COLLECTIVE SIGNATURE PROTOCOLS FOR SIGNING GROUPS BASED ON PROBLEM OF FINDING ROOTS MODULO LARGE PRIME NUMBER

International Journal of Network Security & Its Applications (IJNSA) Vol.13, No.4, July 2021
DOI: 10.5121/ijnsa.2021.13405 59
COLLECTIVE SIGNATURE PROTOCOLS FOR SIGNING
GROUPS BASED ON PROBLEM OF FINDING ROOTS
MODULO LARGE PRIME NUMBER
Tuan Nguyen Kim1
, Duy Ho Ngoc2
and Nikolay A. Moldovyan3
1
Faculty of Information Technology, Duy Tan University, Da Nang 550000, Vietnam
2
Department of Information Technology, Ha Noi, Vietnam
3
St. Petersburg Institute for Informatics and Automation of
Russian Academy of Sciences, St. Petersburg, Russia
ABSTRACT
Generally, digital signature algorithms are based on a single difficult computational problem like prime
factorization problem, discrete logarithm problem, elliptic curve problem. There are also many other
algorithms which are based on the hybrid combination of prime factorization problem and discrete
logarithm problem. Both are true for different types of digital signatures like single digital signature,
group digital signature, collective digital signature etc. In this paper we propose collective signature
protocols for signing groups based on difficulty of problem of finding roots modulo large prime number.
The proposed collective signatures protocols have significant merits one of which is connected with
possibility of their practical using on the base of the existing public key infrastructures.
KEYWORDS
Collective digital signature, group digital signature, signing group, finding roots modulo.
1. INTRODUCTION
Digital signature (DS) protocols are widely used in information technologies to process electronic
legal messages and documents. The DS protocols are based on DS schemes that represent a
mathematical technique applied in public-key cryptography to validate the authenticity of digital
messages or documents. Such validation is connected with the fact that DS as some redundant
information can be computed only with using the private key that is known only to one person,
i.e. to the signer. Verification of the signature validity is performed with signer’s public key that
is known publicly. To solve a variety of different practical tasks, different types of signatures are
proposed: Individual signature [1,8]; Blind signature [2,3]; Collective signature [4,7]; Group
signature [5,10].
The group signature refers to a signature formed on behalf of a group of signers (signing group)
headed by a person called group manager or leader [11].The group digital signature (GDS) to an
electronic message is generated by a group member. To verify the group signature, group public
key needs to be used. Except the group manager, nobody can disclose which particular group
member signed the document. The group signature has the following important properties: Only
group members can sign a document; Group manager, who has both document and valid group
signature can reveal the group members signed the document; And, non-group members could
not reveal the original signers, who generate the group signature [14].

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The collective signature refers to a signature generated with participation of each of the
individual signers included in some declared set of signers. Validity of collective signature to
some electronic document M means that M is signed by each of them. To generate a collective
digital signature (CDS) it is needed each of the mentioned individual signers use his private key.
The procedure of the verification of CDS is performed using public keys of each signer. The CDS
protocols can be practically used on the base of the public key infrastructure (PKI) existing on
practice to support the widely used individual DS protocols. In addition, another merit of the
CDS protocols relates to possibility to implement them using many official DS standards [12], for
example the Russian standard GOST R 34.10–2012 [13].
Combining the main properties of CDS and GDS in the frame of some single DS protocol [15] is
very actual in the following cases: when an electronic document is to be processed and signed by
several different signing groups; when an electronic document is to be processed and signed by
several different signing groups and by several different individual signers. In this paper we
propose the collective signature protocol for both cases, namely: Collective digital signature
protocol for signing groups and Protocol of collective digital signature for group and individual
signers.
Generally, digital signature algorithms are based on a single difficult computational problem like
prime factorization problem, discrete logarithm problem, elliptic curve problem or are based on
the hybrid combination of prime factorization problem and discrete logarithm problem. We based
on difficulty of the problem of finding roots modulo large prime number [16] to design our
proposed collective signature protocols.
2. COLLECTIVE DIGITAL SIGNATURE BASED ON PROBLEM OF FINDING
ROOTS MODULO LARGE PRIME NUMBER
2.1. Digital Signature Protocol
New hard computational problem described in [16] is used in the digital signature scheme (DSS)
described below. It uses the prime modulus having the structure p = Nk2
+ 1, where k is a large
prime (|k|  160 bits) and N is such even number that the size of p satisfies the condition
|p|  1024 bits.
A random value x is selected as a private key. The public key y is computed using the formula
y = xk
mod p. The signature represents a pair of the numbers S and E. The size of S is equal to
|p|  1024 bits and size of E is equal to ||  160 bits, where  is some specified prime number.
Suppose a message M is given.
The signature generation procedure is performed as follows:
1. Select at random a value t < p  1 and calculate:
𝑅 = 𝑡𝑘
𝑚𝑜𝑑 𝑝 (1)
2. Using some specified hash function FH(M) calculate the hash value H corresponding to the
message M and compute the first element of the signature
𝐸 = 𝑓(𝑅, 𝑀) = 𝑅𝐻 𝑚𝑜𝑑 , (2)

