In these days there are plenty of signature schemes such as the
Today Internet is an inseparable part of our life and millions of people will be using the Internet. Reading the news, chatting with friends, purchasing a new product, and researching for a paper, the number of uses of the Internet is endless. One of the attractions of the Internet is that one can do almost anything from the comfort of his/her own home and with a relative sense of anonymity.
Unfortunately, the data going across the Internet may not be as secure as we would like to think. It is not especially difficult for a person with the right technical skills to intercept the data going from one computer to another. Usually this is not a problem; people do not really care if someone knows that they went to
Online banking and a host of other services rely heavily upon the security of credit card numbers, PINs, and other private information as it goes across the network. But if it is easy to intercept these numbers, how do these services work? The answer is cryptography.
In today’s commercial environment, establishing a framework for the authentication of computerbased information requires a familiarity with concepts and professional skills from both the legal and computer security fields. Combining these two disciplines is not an easy task because concepts from the information security field often correspond only loosely to concepts from the legal fields, even in situations where the terminology is similar. For example, from the information security point of view, “digital signature” means the result of applying to specific technical processes. The historical legal concept of “signature” is broader. It recognizes any mark made with the intention of authenticating the marked document [
In this research paper, we discuss threshold proxy signature scheme. In a
In the history of proxy signature technological development, the
Rivest et al. [
Recently, many threshold proxy signature schemes were proposed. The history of threshold proxy signature schemes is made up in Table
History of threshold proxy signature schemes.
Serial number  Scheme  Method 

1 
Rivest et al. [ 
The Lagrange interpolating polynomial and linear projective geometry 
2  ElGamal [ 
Discrete logarithms 
3 
Desmedt and Frankel [ 
RSA and Lagrange coefficient 
4  Zhang [ 
Discrete logarithms 
5  Kim et al. [ 
Discrete logarithms 
6  Sun et al. [ 
Discrete logarithms 
7 
Lee 
Discrete logarithms 
8  Hwang et al. [ 
RSA and Lagrange coefficient 
9  Wang et al. [ 
RSA and Lagrange coefficient 
10  Kuo and Chen [ 
RSA and Lagrange coefficient 
11  H. Jiang (2007)  RSA and Lagrange coefficient 
12  Fanyu (2007)  RSA and Lagrange coefficient 
13 
Li et al. [ 
RSA and Lagrange coefficient 
14 
YongJun et al. [ 
RSA and Lagrange coefficient 
The concept of threshold cryptosystems was also brought up by Denmedt and Frankel in 1991. They adapted the ElGamal public key cryptosystem and used the Lagrange interpolation or geometry to produce shadows.
To make proxy signature applicable to grouporiented situations, Sun [
This scheme requires a protocol to generate a random number among the group without the dealer. Let
Each proxy signer
Then, each
After receiving
If the verifications in Step 3 hold, each
The
Then each actual signer uses his proxy signature key to issue a partial proxy signature such that
Everyone can verify the validity of
If the previous verification holds, the signature, on
To verify the validity of the signature, anyone can examine the following equation:
The Kim et al.’s [
In order to remedy the problem of unknown signers, Sun et al. [
Unfortunately, Zhang’s scheme [
In 1991, Desmedt and Frankel proposed a threshold RSA signature scheme. This technique allows
In 1999, Okamoto et al. [
Hwang et al. [
Lee et al. [
Tzeng et al. [
Hwang et al. [
Lu et al. [
Yang et al. [
Tzeng et al. [
Tzeng et al. [
Hwang et al. [
In 2001, Hsu et al. proposed a nonrepudiable threshold proxy signature scheme. Tsai et al. [
Li et al. [
Hwang et al. [
Hwang et al. [
Raman Kumar et al. [
In the HLL scheme, Hwang et al. [
There are three types of participants in the scheme: the original signer, the
The proxy sharing phase;
The proxy issuing phase;
The verification phase.
In the proxy generation phase, the original signer computes the partial proxy signing keys from his private key and sends them to each designated proxy signer. In the proxy signature issuing phase, the proxy signers cooperatively create a valid signature on a message
Assume that an original signer
Then
When proxy signer
When the combiner receives all partial signature
So
Finally, the combiner generates the signature
The result of proxy signature is
The verifier can verify the signature signed on behalf of the original signer by the following equation:
The original signer can differentiate the actual signer from the signature
All analyses indicated that the scheme fails to satisfy all the requirements except one or two. So, an enhanced threshold proxy signature scheme must satisfy all of the following basic requirements which can be called proxy requirements [
The concept of threshold cryptosystems was first proposed by Katzenbeisser [
We compare the performance of four schemes, Hwang et al. [
A comparison of threshold proxy signature schemes based on proxy requirements.
Serial number  Proxy signature scheme/requirements  Kim et al.  Sun et al.  HLL  Wen et al.  Enhanced scheme 

