In this paper, a variant of the Hill cipher is proposed. In the classical Hill cipher, an inverti... more In this paper, a variant of the Hill cipher is proposed. In the classical Hill cipher, an invertible matrix is used for encryption but the scheme is vulnerable to the known-plaintext attack which can reveal the matrix. In our proposed cryptosystem, each plaintext block is encrypted by a new invertible key matrix that thwarts the known-plaintext attack. To generate the invertible matrices which serve as the dynamic keys we make use of the vector spaces, randomly generated basis and non-singular linear transformation. Resulting cipher is secure against the known-plaintext attack.
In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First... more In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First one is Elliptic ElGamal type group ring public key cryptosystem whose security is greater than security of cryptosystems based on elliptic curves discrete logarithmic problem (ECDLP). Second is ElGamal type group ring public key cryptosystem, which is analogous to ElGamal public key cryptosystem but has comparatively greater security. Examples are also given for both the proposed cryptosystems. Mathematics Subject Classification (2010): 94A60, 20C05, 20C07
In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimate... more In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimates for determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces. For linear inverse problems, we obtain the convergence rates without the use of the classical concept of spectral theory and for non-linear inverse problems, we obtain the results by incorporating both Holder stability estimates as well as the inhomogeneous variational inequality. We also consider Lavrentiev's regularization method for non-linear inverse problems in Hilbert spaces and find the convergence rates of its regularized solutions by incorporating Holder stability estimates as well as inhomogeneous variational inequality. The co-action between the variational inequalities and the Holder stability estimates is also discussed for both linear and non-linear inverse problems.
In this paper, we apply a new kind of smoothness concept, i.e. H\"older stability estimates ... more In this paper, we apply a new kind of smoothness concept, i.e. H\"older stability estimates for the determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces. For linear inverse problems, we obtain the convergence rates without incorporating the classical concept of spectral theory and for non-linear inverse problems, we obtain the convergence rates without incorporating any additional non-linearity estimate. Further, we employ the smoothness concept of inhomogeneous variational inequalities to deduce the convergence rates for non-linear inverse problems. In addition to Tikhonov regularization, we also consider Lavrentiev's regularization method for non-linear inverse problems and determine its convergence rates by incorporating the H\"older stability estimates as well as inhomogeneous variational inequalities. And finally, we discuss the co-action between the variational inequalities and the H\"olde...
In this paper, we analyze the convergence rates of well known Tikhonov regularization scheme for ... more In this paper, we analyze the convergence rates of well known Tikhonov regularization scheme for solving the nonlinear ill-posed problems by using the Holder stability estimates. In particular, we obtain the convergence rates via two different approaches. The first approach is the standard one via Bregman distances and the second one is in terms of weaker norms. The important aspect in the second approach is that the regularization is only used to constrain the regularized solutions to a set where stability holds. Moreover, a novel smoothness condition is introduced termed as approximate Holder estimates which is further exploited to obtain the convergence rates. At the end, we also discuss the convergence rates for Tikhonov regularization with sparsity constraints by incorporating the Holder stability estimates.
In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems ... more In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems in Banach spaces and study its convergence analysis. The method is based on the well known iteratively regularized Landweber iteration method together with an extrapolation strategy. The general formulation of iteratively regularized Landweber iteration method in Banach spaces excludes the use of certain functions such as total variation like penalty functionals, L 1 functions etc. The novel scheme presented in this paper allows to use such non-smooth penalty terms that can be helpful in practical applications involving the reconstruction of several important features of solutions such as piecewise constancy and sparsity. We carefully discuss the choices for important parameters, such as combination parameters and step sizes involved in the design of the method. Additionally, we discuss an example to validate our assumptions.
International Journal of Mathematical Sciences and Computing, 2021
In this paper, we propose a novel variant of the Hill cipher based on vector spaces. In the class... more In this paper, we propose a novel variant of the Hill cipher based on vector spaces. In the classical Hill cipher, a non-singular matrix is used for encryption but it is well known that this cipher is vulnerable to the known-plaintext attack. In our proposed cryptosystem, we eradicate this problem by encrypting each plaintext block with a new invertible key matrix. This makes our scheme immune to all existing attacks in literature on this type of ciphers and so the resulting cipher can be used as other state-of-art block cipher. To generate the invertible matrices which serve as the dynamic keys, we make use of the vector spaces along with randomly generated basis and non-singular linear transformation. In addition to this, we also study the computational complexity of the proposed cryptosystem and compare this with the computational complexities of other schemes based on Hill cipher.
In this paper, the local convergence of Iteratively regularized Landweber iteration method is inv... more In this paper, the local convergence of Iteratively regularized Landweber iteration method is investigated for solving non-linear inverse problems in Banach spaces. Our analysis mainly relies on the assumption that the inverse mapping satisfies the Hölder stability estimate locally. We consider both noisy as well as non-noisy data in our analysis. Under the a-priori choice of stopping index for noisy data, we show that the iterates remain in a certain ball around exact solution and obtain the convergence rates. The convergence of the Iteratively regularized Landweber iterates to the exact solution is shown under certain assumptions in the case of non-noisy data and as a by-product, under different conditions, two different convergence rates are obtained.
