A Public-Key Cryptosystem Based On Discrete Logarithm Problem over Finite Fields 퐅퐩 퐧
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This document presents a public-key cryptosystem based on the discrete logarithm problem over finite fields. The security of the system relies on the difficulty of finding discrete logarithms in the function field Fp^n. The system uses a primitive polynomial f(x) of degree n over the finite field Fp as the base polynomial. The public key consists of f(x) and another primitive polynomial fk(x) with a root that is the kth power of the root of f(x), where k is the secret key. Encryption involves multiplying the message polynomial by a term related to fk(x) or f(x) depending on the secret key. Decryption uses the secret key to recover the message
![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 01-03
www.iosrjournals.org
DOI: 10.9790/5728-11130103 www.iosrjournals.org 1 |Page
A Public-Key Cryptosystem Based On Discrete Logarithm
Problem over Finite Fields 𝐅𝐩 𝐧
Saju M I1
, Lilly P L2
1Assistant Professor, Department of Mathematics, St. Thomas’ College, Thrissur, India
2Associate professor, Department of Mathematics, St. Joseph’s College, Irinjalakuda, India
Abstract: One of the classical problems in mathematics is the Discrete Logarithm Problem (DLP).The
difficulty and complexity for solving DLP is used in most of the cryptosystems. In this paper we design a public
key system based on the ring of polynomials over the field 𝐹𝑝 is developed. The security of the system is based
on the difficulty of finding discrete logarithms over the function field 𝐹 𝑝 𝑛 with suitable prime p and sufficiently
large n. The presented system has all features of ordinary public key cryptosystem.
Keywords: Discrete logarithm problem, Function Field, Polynomials over finite fields, Primitive polynomial,
Public key cryptosystem.
I. Introduction
For the construction of a public key cryptosystem, we need a finite extension field Fpn overFp. In our
paper [1] we design a public key cryptosystem based on discrete logarithm problem over the field F2. Here, for
increasing the complexity and difficulty for solving DLP, we made a proper additional modification in the
system. A cryptosystem for message transmission means a map from units of ordinary text called plaintext
message units to units of coded text called cipher text message units.
The face of cryptography was radically altered when Diffie and Hellman invented an entirely new type
of cryptography, called public key [Diffie and Hellman 1976][2]. At the heart of this concept is the idea of
using a one-way function for encryption. The most common purposes for which public key cryptography has
been applied are confidential message transmission, authentication, key exchange, coin flip, secret sharing and
zero knowledge proof. There are public key cryptosystems and digital signature systems based on the discrete
logarithm problem (DLP) such as Digital Signature Standard (DSS) [3], ElGamal cryptosystem and Diffie-
Hellman key exchange system. The security of the new cryptosystem is based on DLP [4][5]. The main feature
of the new system is that its public key encryption is computationally equivalent to EIGamal public key
encryption.
II. Public Key Cryptosystem
In this system we take a finite field
Fp x
f x
, where f(x) is a primitive polynomial of degree n will be
considered as the base polynomial of the system [6] [7]. Let α be a root of f(x), k be any random number less
than pn
− 1where k, pn
− 1 = 1and let fk(x) be a primitive polynomial with the root αk
. Let k be the secret
parameter of the system and polynomials f(x) and fk(x) be public polynomials of the system. Using the
algorithm [8] we can compute αk
, then we can express αk
as a polynomial g α . However for a given g α to
find k where αk
= g α is a DLP.