1  Secrecy  Yes  Yes  No  No  Yes 
2  Proxy protection  No  No  No  No  Yes 
3  Unforgeability  Yes  No  No  No  Yes 
4  Nonrepudiation  Yes  Yes  No  No  Yes 
5  Timeconstraint  No  No  Yes  Yes  Yes 
6  Known signers  No  Yes  No  No  Yes 
Factoring
Tables
(a) The number of operations needed to factor
Digits 
Number of operations  Time 

100  9.6 × 108  16 minutes 
200  3.3 × 1012  38 days 
300  1.3 × 1015  41 years 
400  1.7 × 1017  5313 years 
500  1.1 × 1019  3.5 × 105 years 
1024  1.3 × 1026  4.2 × 1012 years 
2048  1.5 × 1035  4.9 × 1021 years 




1  885  29.7489 
2  888  29.7993 
3  893  29.8831 
4  900  30 
Computing
assume that
since
then
guess
Suppose
So,
The following attacks have been tested for RSA modules:
Hastad’s attack;
FranklinReiter attack;
extension to Wiener’s attack.
Given a set of
GramSchmidt process [
Consider
A basis
For all nonzero,
The following are the creation of key in seconds for different security levels which can be used for encryption and decryption.
The fields in Tables
(a) Security levels of the RSA module on a 90 MHz Pentium platform. (b) Security levels of the RSA module on a 255 MHz digital AlphaStation.
Security 
Encrypt 
Decrypt 
Create 

512 bit  370  42  0.45 
768 bit  189  15  1.5 
1024 bit  116  7  3.8 
Security 
Encrypt 
Decrypt 
Create 

512 bit  1020  125  0.26 
768 bit  588  42  0.59 
1024 bit  385  23  1.28 
Though our modification can withstand the forgery attack suffered by the said to be [
The probability of catching a user in enhanced threshold proxy signature scheme depends on the number of identity pairs used in it. The more the pairs used, the greater the chance of catching the anonymous user. The probability of catching the anonymous user is
For example, if
The algorithm is as follows.
Input: integer
If
Find the smallest
If
If
For
Output
Here
If
Conversely, if
The Fermat primality test is a probabilistic test to determine if a number is a probable prime. The algorithm is as follows.
Inputs:
Output:
repeat
pick
if
Here are some of the more commonly supported and used algorithms in protocol (see Table
The commonly supported and used algorithms used in protocol.
DiffieHellman key exchange algorithm  First published public key algorithm, 
Uses recipient’s public key to generate a secret key; public data is then sent to recipient who can now generate the secret key 
 
DSA 
Does not encrypt data, but produces a signature that can be verified 
Signing: input is data to be signed, private key, a random number; output is a signature, comprising 2 numbers called 
 
SHA 
US government standard produced by NIST 
Takes a message of less than 2^{64} bits and produces a message digest/fingerprint of 160 bits 
 
DSS 
US government standard method  Uses DSA to sign a message digest/fingerprint produced by SHA 
 
