![ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011
Different Attacks on Selective Encryption in RSA
based Singular Cubic Curve with AVK and Their
Possible Solutions
Kalpana Singh1 and Shefalika Ghosh Samaddar2
Corresponding Author & M.Tech IV Semester Student1, Faculty Member2
Department of Computer Science and Engineering
Motilal Nehru National Institute of Technology Allahabad, Uttar Pradesh -211004, India
Email: Kalpana08.mnnit@gmail.com1, shefalika99@yahoo.com2
Abstract- In this paper, the security of Selective Encryptionin Later, Seng et al.[1] have shown that all three Koyama
RSA based Singular Cubic Curve with Automatic Variable Key schemes are equivalent to each other by an proposed
(AVK) for some well known attacks are analysed. It is proved isomorphism and becomes insecure in the case of some known
that this cryptosystem is more secure than Koyama scheme attacks like partially known attack [2][3], linearly related plain
from which the algorithm has been generated. The proposed text attack [4][5] isomorphism attack [4], homomorphism attack
cryptographic algorithm makes justified use of Koyama [6], Wiener’s attack [7], and Hastad’s attack [7]. Some other
Schemes. Koyama scheme is not semantically secure. The security notions are non-malleability [8] and Plain text-
proposed Scheme is efficient and semantically secure public
Awareness [8]. Non- malleability implies that any attacker
key cryptosystem based on Singular Cubic Curve with AVK.
Further, the partially known attacks, linearly related plain text cannot modify a cipher text while keeping any control over the
attacks, isomorphism attacks, low exponent attacks, Wiener’s relation between the resulting plain text and original one. The
attack and Hastad’s attack are analyzed for effect with the Plaintext-Awareness ensures that no one can produce a valid
proposed scheme. The Selective Encryption in RSA based cipher text without knowing the corresponding Plain text. There
Singular Cubic Curve with AVK for text based documents is are a number of mathematically induced attacks on RSA based
found to be robust enough to encounter all these attacks. cryptosystems.
These are classified into three categories:
Keywords- Singular Cubic Curves, Koyama Public Key • Attacks exploiting the polynomial structure of RSA.
Cryptosystem (PKC), Automatic Variable Key (AVK), Seman- • Attacks based on its homomorphism nature.
tic Security.
• Attacks due to a bad choice of parameters employed in RSA.
There are few attacks on RSA which do not require to
I. INTRODUCTION
factor the modulus. Such attacks are sometimes possible when
Standard RSA public key cryptosystem based on Singular the cipher texts and some additional information are known,
Cubic Curve has given different variants as investigated by for example,
different researchers. Three public key cryptosystem were • When some parts of the plain text is also known,
proposed by Koyama. These are called Koyama schemes. The • The encryption of the same or related plain text is
concept of Singular Cubic Curve in Koyama scheme is in- sent to different users (e.g. in a broadcast
troduced for the first time to deliver a public key application) or
cryptosystem. Singular Cubic Curve is a mathematical tool, • When the encryptions of two related plain texts are
which was first time used by Koyama for the construction of sent to the same user. This cryptosystem is most
public key cryptosystem (PKC). The Singular Cubic Curve commonly used for providing privacy and ensuring
is an important concept in number theory because of its wide authenticity of data.
range of applications. The property of Singular Cubic Curve The efficiency and security [8] are two key factors of
is helpful in cryptography as it forms an abelian group over any cryptosystem. The cryptosystem proposed by Koyama is
finite field. In these schemes two plain texts are are not semantically secure [3]. The cipher text should not leave
any useful information about the plain text in an ideal situation.