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where  is a large prime that is a parameter of the signature generation algorithm. For example, it
is acceptable to use a randomly selected prime  such that||= 160. The function FH(M) is also a
part of the DSS. For example, one can use the hash function SHA-1 recommended by US
National Institute of Standards and technology (NIST).
3. Calculate the second element of the signature:
𝑆 = 𝑥𝑓(𝑅,𝑀)
𝑡 𝑚𝑜𝑑 𝑝 (3)
The signature verification procedure is performed as follows:
1. Using the signature (E, S) compute:
𝑅 = 𝑆𝑘
𝑦𝐸
2. Calculate:
𝐸′
= 𝑓(𝑅; 𝑀) = 𝑅𝐻 𝑚𝑜𝑑  (5)
3. Compare E with E. If E= E, then the signature is valid.
The signature length is equal to
|𝐸| + |𝑆| = || + |𝑝| |𝑝|.
The random value t plays the role of one-time secret key. It is unacceptable to use the same value
t for the formation of signatures to two different documents, since in this case the private key can
be calculated. Indeed, let (R, S1) and (R, S2) are the signature to the messages M1 and M2,
respectively. We have
𝑆1 = 𝑦𝑓(𝑅,𝑀1)
. 𝑅 𝑚𝑜𝑑 𝑝 (3a)
and
𝑆2 = 𝑦𝑓(𝑅,𝑀2)
. 𝑅 𝑚𝑜𝑑 𝑝 (3b)
Therefore
𝑆1
𝑘
𝑆2
𝑘 = 𝑦𝑓(𝑅,𝑀1)−𝑓(𝑅,𝑀2)
(6)
therefore 𝑥 = (
𝑆1
𝑆2
)
1/(𝑓(𝑅,𝑀1)−𝑓(𝑅,𝑀2))
2.2. Collective Signature Protocol
Using the previously described digital signature scheme one can propose the following collective
signature protocol.
Suppose the j-th user owns the public key yj depending on his private key xj< p as follows: 𝑦𝑗 =
𝑥𝑗
𝑘
mod p, where j = 1,2,…, s.