ElGamal 
Variant of DiffieHellman for encryption and decryption as well as key exchanges  Sometimes known as DiffieHellman in earlier versions of PGP 
 
RSA 
First main, and still, the most widely used general purpose publickey encryption algorithm  Encrypt message with public key to obtain confidentiality 
 
3DES 
DES was the first widespread symmetric key encryption algorithm 
DES is a 56bit key, 64bit block cypher using multiple rounds of permutations and substitutions 
 
CAST128 
Modern symmetric key encryption algorithm 
Uses key sizes of 40 to 128 bits (in 8bit increments) with 16 rounds of 64bit blocks of plaintext 
 
IDEA 
Modern symmetric key encryption algorithm, designed as a replacement for DES  128bit key block cypher encrypting 64bit blocks of plaintext 
 
AES/Rijndael 
Selected for the new “Advanced Encryption Standard” by NIST to replace DES  High performance and very secure algorithm, using key sizes of 128, 192, and 256 bits 
 
RSA  Selected for the new “Advanced Encryption Standard” by NIST  High performance and very secure algorithm 
The analysis reports of the proposed hypothesis for enhanced threshold proxy signature scheme are given as the following.
In this case, the value of entropy is the measure of the tendency of a process, to be entropically
Figure
Compression ratio (in %) for threshold proxy signature schemes.
Threshold proxy 
Compression 

HLL  66 
KUOCHEN  66 
GENGVRF  66 
FNGVERF  67 
THRSPROX  72 
Entropy for the enhanced threshold proxy signature scheme.
Radar chart showing compression ratio required in each scheme.
Floating frequencies/intuitive synthesis in its completed three parts entirety takes full advantage of the time complexity, space complexity, and communication overhead provided by the digital medium. We have calculated floating frequency of the threshold proxy signature scheme. Figure
Floating frequencies/intuitive synthesis for the enhanced threshold proxy signature scheme.
The ASCII histogram proved to be very useful since it helped enormously in debugging code involving probability calculations with simple print statements. Probabilistic simulations are extremely hard to test because the results of a given operation are never strictly the same. However, they should have the same probability distribution, so by looking at the rough shape of the histogram, it tells you if your calculations are going in the right direction. In this context, we have calculated ASCII histogram for our threshold proxy signature scheme. Figure
ASCII histogram for enhanced threshold proxy signature scheme.
This is mathematical representation of the degree of similarity between a given time series and a lagged version of itself over successive time intervals. It is the same a calculating the correlation between two different time series, except that the same time series is used twice—once in its original form and once lagged one or more time periods. The term can also be referred to as “lagged correlation” or “serial correlation”. In this, we have calculated autocorrelation for threshold proxy signature scheme. Figure
Autocorrelation for enhanced threshold proxy signature scheme.
A histogram is a graphical representation showing a visual impression of the distribution of data. We have analyzed histogram of for all threshold proxy signature schemes.
Histogram analysis of
Figure
Histogram analysis for the HLL threshold proxy signature scheme.
Number  Substring  Frequency (in %)  Frequency 

1  N  11.0343  654 
2  I  9.1277  541 
3  T  8.824  523 
4  E  8.6216  511 
5  S  7.4405  441 
6  R  7.1368  423 
7  A  5.0785  301 
8  O  4.6567  276 
9  C  4.1842  248 
10  D  3.6612  217 
11  U  3.5937  213 
12  F  3.2732  194 
13  P  3.1213  185 
14  G  3.0707  182 
15  L  3.0201  179 
16  H  2.8682  170 
17  Y  2.6489  157 
18  M  2.4296  144 
19  X  1.4341  85 
20  V  1.0798  64 
21  W  0.9954  59 
22  J  0.8267  49 
23  B  0.7761  46 
24  K  0.6917  41 
25  Q  0.3374  20 
26  Z  0.0675  4 
Radar chart showing histogram analysis for the HLL threshold proxy signature scheme.
Histogram analysis of
Figure
Histogram analysis for the KUOCHEN threshold proxy signature schemes.
Number  Substring  Frequency (in %)  Frequency 