used to form a point on the Singular Cubic
The cryptosystem proposed by Koyama is two times faster than
Curve Curve over , and the cipher text is a point C = em that of standard RSA [7] scheme. But it is also not secure against
on the same Curve. Singular Cubic Curve over the finite field partially known plain text attack, linearly related plain text attack,
and the ring is used. Here n is the product of two distinct isomorphism attack, homomorphism attack and These attacks
odd primes greater than 3. are not possible in Singular Cubic Curve based RSA with AVK
A congruence equation of the form: technique. This scheme is claimed to be semantically secure
and also prevents those attacks which are prevalent in Koyama’s
where may produce a number of solution. The set scheme. The rest of this paper is organized as follows:
of all solutions to (1), is called the solution Section 2 presents the related work in this field. Sec-
space of the given singular cubic curve. tion 3 describes some known attacks and their proposed
© 2011 ACEEE 5
DOI: 01.IJIT.01.01.41](https://image.slidesharecdn.com/41-120913000425-phpapp02/85/Different-Attacks-on-Selective-Encryption-in-RSA-based-Singular-Cubic-Curve-with-AVK-and-Their-Possible-Solutions-1-320.jpg)
![ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011
solution in an elaborate manner proving the proposed scheme Step 1: Key Generation
as semantically secure. Section 4 further probes into other kinds
of attacks that may happen in such scheme. This section is ac-
tually paving the way for future research for finding solution of
other attacks. Countermeasures are suggested in Section 5. Ef-
ficiency and security analysis of the suggested solutions are the
points of concern in Section 6. Section 7 concludes the paper
with a future direction of work.
II. RELATED WORK
Singular Cubic Curve RSA with Automatic Variant Key
(AVK) scheme is based on selective encryption with automatic
time variant key (AVK). Selective encryption [9] provides a
number of advantages in secured communication process. In
the selective encryption, only a random part (r) of whole
message/plain text is encrypted. Let the selected text be
Figure.1: Selective RSA based Singular cubic curve
Using isomorphic mapping, following can be obtained
manipulation can be applied for any cryptographic exploi-
tation as as per valid matrix operation. To construct such a
scheme, random part of plain text is chosen for encryption/
decryption. By applying AVK in this selective text, the de-
sired result is achieved. The key has been automated by de-
sign to get changed in every session. The scheme proposed
is a generalization of the Koyama scheme. The paper con-
tributes towards the robustness by applying Time Variant
key [6] or Automatic Variant Key (AVK) . The concepts of
AVK [12][13] is illustrated in [17]. Following basic opera-
tions are used in the proposed PKC.
A. Proposed Generalization of Koyama Scheme-II using
AVK with application of Selective Encryption
The algorithm demands the implementation of the steps By application of Chinese Remainder Theorem [14] on
of key generation, encryption and decryption. The steps can following equations:
be elaborated as follows:
• Select a part from M.
© 2011 ACEEE 6
DOI: 01.IJIT.01.01.41](https://image.slidesharecdn.com/41-120913000425-phpapp02/85/Different-Attacks-on-Selective-Encryption-in-RSA-based-Singular-Cubic-Curve-with-AVK-and-Their-Possible-Solutions-2-320.jpg)
![ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011
Resolving
Encryption has been applied on selected part of the text
message M only and therefore named as selective encryption
technique. Decryption has been applied on selected part using
the similar algorithm by the receiver. Finally, the full text
document can be constructed by merging the decrypted
selected part of M namely
with the remaining part of the message.
Thus
Can be obtained.
III. SOME KNOWN ATTACKS AND THEIR
PROPOSED SOLUTIONS
A.. Partially known plain text attack and its solution:
obtained:
if ve 0 (mod n), and if is known, = c, d—” c =
M (original plain text) If attacker hacks this data, and wants
to decrypt the data due to AVK, the attacker can not find the
decrypted key applied over the text iteratively even if he/
she is successful in getting hold of immediate key that gets
changed in the next iteration, Hence this attack cannot be
successful.
