defers from what is specified by Pinkas
and Housley (2002). All the signcrypted texts will be directed to the DV
server through the communication network. The DV server accomplishes the certificate
and public key validations for both sender and recipient. It queries the database
server for the certificates of both sender and recipient through their identifiers.
It obtains the revocation statuses by getting OCSP responses from the OCSP server.
It also checks the validity of point R. The DV server will contact the designated
recipient after a successful validation. If any error occurs, the DV server
will send an error message to the initiator and saves a copy in its log file.
According to the security policies, all the transmitted messages may be separately
saved by the DV server for the possible disputes. The DV server digitally signs
its responses, unless an error is occurred. The signed responses should include
a hash value of all the transmitted parameters in addition to the identifiers
of both sender and recipient.
In the optimized configuration, Bob will not check the validity of point R
because it is validated by the DV server. He just checks the DV servers
signature. Alice does not need to check the revocation status of the Bobs
certificate. She just needs to have the public key of Bob that can be simply
queried from a database server through the Bobs identifier when Alice
does not have it. She can then save the public key of Bob for the future uses.
Using timestamps: Timestamps have an unavoidable role in many
security applications when the true order of occurrence and freshness
of messages are important. Alice can produce her timestamp TA
according to her local time or a predefined international time zone. It
can be obtained from a trusted time server. The only required modifications
to the proposed scheme are to modify the session key derivation function
as k = H(xK||IDA||yK||IDB||TA),
and parameter t of signature generation function as t = HMACk
Alice clearly includes her timestamp in her signcrypted text as (TA,
R, C, s) and sends it to Bob. Bob upon receiving a new message, produces
his timestamp TB and checks whether 0<TB-TA<ω
in which ω is the acceptable time window that is determined by estimating
the usual delay time between sending and receiving of a message in the
deployed communication link. Hereby, Bob can easily neglect the old messages.
This can simply thwart the potential replay attacks.
|| An optimized configuration for the proposed scheme
when it is used for the resource-constrained devices
the deployed encryption and signature generation functions will be time-dependent
and the freshness of messages will be enhanced. The timestamp can also
be used as a parameter in the seed of random number generators. However,
exploiting timestamps has its own problems (e.g. the synchronization)
and cannot be deployed in some systems.
Towards directly public verification: In the proposed scheme,
the signature can be verified by any third party provided that (R, C,
s, M, k) is provided by either of participants (Alice or Bob). However,
in some cases, it may be desirable to keep the message confidentiality
so that any third party can verify the senders signature without
any need to the knowledge of plaintext and the corresponding session key.
If necessary, the proposed scheme can be modified so that the signature
can be directly verified by anyone who observes the transmitted pairs
of (R, C, s) and without any for the private keys of the participants.
This can be simply accomplished by modifying the parameter t in all phases
of the proposed scheme as t = H(C||xR||IDA||yR||IDB).
Consequently, anyone who observes the transmitted (R, C, s) can compute
t and directly verify the Alices signature by following the below
||Check the validity of CertA and use it for verifying
||Verify the Alices signature by checking the sG+R = tWA condition.
In this paper, a new elliptic curve-based signcryption scheme is introduced
that simultaneously provides the security attributes of message confidentiality,
authentication, integrity, unforgeability and non-repudiation. It also
has the attribute of public verifiability so any third party can verify
the signature without any need for the private keys of the participants.
It also has the attribute of forward secrecy of message confidentiality
so even if the senders private key is revealed, no one else can
extract the plaintext of the previously signcrypted texts. Since it is
based on elliptic curve cryptography and uses symmetric ciphering for
encrypting messages, it has great advantages to be deployed in resource-constrained
devices such as mobile phones. As a one-pass scheme, it is also so attractive
for security establishment in store-and-forward applications such as E-mail
and Short Message Service (SMS).
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