A Novel Provable Secured Signcryption Scheme : A Hyper-Elliptic Curve-Based Approach
Abstract
:1. Introduction
1.1. Preliminaries
- 𝗁(α) ∈ [α] is a polynomial and the degree is 𝗁(α) ⩽ g
- 𝘧(α) ∈ 𝒻[α] is the monic polynomial, and the degree is 𝘧(α) ⩽ 2g + 1
- The points on the hyper-elliptic curve do not form a group unlike an elliptic curve
- The hyper-elliptic curve works on divisor which is branded as the formal and finite sum of points on a hyper-elliptic curve that can be further symbolized by Mumford as:
1.2. Hyper-Elliptic Curve Discrete Logarithm (HECDLP)
1.3. Basic Notations
Represents a hyper-elliptic curve over the field | |
: | Is a large prime number and the value of |
: | Is the divisor of the generalized elliptic curve |
𝒽1, 𝒽2, 𝒽3: | Demonstrate the hash functions |
𝓒: | Epitomizes the ciphertext |
m: | Epitomizes the plaintext or message |
M = 𝓂.: | Represents the message concatenation with divisor |
: | Represents the private key of the signcrypter |
: | Represents the public key of the signcrypter |
: | Represents the private key of the unsigncrypter |
: | Represents the public key of the unsigncrypter |
Is a fresh nonce | |
: | Represent the secret key |
, : | Representing the subdivided secret key |
E/D: | Represents the encryption and decryption functions |
Eyk (Exk(M)): | Represents double encryption |
Dyk (Dxk(𝓒)): | Represents double decryption |
2. Formal Model of the Proposed Scheme
2.1. Proposed Scheme Construction
2.1.1. Key Generation
2.1.2. Signcryption
Algorithm 1. Algorithm () |
Randomly select a number γ from {1,…‥‥,q − 1}. |
Select a nonce |
Compute where is the divisor on a hyper-elliptic curve. |
Divide into , |
Compute |
Compute |
Divide into , |
Compute |
Compute 𝓒 = |
Compute 𝓣 = (𝓒) |
Compute 𝓢 = |
Send ω = (𝓒, ) to Bob or Unsigncrypter |
2.1.3. Unsigncryption
Algorithm 2. Algorithm () |
Compute = () |
Compute (.) |
Divide into , |
Divide into , |
Use the double decryption method |
Compute |
Compare if equality holds then there is no change in |
Verification of signature is done through 𝒵. + . |
3. Correctness
4. Security Analysis
4.1. Replay Attack
4.2. Confidentiality
4.3. Integrity
4.4. Authenticity
4.5. Unforgeability
4.6. Non-Repudiation
4.7. Forward Secrecy
4.8. Public Verifiability
- Compute ()
- Compute
- Compare if equality holds, then there is no change in
- Verification of signature is done through 𝒵. + ., if satisfy then the message from signcrypter otherwise not.
5. Computational Cost
- Intel Core i74510UCPU
- 2.0 GHz processor
- 8 GB of memory
- Windows 7 Home Basic
- Multi-precision Integer and Rational Arithmetic C Library (MIRACL)
6. Communication Cost
Generalized Formulas for the Reduction of Communication Cost
7. Applications
- Confirm payment order validity
- If not valid then EXIT
8. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
- → H, HH, HHH: hash functions
- M: Is plain text or message
- → M’: Is the cipher text
- → Ju: public key of unsigncrypter
- → inv (Js): private key of signcrypter
- → inv (Ju): private key of unsigncrypter
- → Js: public key of signcrypter
- →(Signcrypter.{H1(M’.Xz’.Nr)}_inv(Js).{H2(Alpa’.H1(M’.Xz’.Nr).inv(Js)).Ju}_Eyk’.{M’.Nr.Exk’}_Eyk’): encryption
- → (Signcrypter.{H1(M’.Xz’.Nr)}_inv(Js).{H2(Alpa’.H1(M’.Xz’.Nr).inv(Js) ).Ju}_Eyk’.{M’.Nr.Exk’}_Eyk’): decryption.