B. Security against Linearly related plain text attack:
Koyama scheme is insecure if two linearly related plain texts
are encrypted with same public key. This attack is explained
as follows: Let M = and M’=
be two plain texts linearly related by the known relations:
By using the property of Singular elliptic curve over field
and Chinese Remainder Theorem, the following isomor-
phic property of Singular Cubic Curve over ring is shown
[15] as follows:
7
© 2011 ACEEE
DOI: 01.IJIT.01.01.41](https://image.slidesharecdn.com/41-120913000425-phpapp02/85/Different-Attacks-on-Selective-Encryption-in-RSA-based-Singular-Cubic-Curve-with-AVK-and-Their-Possible-Solutions-3-320.jpg)
![ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011
E. Common Modulo attack
then M2 can be easily obtained by computing the congru-
ence (4). It is not difficult to check whether congruence
(2) holds. Suppose, an attacker A wants to victimize B by
forging signature on a plain text M
F. Chosen Message attack
The attacker can not find the secret key of B in this com-
putation. Hence Isomorphic attack is not possible in case of
selective encryption with AVK.
D. Security Security against Homorphic attack
This attack originated from homomorphic propert
(k⊗[P + Q] = k⊗[P] ⊕ k⊗[Q]), Some known attacks of
this typ come under homomorphic attacks. Using
homomorphic property such as common modulus attack,
chosen message attack, garbage man-in-the-middle attack,
the Koyama schemes
So, attacker can not compute the signature as the function
AVK is not giving predictable output in a single step.
© 2011 ACEEE 8
DOI: 01.IJIT.01.01.41](https://image.slidesharecdn.com/41-120913000425-phpapp02/85/Different-Attacks-on-Selective-Encryption-in-RSA-based-Singular-Cubic-Curve-with-AVK-and-Their-Possible-Solutions-4-320.jpg)
![ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011
By this equation, it is clear that plain text can not be identified
as it is the functional output of AVK. Output is not confined
in a stepwise manner and very much dependent on data and,
therefore the attack is ineffective in the proposed algorithm
of proposed approach in this paper.
IV. SOME OTHER KNOWN ATTACKS
A. Wiener’s attack
To reduce decryption time, one may wish to use a small
value of d rather than a random d. Since modular
exponentiation takes time linear in log2d , a small d can
improve time reduction or increase in speed to a great extent.
Wiener[7] shows that a small d results in a total break of the
cryptosystem. General description of Wiener’s attack as
follows:
© 2011 ACEEE 9
DOI: 01.IJIT.01.01.41](https://image.slidesharecdn.com/41-120913000425-phpapp02/85/Different-Attacks-on-Selective-Encryption-in-RSA-based-Singular-Cubic-Curve-with-AVK-and-Their-Possible-Solutions-5-320.jpg)
![ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011
V. COUNTER MEASURES AND DISCUSSION ON known attacks are not possible due to the design of the algorithm.
THEORETICAL SIMULATION Application oriented simulation may be taken up to establish
the results beyond any doubt. A future direction of work in this
Randomized key concept is applied in AVK in plain text
area is to develop a generic algorithm which will be equally
after encryption and at the time before decryption by the
applicable to text based as well as image based documents.
receivers should not be chosen too small, since a small
REFERENCES
would give yet other small modular equations over the plain
text that can be combined with for [1] Seng Kiat Chua , Ka Hin Leung, San Ling, “Attack on RSA
typecryptosystem based on Singular Cubic Curve over
even more effective attacks. Since the degree of the equations cryptosystem based on Singular Cubic Curve over science, Z/
resulting from division polynomials is e2 it is suggested to nZ*Theoretical Computer science, Vol.220 19-27 (1999).
choose e at least 16 bits long. These propositions require, a [2] Sahadeo Padhye, “Partial Known Plaintext Attack on Koyama
careful analysis in a case to case basis depending upon the Scheme,” Information Processing Letters, Vol.96 No.3 pp. 96-
100 (2005).