2.1 Encryption For a randomly generated N with n bits we have
xN
≡ T x mod f(x) (1)
and
xN
≡ Tk x modfk(x). (2)
It is easy to show that
Tk x ≡ (T(xk−1
))k
mod fk(x). (3)
or
T(x) ≡ (Tk(xk
))k−1
modf(x) (4)
Where k−1
k ≡ 1 mod (pn
− 1)
Suppose we want to encrypt the message M. We can express the message M as a polynomial M(x) of
degree n overFp. The encryption process is the following:](https://image.slidesharecdn.com/a011130103-151123093248-lva1-app6892/85/A-Public-Key-Cryptosystem-Based-On-Discrete-Logarithm-Problem-over-Finite-Fields-1-320.jpg)
![A public-key cryptosystem based on discrete logarithm problem over finite fields 𝐹 𝑝 𝑛
DOI: 10.9790/5728-11130103 www.iosrjournals.org 3 |Page
III. Security Of The System
The security of the system is based on the discrete logarithm problem (DLP) over the function
field Fpn . Assuming that α is the root of the base primitive polynomial f(x) and αk
is the root of the primitive
polynomialfk(x). For a given αk
it is quite easy to construct its minimal polynomial fk(x) [8]. For a polynomial
fk(x) its root as a polynomial g(α) can be found using the algorithm presented in [8]. The complexity of the
algorithm is not more than O (t3
). However for a given g(α) to find αk
= g α is a DLP. The decryption process
is difficult when we work in the field of size with prime extension to be equal at least to 2048..
IV. Implementation Aspects Of The System
An encryption operation of this system according to the formulae (5) or (6) is just multiplication. The
encryption or decryption process of this system has the same complexity as for the ElGamal type encryption or
decryption. When comparing decryption operations we can conclude that the system presented here has about
the same complexity compared with both RSA and ElGamal type decryption since both require one regular
exponentiation.
V. Conclusion
In this paper a new public key system which is based on DLP is developed. The complexity of this
system is based on the selection of the Function Field Fpn . In this system the selection of the prime number p
and the exponent n has an important role. All public key operations of the presented system can be implemented
virtually with the same complexity compared with existing systems. The concept used in this system may
useful in the construction of digital signature and the process of hashing.
References
[1]. Lilly P.L, Saju M.I., A method of designing a public-key cryptosystem based on discrete logarithm problem, IRJPA-4(11), 2014,
628-630.
[2]. Diffie W., Helman M.E., New Directions in Cryptography, IEEE Transactions on information theory, Vol. IT-22, Nov.1976, 644-
654.
[3]. Digital Signature Standard, Federal Information Processing Standards Publication 186, May 1994.
[4]. Odlyzko A., Discrete logarithms: The past and the Future; Designs, Codes and Cryptography, (2000), 129-145.
[5]. McCurley K., The discrete logarithm problem, Proceedings of Symposia in Applies Mathematica, Vol.42, 1990, 49-74.
[6]. Lidl, Niederreiter (1997), Finite Fields (2nd ed.), Cambridge University, Press.
[7]. Neal Koblitz, Algebraic Aspects of Cryptography, Springer.
[8]. Taher ElGamal, A public-key cryptosystem and a signature scheme based on discrete logarithms, IEEE, Transactions on
Information Theory, Vol. IT-31, n.4, 1985, 469-472, also in CRYPTO 84, 10-18, Springer-Verlag.](https://image.slidesharecdn.com/a011130103-151123093248-lva1-app6892/85/A-Public-Key-Cryptosystem-Based-On-Discrete-Logarithm-Problem-over-Finite-Fields-3-320.jpg)
A Public-Key Cryptosystem Based On Discrete Logarithm Problem over Finite Fields 퐅퐩 퐧
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 01-03 www.iosrjournals.org DOI: 10.9790/5728-11130103 www.iosrjournals.org 1 |Page A Public-Key Cryptosystem Based On Discrete Logarithm Problem over Finite Fields 𝐅𝐩 𝐧 Saju M I1 , Lilly P L2 1Assistant Professor, Department of Mathematics, St. Thomas’ College, Thrissur, India 2Associate professor, Department of Mathematics, St. Joseph’s College, Irinjalakuda, India Abstract: One of the classical problems in mathematics is the Discrete Logarithm Problem (DLP).The difficulty and complexity for solving DLP is used in most of the cryptosystems. In this paper we design a public