1  N  11.3387  631 
2  I  8.841  492 
3  E  8.6253  480 
4  T  8.4636  471 
5  S  7.8886  439 
6  R  7.044  392 
7  O  4.8697  271 
8  A  4.6361  258 
9  C  4.4205  246 
10  U  3.8455  214 
11  G  3.2884  183 
12  P  3.2165  179 
13  L  3.1626  176 
14  F  3.0189  168 
15  H  2.9111  162 
16  D  2.8392  158 
17  M  2.6954  150 
18  Y  1.8509  103 
19  X  1.4196  79 
20  W  1.2579  70 
21  J  1.15  64 
22  V  1.0422  58 
23  B  0.9164  51 
24  K  0.7907  44 
25  Q  0.3953  22 
26  Z  0.0719  4 
Radar chart showing histogram analysis for the KUOCHEN threshold proxy signature scheme.
Histogram Analysis of
Figure
Histogram analysis for the GENGVRF threshold proxy signature schemes.
Number  Substring  Frequency (in %)  Frequency 

1  N  10.9658  587 
2  I  9.4153  504 
3  T  9.079  486 
4  S  8.3878  449 
5  E  7.9208  424 
6  R  7.1175  381 
7  O  4.7076  252 
8  A  4.5769  245 
9  C  3.9978  214 
10  U  3.6802  197 
11  F  3.5681  191 
12  P  3.5494  190 
13  G  3.4747  186 
14  L  2.989  160 
15  D  2.8956  155 
16  H  2.7835  149 
17  M  2.3538  126 
18  Y  1.8121  97 
19  X  1.4945  80 
20  V  1.3824  74 
21  J  0.9714  52 
22  B  0.8967  48 
23  W  0.7659  41 
24  K  0.7286  39 
25  Q  0.411  22 
26  Z  0.0747  4 
Radar chart showing histogram analysis for the GENGVRF threshold proxy signature scheme.
Histogram ANALYSIS of
Figure
Histogram analysis for the FNGVERF threshold proxy signature schemes.
Number  Substring  Frequency (in %)  Frequency 

1  N  10.947  630 
2  I  9.1573  527 
3  T  8.7576  504 
4  S  8.3927  483 
5  E  8.2711  476 
6  R  6.9505  400 
7  O  4.7089  271 
8  A  4.6568  268 
9  C  4.066  234 
10  U  3.8401  221 
11  F  3.5274  203 
12  G  3.5274  203 
13  P  3.4231  197 
14  L  3.3189  191 
15  D  2.7454  158 
16  H  2.6933  155 
17  M  2.6759  154 
18  Y  1.7724  102 
19  X  1.4248  82 
20  V  1.1816  68 
21  W  1.0252  59 
22  B  0.8862  51 
23  J  0.8688  50 
24  K  0.7298  42 
25  Q  0.3823  22 
26  Z  0.0695  4 
Radar chart showing histogram analysis for the FNGVRF threshold proxy signature scheme.
Histogram analysis of
Figure
Figure
Histogram analysis for the enhanced threshold proxy signature schemes.
Number  Substring  Frequency (in %)  Frequency 