Cipher Suite’s requirement. [3] Sahadeo Padhye and B. K. Sharma, “A Fast Semantically
Secure Publication Key Cryptosystem Based on Factoring,”
VI. EFFICIENCY AND SECURITY ANALYSIS International Journal of Network Security, Vol.3, No.2,
PP.144150, Sept, (2006), retrieved
In the scheme given by Koyama, eth power of under [4] Sahadeo Padhye, “Cryptanalysis of Koyama Scheme,”
modulo n is computed during the encryption process. In case International Journal of Network Security, Vol.2, No.1, pp.
of proposed extension of Koyama scheme as presented by Singh 73-80, (2006).
and Samaddar [17] algorithm, selective encryption has been [5] D. Blichenbacher, “On the security of KMOV public key
cryptosystem,” Crypto’97 LNCS Springer- Verlag Vol.1294,
applied, then AVK has been used for security purpose. This 235-348, (1997).
feature increases the efficiency of encryption. But, the decryption [6] Marc Joye and Jean- Jacques Quisquater, “Cryptanalysis of
is approximately of similar efficiency of the schemes given by RSA-Type Cryptosystem: A Visit,” American Mathematical
Koyama. Society, vol. 38, pp. 21- 31, (1998).
From Koyama’s analysis, let x and y the coordinates of 2 [7] Dan Boneh, “ Twenty Years of Attacks on the RSA
log n-bit plain text be transformed to a log n-bit plain text by Cryptosystem,” American Mathematical Society (AMS), Vol.
46, No. 2, pp. 203-213, (1999).
isomorphic mapping. This message of log n bit length is then
encrypted by using encryption process. The obtained cipher [8] Sahadeo Padhye, “A public key cryptosystem based on singular
cubic curve”, Eprint Archive-2005/109, http: //eprint.iacr.org/
text is decrypted by using decryption key over which is n 2005/109.pdf, 2002.
the transformed message. By using the inverse transformation, [9] Prof C T Bhunia, Gourchari Mondal and S Samaddar, “Theory
the origional 2 log n bit length message is obtained. But from and application of time variant key in RSA and that with
selective encryption in AES “, 2006.
the analysis of Koyama schemes, a number of attacks are
[10] D. Galindo, S. Mortin, J. L. Villar, “An efficient semantically
permissible like Partial known plain text, Linear related plain
secure elliptic curve cryptosystem based on KMOV scheme”,
text attack, Isomorphic attack, Homomorphic attack (Common Eprint Archiecve2002.1037/ http//erpint.iacr.org/2002/1037,
modulo attack, common modulo attack, Garbage Man-in-the - 2002.
middle attack ), Wiener’s attack and Hastad’s attack. This proves [11] C.T.Bhunia “Application of avk and selective encryption in
that Koyama scheme is not semantically secure enough and improving performance of quantum cryptography and networks,”
United Nations Educational Scientific and Cultural Organization
therefore require enhancement in terms of robustness that has and International Atomic Energy Agency, (2006), retrieved 10/
been pointed out here by the introduction of AVK. In case of 12/2009, from http://users.ictp.it/ pub off/preprints-sources/2006/
Selective encryption Singular Cubic Curve RSA with AVK, IC2006045P.pdf.
these attacks are not possible. This algorithm is also proved to [12] P. Chakrabarti, B Bhuyan, A.Chowdhuri C.T.Bhunia, “A novel
approach towards realizing optimum data transfer and
be semantically secure due to use of AVK concepts. The automatic variable key (AVK)” IJCSNS International Journal
proposed system has already been substantiated [17]. The results of Computer Science and Network Security, VOL.8 No.5, May
obtained in this paper on the basis of theoretical simulation and 2008.
analysis can be substantiated further by computational [13] Deepak Garg, Seema Verma, “Improvement over public key
cryptographic algorithm”, IEEE, International Advance
simulation. This has been taken up for application oriented study Computing Conference (IACC), 2009.
in future course of research. Due the page limitation the [14] Koyama K, “Fast RSA -type schemes based on Singular Cubic
concerned graph of simulation results could not be produced Curves +axy,” Proceeding in LNCS EUROCYPT 95,Volume
here that echo the theoretical implications obtained in this paper. - 921 , PP. 329-340.Springer Verlag (1995).
[15] Don Coppersmith, Matthew Franklin, Jacques Patarin,
VII. CONCLUSION AND FUTURE WORK TO BE Michael Reitert, “Low-Exponent RSA with Related
Messages,” Advances in Cryptology - EUROCRYPT ’96,
UNDERTAKEN LNCS 1070, pp. 1-9, (1996).