key system based on the ring of polynomials over the field 𝐹𝑝 is developed. The security of the system is based on the difficulty of finding discrete logarithms over the function field 𝐹 𝑝 𝑛 with suitable prime p and sufficiently large n. The presented system has all features of ordinary public key cryptosystem. Keywords: Discrete logarithm problem, Function Field, Polynomials over finite fields, Primitive polynomial, Public key cryptosystem. I. Introduction For the construction of a public key cryptosystem, we need a finite extension field Fpn overFp. In our paper [1] we design a public key cryptosystem based on discrete logarithm problem over the field F2. Here, for increasing the complexity and difficulty for solving DLP, we made a proper additional modification in the system. A cryptosystem for message transmission means a map from units of ordinary text called plaintext message units to units of coded text called cipher text message units. The face of cryptography was radically altered when Diffie and Hellman invented an entirely new type of cryptography, called public key [Diffie and Hellman 1976][2]. At the heart of this concept is the idea of using a one-way function for encryption. The most common purposes for which public key cryptography has been applied are confidential message transmission, authentication, key exchange, coin flip, secret sharing and zero knowledge proof. There are public key cryptosystems and digital signature systems based on the discrete logarithm problem (DLP) such as Digital Signature Standard (DSS) [3], ElGamal cryptosystem and Diffie- Hellman key exchange system. The security of the new cryptosystem is based on DLP [4][5]. The main feature of the new system is that its public key encryption is computationally equivalent to EIGamal public key encryption. II. Public Key Cryptosystem In this system we take a finite field Fp x f x , where f(x) is a primitive polynomial of degree n will be considered as the base polynomial of the system [6] [7]. Let α be a root of f(x), k be any random number less than pn − 1where k, pn − 1 = 1and let fk(x) be a primitive polynomial with the root αk . Let k be the secret parameter of the system and polynomials f(x) and fk(x) be public polynomials of the system. Using the algorithm [8] we can compute αk , then we can express αk as a polynomial g α . However for a given g α to find k where αk = g α is a DLP. 2.1 Encryption For a randomly generated N with n bits we have xN ≡ T x mod f(x) (1) and xN ≡ Tk x modfk(x). (2) It is easy to show that Tk x ≡ (T(xk−1 ))k mod fk(x). (3) or T(x) ≡ (Tk(xk ))k−1 modf(x) (4) Where k−1 k ≡ 1 mod (pn − 1) Suppose we want to encrypt the message M. We can express the message M as a polynomial M(x) of degree n overFp. The encryption process is the following:
- 2. A public-key cryptosystem based on discrete logarithm problem over finite fields 𝐹 𝑝 𝑛 DOI: 10.9790/5728-11130103 www.iosrjournals.org 2 |Page {M. T x −1 , Tk xk } (5) or {M. Tk x −1 , T xk−1 } (6) and the encrypted message is a pair as represented in (5) or (6). 2.2 Decryption Using the secret key k, compute either (Tk(xk ))k−1 or (T(xk−1 ))k and can get M by multiplying the respective element with the first part of the encrypted message. 1. Example Here p=2 and we take the finite field F2 x f x , where f x = x3 + x2 + 1 be the base polynomial of the system and we will denote by a root of f(x). Let k = 3 and let f3 x = x3 + x + 1 be the primitive polynomial with the root α3 . Let k = 3 be the secret parameter of the system and polynomials f x and f3 x be public polynomials of the system. Take N = (101)2 = 5, we have, x5 ≡ (x + 1)mod(x3 + x2 + 1) and x5 ≡ (x2 + x + 1)mod(x3 + x + 1). Here, T x = x + 1 and T3 x = x2 + x + 1. Then, x2 + x + 1 ≡ (x5 + 1)3 mod(x3 + x + 1) or x + 1 ≡ (x6 + x3 + 1)5 mod(x3 + x2 + 1), where 3−1 = 5 (mod7). Also, T x −1 = x2 or T3 x −1 = x2 + 1. Let the message M that needs to be encrypted be represented as a polynomial M x = x3 + x + 1. Then compute, {M(x). T x −1 , T3 x3 }={x5 + x3 + x2 , x6 + x3 + 1} (7) Or {M(x). T3 x −1 , T x5 }={x5 + x2 + x + 1, x5 + 1} (8) The encrypted message is a pair as represented in (7) or (8). Decryption is based on the fact that only the owner of the system knows the secret number 3 or 5 and having T3 x = x2 + x + 1 or T x = x + 1 he can calculate either (T3 x3 )5 or (T x5 )3 and get M by multiplying the respective results with the first part of the encrypted message. 