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Suppose an electronic document M is given and m(m < s) users owning the public keys y1, y2,
…, ym should sign it simultaneously.
The following protocol produces the collective digital signature:
1. Each jth user selects at random a value𝑡α𝑗
< 𝑝and computes the public value:
𝑅𝛼𝑗
= 𝑡𝛼𝑗
𝑘
𝑚𝑜𝑑𝑝 (8)
where j = 1,2, …, m.
2. Some of the users (or one of them) calculate the common randomization value:
𝑅 = ∏ 𝑅𝛼𝑗
𝑚𝑜𝑑𝑝
𝑚
𝑗=1 (9)
and then calculate the first part of the CDS:
𝐸 = 𝑓(𝑅, 𝑀)
where f is a specified compression function. For example, we will use the following function:
𝐸 = 𝑅𝐻 𝑚𝑜𝑑  (10)
where  is a large prime having length ||= 160 bit and H is a hash value computed from the
message M.
3. Using the values R and 𝑡α𝑗
each jth user computes its share in the CDS:
𝑆𝛼𝑗
= 𝑥𝛼𝑗
𝑓(𝑅,𝑀)
𝑡𝛼𝑗
that is supposed to be available to all users of the group.
4. Calculate the second element of the CDS:
𝑆 = ∏ 𝑆𝛼𝑗
𝑚𝑜𝑑 𝑝
𝑚
𝑗=1 (12)
Thus, the CDS is computed with 2m modulo exponentiations. The CDS length is fixed and equals
to|𝑆| + |𝛿|.
The CDS verification procedure is performed as follows.
1. Compute the collective public key y:
𝑦 = ∏ 𝑦𝛼𝑗
𝑚𝑜𝑑 𝑝
𝑚
𝑗=1 (13)
2. Using the CDS (E; S) compute value 𝑅′
𝑅′ = 𝑆𝑘
𝑦𝐸
3. Compute 𝐸′ = 𝑓(𝑅′, 𝑀) = 𝑅′𝐻 𝑚𝑜𝑑 
4. Compare values Eand E.
If E= E, then the signature is valid. Otherwise the signature is false.

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3. COLLECTIVE DIGITAL SIGNATURE FOR SIGNING GROUPS BASED ON
PROBLEM OF FINDING ROOTS MODULO LARGE PRIME NUMBER
The GDS protocol presupposes the formation of a digital signature to some electronic document
on behalf of some collegial body (group of signers, i.e. signing group), which is headed by a
group manager. Each representative of a group of signers generates his private key x and his
public key 𝑦 = 𝑥𝑘
𝑚𝑜𝑑 𝑝. The public key Y of the group manager is a public key of the group
and is calculated as follows 𝑌 = 𝑋𝑘
𝑚𝑜𝑑 𝑝, where X is manager’s private key. The value Y is
also the public key of the group, i.e. the value Y is used to verify authenticity of the GDS.
Let m group members (having public keys 𝑦𝑖 = 𝑥𝑖
𝑘
𝑚𝑜𝑑 𝑝and corresponding private keys xi , i
= 1, 2, …, m) wish to sign the document M.
The group signature protocol is described as follows:
 Signature generation:
1. The group manager computes hash value from document 𝐻 = 𝐹𝐻(𝑀), where 𝐹𝐻 is some
specified hash function, calculates masking coefficients
𝑖 = 𝐹𝐻(𝐻 || 𝑦𝑖|| 𝐹𝐻(𝐻 ||𝑦𝑖|| 𝑋)) (15)
and sends each value i to the corresponding i-th group member, for i=1, 2, .., m. Then the group
manager computes the first element of the group signature:
𝑈 = ∏ 𝑦𝑖
𝑖
𝑚𝑜𝑑 𝑝
𝑚
𝑖=1 (16)
2. Each i-th group member (i = 1, 2, …, m) generates a random number ti < p-1, computes
the value:
3.
𝑅𝑖 = 𝑡𝑖
𝑘
and sends Ri to the group manager.
4. The group manager generates the random number T < p-1 and computes the values
5.
𝑅′ = 𝑇𝑘
𝑚𝑜𝑑 𝑝, (18)
𝑅 = 𝑅′ ∏ 𝑅𝑖
𝑚
𝑖=1 𝑚𝑜𝑑 𝑝 = (𝑇. ∏ 𝑡𝑖
𝑚
𝑖=1 )𝑘
, (19)
and
𝐸 = 𝐹𝐻(𝑀||𝑅||𝑈) 𝑚𝑜𝑑 , (20)
where is a large prime having length ||= 160 bit,E is the second element of the group signature.
Then he sends value E to all group members who have initiated the protocol.
6. Each i-th group member (i = 1, 2, …, m) computes his signature share
𝑆𝑖 = 𝑥𝑖
𝐸𝑖
.𝑡𝑖 𝑚𝑜𝑑 𝑝 (21)