- → Eyk, Exk: Symmetric key
- → Alpa’: random number
- , → Xz,Yz: Dividing
- Ki → intruder public key
- inv (ki) → intruder private key
- → {M’.Nr.Exk’}_Eyk’)
- → {H1(M’.Xz’.Nr)}_inv(Js)
- → {H2(Alpa’.H1(M’.Xz’.Nr).inv(Js) ).Ju}_Eyk’
HLPSL Code () |
role |
%%start of the protocol by Signcrypter already knows the Unsigncrypter’s |
role_Signcrypter(Signcrypter:agent,Unsigncrypter:agent,Ju:public_key,Js:public_key,Nr:text,Ns:text,SND,RCV:channel(dy)) |
played_by Signcrypter |
def= |
local |
%%start of the protocol by Signcrypter already knowing the %%Unsigncrypter’s public key |
State:nat,Alpa:text,H1:hash_func,Xz:text,H2:hash_func,Eyk:symmetric_key,M:text,Exk:symmetric_key |
init |
State:= 0 |
transition |
%% signcrypter receives challenge from Unsigncrypter by using his public %%key and nonce, sends message to unsigncrypter in response |
1. State=0/\ RCV(Unsigncrypter.{Ns}_Js) =|> State’:=1/\ Eyk’:=new()/\ Exk’:=new()/\ M’:=new()/\ Xz’:=new()/\ Alpa’:=new()/\ SND(Signcrypter.{H1(M’.Xz’.Nr)}_inv(Js).{H2(Alpa’.H1(M’.Xz’.Nr).inv(Js)).Ju}_Eyk’.{M’.Nr.Exk’}_Eyk’)/\ SND(Signcrypter.{H2(Alpa’.H1(M’.Xz’.Nr).inv(Js)).Ju}_Eyk’)/\ SND(Signcrypter.{M’.Nr.Exk’}_Eyk’) |
end role |
role |
%%defining the role played by signcrypter by using its keys… |
role_Unsigncrypter(Signcrypter:agent,Unsigncrypter:agent,Ju:public_key,Js:public_key,Nr:text,Ns:text,SND,RCV:channel(dy)) |
played_by Unsigncrypter |
def= |
local |
State:nat,Alpa:text,H1:hash_func,Xz:text,H2:hash_func,Eyk:symmetric_key,M:text,Exk:symmetric_key |
init |
State:= 0 |
transition |
1. State=0/\ RCV(start) =|> State’:=1/\ SND(Unsigncrypter.{Ns}_Js) |
2. State=1/\ RCV(Signcrypter.{H1(M’.Xz’.Nr)}_inv(Js).{H2(Alpa’.H1(M’.Xz’.Nr).inv(Js)).Ju}_Eyk’.{M’.Nr.Exk’}_Eyk’) =|> State’:=2 |
3. State=2/\ RCV(Signcrypter.{H2(Alpa.H1(M.Xz.Nr).inv(Js)).Ju}_Eyk) =|> State’:=3 |
4. State=3/\ RCV(Signcrypter.{M.Nr.Exk}_Eyk) =|> State’:=4 |
end role |
role |
%%session1 between agents signcrypter and unsigncrypter |
session1(Signcrypter:agent,Unsigncrypter:agent,Ju:public_key,Js:public_key,Nr:text,Ns:text) |
def= |
local |
SND2,RCV2,SND1,RCV1:channel(dy) |
composition |
role_Unsigncrypter(Signcrypter,Unsigncrypter,Ju,Js,Nr,Ns,SND2,RCV2)/\ role_Signcrypter(Signcrypter,Unsigncrypter,Ju,Js,Nr,Ns,SND1,RCV1) |
end role |
role |
%%session2 between agents signcrypter and unsigncrypter |
session2(Signcrypter:agent,Unsigncrypter:agent,Ju:public_key,Js:public_key,Nr:text,Ns:text) |
def= |
local |
SND1,RCV1:channel(dy) |
composition |
role_Signcrypter(Signcrypter,Unsigncrypter,Ju,Js,Nr,Ns,SND1,RCV1) |
end role |
role environment() |
def= |
const |
hash_0:hash_func,nr:text,ju:public_key,alice:agent,bob:agent,js:public_key,ns:text,const_1: agent,const_2:public_key,const_3:public_key,const_4:text,const_5:text,auth_1:protocol_id, sec_2:protocol_id |