The robustness of the algorithm [17] over the extension of [16] Singh, Kalpana and Samaddar, Shefalika Ghosh “Selective
Encryption Technique in RSA based Singular Cubic Curve
Koyama Scheme with selective encryption in RSA Singular with AVK for Text Based Documents: Enhancement of Koyama
Cubic Curve with AVK is considered here. The algorithm Approach,” 2010 International Conference on Networking and
proposed by Singh and Samaddar [17] is investigated from Information Technology (ICNIT 2010) Manila, Philippines,
June 11 - 12, 2010.
different angles to come to the conclusion that a number of
© 2011 ACEEE 10
DOI: 01.IJIT.01.01.41](https://image.slidesharecdn.com/41-120913000425-phpapp02/85/Different-Attacks-on-Selective-Encryption-in-RSA-based-Singular-Cubic-Curve-with-AVK-and-Their-Possible-Solutions-6-320.jpg)
Different Attacks on Selective Encryption in RSA based Singular Cubic Curve with AVK and Their Possible Solutions
- 1. ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011 Different Attacks on Selective Encryption in RSA based Singular Cubic Curve with AVK and Their Possible Solutions Kalpana Singh1 and Shefalika Ghosh Samaddar2 Corresponding Author & M.Tech IV Semester Student1, Faculty Member2 Department of Computer Science and Engineering Motilal Nehru National Institute of Technology Allahabad, Uttar Pradesh -211004, India Email: Kalpana08.mnnit@gmail.com1, shefalika99@yahoo.com2 Abstract- In this paper, the security of Selective Encryptionin Later, Seng et al.[1] have shown that all three Koyama RSA based Singular Cubic Curve with Automatic Variable Key schemes are equivalent to each other by an proposed (AVK) for some well known attacks are analysed. It is proved isomorphism and becomes insecure in the case of some known that this cryptosystem is more secure than Koyama scheme attacks like partially known attack [2][3], linearly related plain from which the algorithm has been generated. The proposed text attack [4][5] isomorphism attack [4], homomorphism attack cryptographic algorithm makes justified use of Koyama [6], Wiener’s attack [7], and Hastad’s attack [7]. Some other Schemes. Koyama scheme is not semantically secure. The security notions are non-malleability [8] and Plain text- proposed Scheme is efficient and semantically secure public Awareness [8]. Non- malleability implies that any attacker key cryptosystem based on Singular Cubic Curve with AVK. Further, the partially known attacks, linearly related plain text cannot modify a cipher text while keeping any control over the attacks, isomorphism attacks, low exponent attacks, Wiener’s relation between the resulting plain text and original one. The attack and Hastad’s attack are analyzed for effect with the Plaintext-Awareness ensures that no one can produce a valid proposed scheme. The Selective Encryption in RSA based cipher text without knowing the corresponding Plain text. There Singular Cubic Curve with AVK for text based documents is are a number of mathematically induced attacks on RSA based found to be robust enough to encounter all these attacks. cryptosystems. These are classified into three categories: Keywords- Singular Cubic Curves, Koyama Public Key • Attacks exploiting the polynomial structure of RSA. Cryptosystem (PKC), Automatic Variable Key (AVK), Seman- • Attacks based on its homomorphism nature. tic Security. • Attacks due to a bad choice of parameters employed in RSA. There are few attacks on RSA which do not require to I. INTRODUCTION factor the modulus. Such attacks are sometimes possible when Standard RSA public key cryptosystem based on Singular the cipher texts and some additional information are known, Cubic Curve has given different variants as investigated by for example, different researchers. Three public key cryptosystem were • When some parts of the plain text is also known, proposed by Koyama. These are called Koyama schemes. The • The encryption of the same or related plain text is concept of Singular Cubic Curve in Koyama scheme is in- sent to different users (e.g. in a broadcast troduced for the first time to deliver a public key application) or cryptosystem. Singular Cubic Curve is a mathematical tool, • When the encryptions of two related plain texts are which was first time used by Koyama for the construction of sent to the same user. This cryptosystem is most public key cryptosystem (PKC). The