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and sends it to the group manager.
7. The group manager verifies the correctness of each share Si by checking equality
8.
𝑅𝑖 = 𝑆𝑖
𝑘
𝑦𝑖
−𝐸𝑖
If all signature shares Si satisfy the last verification equation, then he computes his share
𝑆′ = 𝑋𝐸
. 𝑇 𝑚𝑜𝑑 𝑝 (23)
and the third element of the group signature
𝑆 = 𝑆′. ∏ 𝑆𝑖 𝑚𝑜𝑑 𝑝
𝑚
𝑖=1 (24)
 Signature verification:
The verification procedure includes the following steps:
1. The verifier computes the hash-function value from the document M: H = FH(M). Using
the group public key Y and signature (U, E, S) he computes value:
𝑅* = 𝑆k
(𝑌𝑈)-E
2. He computes value
3.
𝐸* = 𝐹H(𝑀 || 𝑅* || 𝑈) (26)
4. Compares the values E and E.
If E* = E, then the verifier concludes that the group signature is valid. Otherwise, he rejects the
signature.
 Proof of correctness:
Let us show that the proposed protocol generating the CDS (U, e, s) works correctly.
Substituting the value:
𝑆 = 𝑆′. ∏ 𝑆𝑖
𝑚
𝑖=1 𝑚𝑜𝑑 𝑝,
𝑌 = 𝑋𝑘
𝑚𝑜𝑑 𝑝
And 𝑈 = ∏ 𝑦𝑖
𝑖
𝑚𝑜𝑑 𝑝
𝑚
𝑖=1
in the right part of the verification equation (25):
𝑅* = 𝑆k
(𝑌𝑈)-E
𝑚𝑜𝑑 𝑝
we get: 𝑅∗
= 𝑆𝑘
(𝑌𝑈)−𝐸
𝑚𝑜𝑑 𝑝
= (𝑋𝐸
. 𝑇 ∏ 𝑥𝑖
𝐸𝑖
.𝑡𝑖)
𝑚
𝑖=1
𝑘
(𝑋𝑘
. 𝑇 ∏ 𝑦𝑖
𝑖
)
𝑚
𝑖=1
−𝐸

65
= 𝑋𝑘𝐸
. (𝑇 ∏𝑥𝑖
𝐸𝑖
. 𝑡𝑖)
𝑚
𝑖=1
𝑘
𝑋−𝑘𝐸
. (𝑇 ∏ 𝑦𝑖
𝑘𝑖
)
𝑚
𝑖=1
−𝐸
= 𝑇𝑘
. ∏ 𝑡𝑖
𝑘
𝑚
𝑖=1
𝑚𝑜𝑑 𝑝 = 𝑅
It is easy to see that the value:
𝐷 = ∏ 𝑆𝑖
𝑚
𝑖=1 𝑚𝑜𝑑 𝑝 (27)
can be considered as a "group pre-signature" approving of which is performed by the group
manager with adding his signature share S’. The value is actually calculated analogously to the
computation of the collective signature in the protocols [5,6]. The main difference between the
described GDS protocol and collective DS protocols [5,6] is using the masking coefficients i at
time of generating the collective public key U, which is used as the first element of the GDS. The
value U conserves the information about all group members who participated in the process of
generating the GDS. It is easy to see that only the group manager can open the GDS, using the
value U, since only he can compute the masking values i.
In the protocol developed in this paper it is also used the mechanism of the formation of the
collective DS. Namely, this mechanism is used in the following two ways: i) to form a pre-
signature and ii) to form a collective signature shared by several signing groups.
Let g signing groups with public keys𝑌
𝑗 = 𝑋𝑗
𝑘
𝑚𝑜𝑑 𝑝, where j = 1,2, …, g; Xj is the secret key of
the j-th group manager, have intention to sign the document M.
Suppose also the j-th signing group includes mj active individual signers (persons appointed to act
on behalf of the j-th signing group). The protocol of collective signature for group signers is
described as follows.
The signature generation procedure relating to the proposed collective DS protocol for signing
groups:
1. Within the framework of the GDS protocol described above, the manager of each j-group
of signers (j = 1, 2,.., g) generates masking parameters ji for the signers of his group and
computes the value:
𝑈𝑗 = ∏ 𝑦𝑗𝑖
𝑗𝑖
𝑚𝑗
𝑖=1
(where i = 1,2,…, mj) as the j-th share in the first element of the collective group signature and
the randomizing parameter:
𝑅𝑗 = 𝑅′𝑗 ∏ 𝑅𝑗𝑖
𝑚𝑗
𝑖=1
Then he sends values Uj and Rj to all other managers.
2. Each j-th group manager (j = 1, 2, …, g) computes values
𝑈 = ∏ 𝑈𝑗
𝑔
𝑗=1 𝑚𝑜𝑑 𝑝, 𝑅 = ∏ 𝑅𝑗
𝑔
𝑗=1 𝑚𝑜𝑑 𝑝, (30)