intruder_knowledge = {alice,bob} |
composition |
session2(i,const_1,const_2,const_3,const_4,const_5)/\ session1(alice,bob,ju,js,nr,ns) |
end role |
goal |
%% defining the goals as an aim of protocol |
authentication_on auth_1 |
secrecy_of sec_2 |
end goal |
environment() |
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Schemes | 𝓒𝓞𝓝 | 𝓘𝓝𝓣 | 𝓐𝓤𝓣 | 𝓤𝓝𝓕 | 𝓝𝓡𝓟 | 𝓕𝓡𝓢 | 𝓟𝓥 | 𝓡𝓐𝓡 | 𝓢𝓔𝓒𝓥𝓝 |
---|---|---|---|---|---|---|---|---|---|
Hwang [10] | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓝𝓞 | 𝓝𝓞 |
Toorani [11] | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓝𝓞 | 𝓨𝓢 | 𝓨𝓢 | 𝓝𝓞 | 𝓝𝓞 |
Mohpathra [33] | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓝𝓞 | 𝓝𝓞 |
Elkamchouchi [34] | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓝𝓞 | 𝓝𝓞 |
Proposed | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 | 𝓨𝓢 |
Schemes | Signcryption | Unsigncryption | Total |
---|---|---|---|
Hwang [10] | 2 𝓔𝓒-𝓟𝓜 | 3 𝓔𝓒-𝓟𝓜 | 5𝓔𝓒-𝓟𝓜 |
Toorani [11] | 2 𝓔𝓒-𝓟𝓜 | 3 𝓔𝓒-𝓟𝓜 | 5 𝓔𝓒-𝓟𝓜 |
Mohpathra [33] | 3 𝓔𝓒-𝓟𝓜 | 2 𝓔𝓒-𝓟𝓜 | 5 𝓔𝓒-𝓟𝓜 |
Elkamchouchi [34] | 2 𝓔𝓒-𝓟𝓜 | 4 𝓔𝓒-𝓟𝓜 | 6 𝓔𝓒-𝓟𝓜 |
Proposed | 2 𝓗𝓒-𝓓𝓜 | 4 𝓗𝓒-𝓓𝓜 | 6 𝓗𝓒-𝓓𝓜 |
Schemes | Signcryption | Unsigncryption | Total |
---|---|---|---|
Hwang [10] | 1.94 | 2.91 | 4.85 |
Toorani [11] | 1.94 | 2.91 | 4.85 |
Mohpathra [33] | 2.91 | 1.91 | 4.85 |
Elkamchouchi [34] | 2.91 | 3.88 | 5.86 |
Proposed | 0.96 | 1.92 | 2.88 |
Schemes | Communication Cost |
---|---|
Hwang [10] | |
Toorani [11] | |
Mohpathra [33] | |
Elkamchouchi [34] | |
Proposed |
Schemes | Message Size | 128 bits | 256 bits | 1024 bits |
---|---|---|---|---|
Hwang [10] | 128 + |160| + 2|160| = 608 | 256 + |160| + 2|160| = 736 | 1024 + |160| + 2|160| = 1504 | |
Toorani [11] | 128 + |160| + 2|160| = 608 | 256 + |160| + 2|160| = 736 | 1024 + |160| + 2|160| = 1504 | |
Mohpathra [33] | 128 + |160| + 2|160| = 608 | 256 + |160| + 2|160| = 736 | 1024 + |160| + 2|160| = 1504 | |
Elkamchouchi [34] | 128 + |160| + |160| = 448 | 256 + |160| + |160| = 576 | 1024 + |160| + |160| = 1344 | |
Proposed | 128 + |80| + |80| = 288 | 256 + |80| + |80| = 416 | 1024 + |80| + |80| = 1184 |
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Share and Cite
Ullah, I.; Amin, N.U.; Khan, J.; Rehan, M.; Naeem, M.; Khattak, H.; Khattak, S.J.; Ali, H.
A Novel Provable Secured Signcryption Scheme
Ullah I, Amin NU, Khan J, Rehan M, Naeem M, Khattak H, Khattak SJ, Ali H.
A Novel Provable Secured Signcryption Scheme
Ullah, Insaf, Noor Ul Amin, Junaid Khan, Muhammad Rehan, Muhammad Naeem, Hizbullah Khattak, Shah Jahan Khattak, and Haseen Ali.
2019. "A Novel Provable Secured Signcryption Scheme
Ullah, I., Amin, N. U., Khan, J., Rehan, M., Naeem, M., Khattak, H., Khattak, S. J., & Ali, H.
(2019). A Novel Provable Secured Signcryption Scheme