1  N  11.5549  891 
2  S  8.7278  673 
3  E  8.5981  663 
4  T  8.5722  661 
5  I  8.3258  642 
6  R  6.6399  512 
7  O  5.4597  421 
8  A  4.539  350 
9  C  4.3963  339 
10  U  3.696  285 
11  F  3.4626  267 
12  P  3.3329  257 
13  G  3.307  255 
14  L  3.1384  242 
15  H  3.0865  238 
16  D  2.5937  200 
17  M  2.3603  182 
18  Y  1.8934  146 
19  X  1.634  126 
20  V  1.2061  93 
21  J  0.83  64 
22  W  0.817  63 
23  B  0.7522  58 
24  K  0.7133  55 
25  Q  0.3112  24 
26  Z  0.0519  4 
Radar chart showing histogram analysis for the enhanced threshold proxy signature scheme.
Radar chart showing overall analysis for all threshold proxy signature schemes.
The collusion attack is an action carried out by a given set of malicious users in possession of a copy of protected content that join together in order to obtain at the end of the attack procedure an unprotected asset. The attack is carried out by properly combining the protected copies of the multimedia documents collected by the colluders, according to the type of content and the kind of adopted protection system.
When the protection is assured by a data hiding algorithm, the collusion usually can take place in one of two different application frameworks: multimedia fingerprinting and ownership verification. In multimedia fingerprinting, a content owner, to protect his/her copyright, embeds a different code into each copy of the content distributed to each customer in order to be able to trace possible illegal usage of data and discover the source of the leakage of information; in this case, then, each colluder possesses a slightly different copy of the same multimedia content, and the attack consists in averaging all documents they have, trying to produce a new document in which the watermark is no longer present. If the number of averaged documents is large enough, the attack is very effective even without the introduction of perceptually significant degradation between the averaged multimedia document and the original one. In ownership verification, a content owner, to demonstrate that he/she is the authorized holder of the distributed content, embeds always the same code, linked to his/her identity, into different watermarked documents before they are distributed to the customers in such a way that the hidden code can be used to prove ownership in court if someone will infringe on his/her copyrights; in this case, then, each colluder possesses different multimedia documents, with the same hidden code, so that the attack is carried out by estimating the watermark by means of an average of all the different contents they have (this approach is suitable only for datahiding systems in which the hidden watermark does not depend on the host data). Then the estimated watermark can be removed from all the documents hiding in it or even falsely inserted in other ones to generate fake watermarked documents [
One advantage of enhanced threshold proxy signature schemes is that they can prevent a “collusion attack” in which two key generation servers communicate with each other to get useful information about the user’s private key. In essence, Figure
Existential forgery using a
Choose a random
Compute
We have
Existential forgery using a known message
Suppose
Can check
So
Existential forgery using a
To get a signature for
Query for signatures of
RSA function is a multiplicative homomorphism; for all
The Friedman test is a nonparametric statistical test developed by the US economist Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns. Applicable to complete block designs, it is thus a special case of the Durbin test. The Friedman test is used for oneway repeated measures analysis of variance by ranks. In its use of ranks, it is similar to the KruskalWallis oneway analysis of variance by ranks. We have tested our hypothesis against the Friedman test [
Figure
Chart showing Friedman test for all threshold proxy signature schemes.
The implementation of the schemes has been done.
When determining the time complexity of an algorithm, we measure how fast the computing requirements grow as the size of the input grows.
Readings are taken for two scenarios described below. Each reading shown in Table
Scenario 1: when number of threshold signers,
Scenario 2: when number of threshold signers,
Table
Variation of time with number of signers for scenario 1 (
Number of signers 
Time (in microseconds)  

HLL  KC  Geng  Fengying  Proposed  
1  128.0267  118.5867  123.6233  100.5567  98.52333 
2  135.92  135.7567  135.46  107.42  103.9967 
3  145.9433  142.23  146.5733  110.98  112.6033 
4  163.4033  152.1533  168.17  115.01  116.68 
5  164.8167  157.0067  172.7933  128.6167  122.1067 
6  167.7767  175.15  194.65  130.22  132.5533 
7  176.0667  178.7533  196.7933  135.69  135.77 
8  205.11  188.2633  213.01  138.0067  137.9133 
9  207.7133  189.5567  224.61  155.59  146.92 
10  214.1167  191.36  226.2533  159.64  158.2633 
Variation of time with number of signers for scenario 2 (
Number of signers 
Time (in microseconds)  