66
And
𝐸 = 𝐹𝐻(𝑀||𝑅||𝑈) 𝑚𝑜𝑑 , (31)
where  is a large prime having length ||= 160 bit, E and U are the first and second elements of
the group signature.
3. Each j-th group manager (j = 1, 2, …, g) computes signature share of his group
𝑆𝑗 = 𝑆′𝑗 ∏ 𝑆𝑗𝑖
𝑚𝑗
𝑖=1
𝑚𝑜𝑑 𝑝, (32)
where Sji is the signature share of the ith individual signer in the ith signing group, and sends it to
other group managers.
4. Each j-th group manager can verify the correctness of each share Sj by checking equality
5.
𝑅𝑗 = 𝑆𝑗
𝑘
(𝑌
𝑗𝑈𝑗)−𝐸
𝑚𝑜𝑑 𝑝. (33)
If all shares Sj satisfy the last verification equation, then the third element S of the collective
signature is computed:
𝑆 = ∏ 𝑆𝑗
𝑔
𝑗=1 𝑚𝑜𝑑 𝑝 (34)
The tuple (U, E, S) generated by the above procedure presents the collective signature (to the
document M) shared by g signing groups.
 Signature Verification:
The signature verification procedure relating to the proposed collective DS protocol for
signing groups:
1. Compute the collective public key shared by all signing groups:
𝑌𝑐𝑜𝑙 = ∏ 𝑌
𝑗
𝑔
𝑗=1 𝑚𝑜𝑑 𝑝 = (∏ 𝑋𝑗
𝑔
𝑗=1 )𝑘
2. Compute the value:
𝑅* = 𝑆k
(𝑈𝑌col)-E
3. Compute the value:
𝐸* = 𝐹H(𝑀 || 𝑅* || 𝑈) (37)
4. Compare the values E and E*.
If E* = E, then one concludes that the group signature is valid. Otherwise, the signature is
rejected.
 Proof of correctness:
Substituting the value:
𝑆 = ∏ 𝑆𝑗
𝑔
𝑗=1 𝑚𝑜𝑑 𝑝,𝑈 = ∏ 𝑈𝑗
𝑔
𝑗=1 𝑚𝑜𝑑 𝑝,
𝑌𝑐𝑜𝑙 = ∏𝑌
𝑗
𝑔
𝑗=1
𝑚𝑜𝑑 𝑝