Singular Cubic Curve commonly used for providing privacy and ensuring is an important concept in number theory because of its wide authenticity of data. range of applications. The property of Singular Cubic Curve The efficiency and security [8] are two key factors of is helpful in cryptography as it forms an abelian group over any cryptosystem. The cryptosystem proposed by Koyama is finite field. In these schemes two plain texts are are not semantically secure [3]. The cipher text should not leave any useful information about the plain text in an ideal situation. used to form a point on the Singular Cubic The cryptosystem proposed by Koyama is two times faster than Curve Curve over , and the cipher text is a point C = em that of standard RSA [7] scheme. But it is also not secure against on the same Curve. Singular Cubic Curve over the finite field partially known plain text attack, linearly related plain text attack, and the ring is used. Here n is the product of two distinct isomorphism attack, homomorphism attack and These attacks odd primes greater than 3. are not possible in Singular Cubic Curve based RSA with AVK A congruence equation of the form: technique. This scheme is claimed to be semantically secure and also prevents those attacks which are prevalent in Koyama’s where may produce a number of solution. The set scheme. The rest of this paper is organized as follows: of all solutions to (1), is called the solution Section 2 presents the related work in this field. Sec- space of the given singular cubic curve. tion 3 describes some known attacks and their proposed © 2011 ACEEE 5 DOI: 01.IJIT.01.01.41
- 2. ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011 solution in an elaborate manner proving the proposed scheme Step 1: Key Generation as semantically secure. Section 4 further probes into other kinds of attacks that may happen in such scheme. This section is ac- tually paving the way for future research for finding solution of other attacks. Countermeasures are suggested in Section 5. Ef- ficiency and security analysis of the suggested solutions are the points of concern in Section 6. Section 7 concludes the paper with a future direction of work. II. RELATED WORK Singular Cubic Curve RSA with Automatic Variant Key (AVK) scheme is based on selective encryption with automatic time variant key (AVK). Selective encryption [9] provides a number of advantages in secured communication process. In the selective encryption, only a random part (r) of whole message/plain text is encrypted. Let the selected text be Figure.1: Selective RSA based Singular cubic curve Using isomorphic mapping, following can be obtained manipulation can be applied for any cryptographic exploi- tation as as per valid matrix operation. To construct such a scheme, random part of plain text is chosen for encryption/ decryption. By applying AVK in this selective text, the de- sired result is achieved. The key has been automated by de- sign to get changed in every session. The scheme proposed is a generalization of the Koyama scheme. The paper con- tributes towards the robustness by applying Time Variant key [6] or Automatic Variant Key (AVK) . The concepts of AVK [12][13] is illustrated in [17]. Following basic opera- tions are used in the proposed PKC. A. Proposed Generalization of Koyama Scheme-II using AVK with application of Selective Encryption The algorithm demands the implementation of the steps By application of Chinese Remainder Theorem [14] on of key generation, encryption and decryption. The steps can following equations: be elaborated as follows: • Select a part from M. © 2011 ACEEE 6 DOI: 01.IJIT.01.01.41
- 3. ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011 Resolving Encryption has been applied on selected part of the text message M only and therefore named as selective encryption technique. Decryption has been applied on selected part using the similar algorithm by the receiver. Finally, the full text document can be constructed by merging the decrypted selected part of M namely with the remaining part of the message. Thus Can be obtained. III. SOME KNOWN ATTACKS AND THEIR PROPOSED SOLUTIONS A.. Partially known plain text attack and its solution: obtained: if ve 0 (mod n), and if is known, = c, d—” c = M (original plain text) If attacker hacks this data, and wants to decrypt the data due to AVK, the attacker