HLL  KC  Geng  Fengying  Proposed  
1  128.0267  118.5867  123.6233  100.5567  98.52333 
2  138.9933  147.0233  152.16  115.6033  117.08 
3  163.9733  175.71  191.2133  138.89  139.5033 
4  177.2367  206.82  225.47  166.2267  165.08 
5  196.0533  216.4767  254.1533  190.6533  179.8767 
6  217.9833  257.7233  276.2533  214.4267  209.7767 
7  252.38  282.3267  305.4967  249.96  233.82 
8  261.4233  313.67  337.4967  283.1467  256.4967 
9  304.9767  346.7833  375.7  315.74  270.6067 
10  324.8367  387.2833  415.05  350.9433  303.99 
We plot the graphs with the following:
execution time (in microseconds) on
number of proxy signers (
Scenario 1: when number of threshold signers,
Scenario 2: when number of threshold signers,
Figures
(a) Comparison of schemes based on time complexity when
(a) comparison of schemes based on time complexity when
Comparison of schemes based on time complexity when
Figures
RSA modulus,
Private key,
Public key,
The more the number of proxy signers, the more the computations. The graphs generated also confirm this point. Also, we observe that in the graphs in Figures
Partial proxy signatures,
The Lagrange coefficient,
The more the number of threshold proxy signers, the more the computations as confirmed by the graphs generated.
Scenario 3: when number of threshold signers,
Table
Variation of time with number of signers for scenario 3.
Number of signers 
Time (in microseconds)  

HLL  KC  Geng  Fengying  Proposed  
1  103.296  106.593  107.142  99.45  100.549 
2  124.175  118.132  119.230  121.978  118.132 
3  127.473  137.362  146.450  134.066  128.571 
4  154.945  156.044  154.395  172.527  173.076 
5  179.670  173.626  185.165  181.268  172.428 
6  195.054  203.846  209.890  203.846  196.703 
7  216.043  235.714  231.868  235.164  222.527 
8  256.593  268.329  259.340  249.400  243.956 
9  265.187  278.219  272.967  278.571  276.923 
10  286.264  313.626  311.538  304.285  305.274 
11  347.472  369.010  326.923  318.241  330.769 
12  364.325  371.428  345.114  350  351.648 
13  357.143  356.593  368.021  386.153  353.846 
14  373.143  364.286  367.142  386.505  370.879 
15  378.021  388.461  403.890  384.835  371.538 
Scenario 3: when number of threshold signers,
The space complexity of a program is the number of elementary objects that this program needs to store during its execution. We generate graphs to analyze the space complexity of the schemes.
Table
We plot the graphs with the following:
memory overhead (in terms of number of variables) on
Number of proxy signers (
The graphs have been shown in Figures
Comparison of schemes based on space complexity when
Comparison of schemes based on space complexity when
Figures
Space Requirements of the Schemes.
Serial no.  Scheme  

HLL  KC  Geng  Fengying  Proposed  
Variables  Quantity  Variables  Quantity  Variables  Quantity  Variables  Quantity  Variables  Quantity  
1. 










2. 










3. 










4. 










5. 










 
Total 





Scenario 1: when number of threshold signers,
Scenario 2: when number of threshold signers,
Figures
RSA modulus,
Private key,
Public key,
The more the number of proxy signers, the more the memory requirements. The figures generated also confirm this point.
Also, we observe in the graphs that memory overhead is more in the case of
Partial proxy signatures,
The Lagrange coefficient,
The more the number of threshold proxy signers, the more the memory requirements as confirmed by the graphs generated.
The communication overhead includes two types of communication in the schemes:
number of transmissions;
number of broadcasts.
Table
Figure
Comparison of communication overheads of the threshold proxy signature schemes.
Schemes  HLL  KC  Geng  Fengying  Proposed  

Number of Communications  PS  PSIV  PS  PSIV  PS  PSIV  PS  PSIV  PS  PSIV 
Number of transmissions 










Number of broadcasts  3  0  3  0  3  0  3  0  3  0 
 
Total 





PS: proxy sharing phase.
PSIV: proxy signature issuing and verification phase.
*  Schemes  HLL  Frequency 1  KUOCHEN  Frequency 2  GENGVRF  Frequency 3  FNGVERF  Frequency 4  THRSPROX  Frequency 5 