67
in the right part of the verification equation (36):
𝑅* = 𝑆k
(𝑈𝑌col)-E
𝑚𝑜𝑑 𝑝
we get:
𝑅∗
= 𝑆𝑘
(𝑈𝑌𝑐𝑜𝑙)−𝐸
𝑚𝑜𝑑 𝑝
= (∏ 𝑆𝑗)𝑘
𝑔
𝑗=1
(∏ 𝑈𝑗
𝑔
𝑗=1
∏𝑌
𝑗
𝑔
𝑗=1
)−𝐸
𝑚𝑜𝑑 𝑝
= ∏ 𝑆𝑗
𝑘
(𝑈𝑗𝑌
𝑗)−𝐸
𝑔
𝑗=1
𝑚𝑜𝑑 𝑝
= ∏ 𝑅𝑗
𝑔
𝑗=1
𝑚𝑜𝑑 𝑝 = 𝑅
The first element U of the collective signature contains information about all group members of
each signing group who signed the document M. The identification procedure (the disclosure of
the group signature) is carried out by analogy with the procedure for disclosing the group
signature described in [9]. It should be noted that the procedure for identifying individual signers
requires the participation of the group managers of each group that share the collective signature.
At the same time, the computational complexity of this procedure is relatively high and rapidly
increases with the growth of number of the signing groups that share collective signature.
In the proposed collective DS scheme the signature verification procedure includes the steps of
the verification procedure in the group signature scheme and an additional initial step for
computing the collective public key (step 1). In the signature verification equation it is used the
collective public key Ycol instead of the group public key.
4. PROTOCOL OF COLLECTIVE DIGITAL SIGNATURE FOR GROUP AND
INDIVIDUAL SIGNERS
Another important practical scenario relates to the processing document M by several individual
signers and by several group signers. Construction of the collective signature protocol (in Section
2) for such case can be implemented in full correspondence with the collective signature protocol
for group signers described in Section 3, if it is accepted an agreement that for individual signers
the value Uj is equal to 1.
It is evident that only all group managers act in the procedure of disclosing the collective group
signature (identification of the individual signers acted in the frame of each group signer).
5. CONCLUSION
In paper [16], Nicolay A. Moldovyan based on difficulty of finding the kth
roots in the finite
fields GF(p) such that p = Nk2
+ 1, where k is sufficiently large prime having the size |k|> 160
bits and N is even number such that the size of p is |p|> 1024 bits, to propose a collective digital
signature scheme. This is the basis for us design collective signature protocols for signing groups
based on problem of finding roots modulo large prime numbers: Collective digital signature for
signing groups and Collective digital signature for group and individual signers. Both are
extensions of collective digital signatures that combine the advantages of group digital signatures

68
and collective digital signatures. Their size does not depend on the number of members involved
in the formation of the final signature. In each turn, we presented the signature generation
process, the signature verification process, as well as demonstrate the correctness of this
verification process.
We also set all our hope on our future work to develop the collective signature schemes of the
proposed types, in which the signature contains only two elements E and S.
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69
AUTHORS
Tuan Nguyen Kim was born in 1969, received B.E., and M.E from Hue University of
Sciences in 1994, and from Hanoi University of Technology in 1998. He has been a
lecturer at Hue University since 1996. From 2011 to the present (2021) he is a lecturer at
School of Computer Science, Duy Tan University, Da Nang, Vietnam. His main research
interests include Computer Network Technology and Information Security.
Duy Ho Ngoc was born in 1982. He received his Ph.D. in Cybersecurity in 2007 from
LETI University, St. Petersburg, Russia Federation. He has authored more than 45
scientiﬁc articles in cybersecurity.
Nikolay A. Moldovyan is an honored inventor of Russian Federation (2002), a
laboratory head at St. Petersburg Institute for Informatics and Automation of Russian
Academy of Sciences, and a Professor with the St. Petersburg State Electrotechnical
University. His research interests include computer security and cryptography. He has
authored or co-authored more than 60 inventions and 220 scientiﬁc articles, books, and
reports. He received his Ph.D. from the Academy of Sciences of Moldova (1981).

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