can not find the decrypted key applied over the text iteratively even if he/ she is successful in getting hold of immediate key that gets changed in the next iteration, Hence this attack cannot be successful. B. Security against Linearly related plain text attack: Koyama scheme is insecure if two linearly related plain texts are encrypted with same public key. This attack is explained as follows: Let M = and M’= be two plain texts linearly related by the known relations: By using the property of Singular elliptic curve over field and Chinese Remainder Theorem, the following isomor- phic property of Singular Cubic Curve over ring is shown [15] as follows: 7 © 2011 ACEEE DOI: 01.IJIT.01.01.41
- 4. ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011 E. Common Modulo attack then M2 can be easily obtained by computing the congru- ence (4). It is not difficult to check whether congruence (2) holds. Suppose, an attacker A wants to victimize B by forging signature on a plain text M F. Chosen Message attack The attacker can not find the secret key of B in this com- putation. Hence Isomorphic attack is not possible in case of selective encryption with AVK. D. Security Security against Homorphic attack This attack originated from homomorphic propert (k⊗[P + Q] = k⊗[P] ⊕ k⊗[Q]), Some known attacks of this typ come under homomorphic attacks. Using homomorphic property such as common modulus attack, chosen message attack, garbage man-in-the-middle attack, the Koyama schemes So, attacker can not compute the signature as the function AVK is not giving predictable output in a single step. © 2011 ACEEE 8 DOI: 01.IJIT.01.01.41
- 5. ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011 By this equation, it is clear that plain text can not be identified as it is the functional output of AVK. Output is not confined in a stepwise manner and very much dependent on data and, therefore the attack is ineffective in the proposed algorithm of proposed approach in this paper. IV. SOME OTHER KNOWN ATTACKS A. Wiener’s attack To reduce decryption time, one may wish to use a small value of d rather than a random d. Since modular exponentiation takes time linear in log2d , a small d can improve time reduction or increase in speed to a great extent. Wiener[7] shows that a small d results in a total break of the cryptosystem. General description of Wiener’s attack as follows: © 2011 ACEEE 9 DOI: 01.IJIT.01.01.41
- 6. ACEEE Int. J. on Information Technology, Vol. 01, No. 01, Mar 2011 V. COUNTER MEASURES AND DISCUSSION ON known attacks are not possible due to the design of the algorithm. THEORETICAL SIMULATION Application oriented simulation may be taken up to establish the results beyond any doubt. A future direction of work in this Randomized key concept is applied in AVK in plain text area is to develop a generic algorithm which will be equally after encryption and at the time before decryption by the applicable to text based as well as image based documents. receivers should not be chosen too small, since a small REFERENCES would give yet other small modular equations over the plain text that can be combined with for [1] Seng Kiat Chua , Ka Hin Leung, San Ling, “Attack on RSA typecryptosystem based on Singular Cubic Curve over even more effective attacks. Since the degree of the equations cryptosystem based on Singular Cubic Curve over science, Z/ resulting from division polynomials is e2 it is suggested to nZ*Theoretical Computer science, Vol.220 19-27 (1999). choose e at least 16 bits long. These propositions require, a [2] Sahadeo Padhye, “Partial Known Plaintext Attack on Koyama careful analysis in a case to case basis depending upon the Scheme,” Information Processing Letters, Vol.96 No.3 pp. 96- 100 (2005). Cipher Suite’s requirement. [3] Sahadeo Padhye and B. K. Sharma, “A Fast Semantically Secure Publication Key Cryptosystem Based on Factoring,” VI. EFFICIENCY AND SECURITY ANALYSIS International Journal of Network Security, Vol.3, No.2, PP.144150, Sept, (2006), retrieved In the scheme given by Koyama, eth power of under [4] Sahadeo Padhye, “Cryptanalysis of Koyama Scheme,” modulo n is computed during the encryption process. In case International Journal of Network Security, Vol.2, No.1, pp. of proposed extension of Koyama scheme as presented by Singh 73-80, (2006). and Samaddar [17] algorithm, selective encryption has been [5] D. Blichenbacher, “On the security of KMOV public key cryptosystem,” Crypto’97 LNCS Springer- Verlag Vol.1294, applied, then AVK has been used for security purpose. This 235-348, (1997). feature increases the efficiency of encryption. But, the decryption [6] Marc Joye and Jean- Jacques Quisquater, “Cryptanalysis of is approximately of similar efficiency of the schemes given by RSA-Type Cryptosystem: A Visit,” American Mathematical Koyama. Society, vol. 38, pp. 21- 31, (1998). From Koyama’s analysis, let x and y the coordinates of 2 [7] Dan Boneh, “ Twenty Years of Attacks on the RSA log n-bit plain text be transformed to a log n-bit plain text by Cryptosystem,” American Mathematical Society (AMS), Vol. 46, No. 2, pp. 203-213, (1999). isomorphic mapping. This message of log n bit length is then encrypted by using encryption process. The obtained cipher [8] Sahadeo Padhye, “A public key cryptosystem based on singular cubic curve”, Eprint Archive-2005/109, http: //eprint.iacr.org/ text is decrypted by using decryption key over which is n 2005/109.pdf, 2002. the transformed message. By using the inverse transformation, [9] Prof C T Bhunia, Gourchari Mondal and S Samaddar, “Theory the origional 2 log n bit length message is obtained. But from and application of time variant key in RSA and that with selective encryption in AES “, 2006. the analysis of Koyama schemes, a number of attacks are [10] D. Galindo, S. Mortin, J. L. Villar, “An efficient semantically permissible like Partial known plain text, Linear related plain secure elliptic curve cryptosystem based on KMOV scheme”, text attack, Isomorphic attack, Homomorphic attack (Common Eprint Archiecve2002.1037/ http//erpint.iacr.org/2002/1037, modulo attack, common modulo attack, Garbage Man-in-the - 2002. middle attack ), Wiener’s attack and Hastad’s attack. This proves [11] C.T.Bhunia “Application of avk and selective encryption in that Koyama scheme is not semantically secure enough and improving performance of quantum cryptography and networks,” United Nations Educational Scientific and Cultural Organization therefore require enhancement in terms of robustness that has and International Atomic Energy Agency, (2006), retrieved 10/ been pointed out here by the introduction of AVK. In case of 12/2009, from http://users.ictp.it/ pub off/preprints-sources/2006/ Selective encryption Singular Cubic Curve RSA with AVK, IC2006045P.pdf. these attacks are not possible. This algorithm is also proved to [12] P. Chakrabarti, B Bhuyan, A.Chowdhuri C.T.Bhunia, “A novel approach towards realizing optimum data transfer and be semantically secure due to use of AVK concepts. The automatic variable key (AVK)” IJCSNS International Journal proposed system has already been substantiated [17]. The results of Computer Science and Network Security, VOL.8 No.5, May obtained in this paper on the basis of theoretical simulation and 2008. analysis can be substantiated further by computational [13] Deepak Garg, Seema Verma, “Improvement over public key cryptographic algorithm”, IEEE, International Advance simulation. This has been taken up for application oriented study Computing Conference (IACC), 2009. in future course of research. Due the page limitation the [14] Koyama K, “Fast RSA -type schemes based on Singular Cubic concerned graph of simulation results could not be produced Curves +axy,” Proceeding in LNCS EUROCYPT 95,Volume here that echo the theoretical implications obtained in this paper. - 921 , PP. 329-340.Springer Verlag (1995). [15] Don Coppersmith, Matthew Franklin, Jacques Patarin, VII. CONCLUSION AND FUTURE WORK TO BE Michael Reitert, “Low-Exponent RSA with Related Messages,” Advances in Cryptology - EUROCRYPT ’96, UNDERTAKEN LNCS 1070, pp. 1-9, (1996). The robustness of the algorithm [17] over the extension of [16] Singh, Kalpana and Samaddar, Shefalika Ghosh “Selective Encryption Technique in RSA based Singular Cubic Curve Koyama Scheme with selective encryption in RSA Singular with AVK for Text Based Documents: Enhancement of Koyama Cubic Curve with AVK is considered here. The algorithm Approach,” 2010 International Conference on Networking and proposed by Singh and Samaddar [17] is investigated from Information Technology (ICNIT 2010) Manila, Philippines, June 11 - 12, 2010. different angles to come to the conclusion that a number of © 2011 ACEEE 10 DOI: 01.IJIT.01.01.41