No.  Substring  Frequency 
Frequency  Frequency 
Frequency  Frequency 
Frequency  Frequency 
Frequency  Frequency 
Frequency 
 
1  N  11.0343  654  11.3387  631  10.9658  587  10.947  630  11.5549  891 
2  I  9.1277  541  8.841  492  9.4153  504  9.1573  527  8.7278  673 
3  T  8.824  523  8.6253  480  9.079  486  8.7576  504  8.5981  663 
4  E  8.6216  511  8.4636  471  8.3878  449  8.3927  483  8.5722  661 
5  S  7.4405  441  7.8886  439  7.9208  424  8.2711  476  8.3258  642 
6  R  7.1368  423  7.044  392  7.1175  381  6.9505  400  6.6399  512 
7  A  5.0785  301  4.8697  271  4.7076  252  4.7089  271  5.4597  421 
8  O  4.6567  276  4.6361  258  4.5769  245  4.6568  268  4.539  350 
9  C  4.1842  248  4.4205  246  3.9978  214  4.066  234  4.3963  339 
10  D  3.6612  217  3.8455  214  3.6802  197  3.8401  221  3.696  285 
11  U  3.5937  213  3.2884  183  3.5681  191  3.5274  203  3.4626  267 
12  F  3.2732  194  3.2165  179  3.5494  190  3.5274  203  3.3329  257 
13  P  3.1213  185  3.1626  176  3.4747  186  3.4231  197  3.307  255 
14  G  3.0707  182  3.0189  168  2.989  160  3.3189  191  3.1384  242 
15  L  3.0201  179  2.9111  162  2.8956  155  2.7454  158  3.0865  238 
16  H  2.8682  170  2.8392  158  2.7835  149  2.6933  155  2.5937  200 
17  Y  2.6489  157  2.6954  150  2.3538  126  2.6759  154  2.3603  182 
18  M  2.4296  144  1.8509  103  1.8121  97  1.7724  102  1.8934  146 
19  X  1.4341  85  1.4196  79  1.4945  80  1.4248  82  1.634  126 
20  V  1.0798  64  1.2579  70  1.3824  74  1.1816  68  1.2061  93 
21  W  0.9954  59  1.15  64  0.9714  52  1.0252  59  0.83  64 
22  J  0.8267  49  1.0422  58  0.8967  48  0.8862  51  0.817  63 
23  B  0.7761  46  0.9164  51  0.7659  41  0.8688  50  0.7522  58 
24  K  0.6917  41  0.7907  44  0.7286  39  0.7298  42  0.7133  55 
25  Q  0.3374  20  0.3953  22  0.411  22  0.3823  22  0.3112  24 
26  Z  0.0675  4  0.0719  4  0.0747  4  0.0695  4  0.0519  4 
Comparison of schemes based on communication overhead.
Overall communication architecture for secure threshold proxy signature scheme based on RSA cryptosystem.
Figure
It achieved the property of nonrepudiation.
Anyone cannot forge the legal proxy signature.
The verifier can identify the actual proxy signer of its group.
It also provides more refined result against its time complexity, space complexity, and communication overhead.
The proposed scheme is secure and efficient against notorious conspiracy attacks.
As the proxy signature is the solution to the delegation of signing capabilities in any electronic environment, in this paper, various threshold proxy signature schemes have been compared based on whether they fulfill the proxy signature requirements or not and proposed an enhanced secure threshold proxy signature scheme. Some of these schemes are based on an RSA cryptosystem for known signers, as the RSA cryptosystem is a popular security technique. In this, we also propose a new scheme which includes the features and benefits of the two schemes: Fengying et al. and Geng et al. The implementation of the encryption and decryption functions justifies the reallife applicability of the proposed scheme. In this, we have analyzed our enhanced threshold proxy signature scheme for various parameters
The authors wish to thank many anonymous referees for their suggestions to improve this paper.