1. Example Here p=2 and we take the finite field F2 x f x , where f x = x8 + x6 + 1 be the base polynomial of the system and we will denote by a root of f(x). Let k = 4 and let f4 x = x4 + x3 + 1 be the primitive polynomial with the root α4 . Let k = 4 be the secret parameter of the system and polynomials f x and f4 x be public polynomials of the system. Take N = (11111)2 = 31, we have, x31 ≡ x mod(x8 + x6 + 1) and x31 ≡ x mod(x4 + x3 + 1). Here, T x = x and T4 x = x. Then, x ≡ (x64 )4 mod(x4 + x3 + 1) Or x ≡ (x4 )64 mod(x8 + x6 + 1) , where 4−1 = 64 (mod255). Also, T x −1 = x7 + x5 or T4 x −1 = x7 + x5 . Let the message M that needs to be encrypted be represented as a polynomial M x = x8 + x7 + x6 + x5 + 1. Then compute, {M(x). T x −1 , T4 x4 }= (x8 + x7 + x6 + x5 + 1 (x7 + x5 ), x4 } (9) Or {M(x). T4 x −1 , T x64 }={ (x8 + x7 + x6 + x5 + 1 (x7 + x5 ), x64 } (10) The encrypted message is a pair as represented in (9) or (10). Decryption is based on the fact that only the owner of the system knows the secret number 4 or 64 and having T4 x = x or T x = x he can calculate either (T4 x4 )64 or (T x64 )4 and get M by multiplying the respective results with the first part of the encrypted message.
- 3. A public-key cryptosystem based on discrete logarithm problem over finite fields 𝐹 𝑝 𝑛 DOI: 10.9790/5728-11130103 www.iosrjournals.org 3 |Page III. Security Of The System The security of the system is based on the discrete logarithm problem (DLP) over the function field Fpn . Assuming that α is the root of the base primitive polynomial f(x) and αk is the root of the primitive polynomialfk(x). For a given αk it is quite easy to construct its minimal polynomial fk(x) [8]. For a polynomial fk(x) its root as a polynomial g(α) can be found using the algorithm presented in [8]. The complexity of the algorithm is not more than O (t3 ). However for a given g(α) to find αk = g α is a DLP. The decryption process is difficult when we work in the field of size with prime extension to be equal at least to 2048.. IV. Implementation Aspects Of The System An encryption operation of this system according to the formulae (5) or (6) is just multiplication. The encryption or decryption process of this system has the same complexity as for the ElGamal type encryption or decryption. When comparing decryption operations we can conclude that the system presented here has about the same complexity compared with both RSA and ElGamal type decryption since both require one regular exponentiation. V. Conclusion In this paper a new public key system which is based on DLP is developed. The complexity of this system is based on the selection of the Function Field Fpn . In this system the selection of the prime number p and the exponent n has an important role. All public key operations of the presented system can be implemented virtually with the same complexity compared with existing systems. The concept used in this system may useful in the construction of digital signature and the process of hashing. References [1]. Lilly P.L, Saju M.I., A method of designing a public-key cryptosystem based on discrete logarithm problem, IRJPA-4(11), 2014, 628-630. [2]. Diffie W., Helman M.E., New Directions in Cryptography, IEEE Transactions on information theory, Vol. IT-22, Nov.1976, 644- 654. [3]. Digital Signature Standard, Federal Information Processing Standards Publication 186, May 1994. [4]. Odlyzko A., Discrete logarithms: The past and the Future; Designs, Codes and Cryptography, (2000), 129-145. [5]. McCurley K., The discrete logarithm problem, Proceedings of Symposia in Applies Mathematica, Vol.42, 1990, 49-74. [6]. Lidl, Niederreiter (1997), Finite Fields (2nd ed.), Cambridge University, Press. [7]. Neal Koblitz, Algebraic Aspects of Cryptography, Springer. [8]. Taher ElGamal, A public-key cryptosystem and a signature scheme based on discrete logarithms, IEEE, Transactions on Information Theory, Vol. IT-31, n.4, 1985, 469-472, also in CRYPTO 84, 10-18, Springer-Verlag.