In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First... more In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First one is Elliptic ElGamal type group ring public key cryptosystem whose security is greater than security of cryptosystems based on elliptic curves discrete logarithmic problem (ECDLP). Second is ElGamal type group ring public key cryptosystem, which is analogous to ElGamal public key cryptosystem but has comparatively greater security. Examples are also given for both the proposed cryptosystems.
Journal of Discrete Mathematical Sciences and Cryptography, 2021
In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First... more In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First one is Elliptic ElGamal type group ring public key cryptosystem whose security is greater than security of cryptosystems based on elliptic curves discrete logarithmic problem (ECDLP). Second is ElGamal type group ring public key cryptosystem, which is analogous to ElGamal public key cryptosystem but has comparatively greater security. Examples are also given for both the proposed cryptosystems.
In this paper we establish compactness results of multiscale and very weak multiscale type for se... more In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in L 2 (0, T ; H 1 0 (Ω)), fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation ε p ∂ t uε(x, t) − ∇ • (a(xε −1 , xε −2 , tε −q , tε −r)∇uε(x, t)) = f (x, t), where 0 < p < q < r. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by p, compared to the standard matching that gives rise to local parabolic problems.
In this paper, a variant of the Hill cipher is proposed. In the classical Hill cipher, an inverti... more In this paper, a variant of the Hill cipher is proposed. In the classical Hill cipher, an invertible matrix is used for encryption but the scheme is vulnerable to the known-plaintext attack which can reveal the matrix. In our proposed cryptosystem, each plaintext block is encrypted by a new invertible key matrix that thwarts the known-plaintext attack. To generate the invertible matrices which serve as the dynamic keys we make use of the vector spaces, randomly generated basis and non-singular linear transformation. Resulting cipher is secure against the known-plaintext attack.
In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First... more In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First one is Elliptic ElGamal type group ring public key cryptosystem whose security is greater than security of cryptosystems based on elliptic curves discrete logarithmic problem (ECDLP). Second is ElGamal type group ring public key cryptosystem, which is analogous to ElGamal public key cryptosystem but has comparatively greater security. Examples are also given for both the proposed cryptosystems. Mathematics Subject Classification (2010): 94A60, 20C05, 20C07
In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimate... more In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimates for determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces. For linear inverse problems, we obtain the convergence rates without the use of the classical concept of spectral theory and for non-linear inverse problems, we obtain the results by incorporating both Holder stability estimates as well as the inhomogeneous variational inequality. We also consider Lavrentiev's regularization method for non-linear inverse problems in Hilbert spaces and find the convergence rates of its regularized solutions by incorporating Holder stability estimates as well as inhomogeneous variational inequality. The co-action between the variational inequalities and the Holder stability estimates is also discussed for both linear and non-linear inverse problems.
In this paper, we apply a new kind of smoothness concept, i.e. H\"older stability estimates ... more In this paper, we apply a new kind of smoothness concept, i.e. H\"older stability estimates for the determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces. For linear inverse problems, we obtain the convergence rates without incorporating the classical concept of spectral theory and for non-linear inverse problems, we obtain the convergence rates without incorporating any additional non-linearity estimate. Further, we employ the smoothness concept of inhomogeneous variational inequalities to deduce the convergence rates for non-linear inverse problems. In addition to Tikhonov regularization, we also consider Lavrentiev's regularization method for non-linear inverse problems and determine its convergence rates by incorporating the H\"older stability estimates as well as inhomogeneous variational inequalities. And finally, we discuss the co-action between the variational inequalities and the H\"olde...
In this paper, we analyze the convergence rates of well known Tikhonov regularization scheme for ... more In this paper, we analyze the convergence rates of well known Tikhonov regularization scheme for solving the nonlinear ill-posed problems by using the Holder stability estimates. In particular, we obtain the convergence rates via two different approaches. The first approach is the standard one via Bregman distances and the second one is in terms of weaker norms. The important aspect in the second approach is that the regularization is only used to constrain the regularized solutions to a set where stability holds. Moreover, a novel smoothness condition is introduced termed as approximate Holder estimates which is further exploited to obtain the convergence rates. At the end, we also discuss the convergence rates for Tikhonov regularization with sparsity constraints by incorporating the Holder stability estimates.
In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems ... more In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems in Banach spaces and study its convergence analysis. The method is based on the well known iteratively regularized Landweber iteration method together with an extrapolation strategy. The general formulation of iteratively regularized Landweber iteration method in Banach spaces excludes the use of certain functions such as total variation like penalty functionals, L 1 functions etc. The novel scheme presented in this paper allows to use such non-smooth penalty terms that can be helpful in practical applications involving the reconstruction of several important features of solutions such as piecewise constancy and sparsity. We carefully discuss the choices for important parameters, such as combination parameters and step sizes involved in the design of the method. Additionally, we discuss an example to validate our assumptions.
International Journal of Mathematical Sciences and Computing, 2021
In this paper, we propose a novel variant of the Hill cipher based on vector spaces. In the class... more In this paper, we propose a novel variant of the Hill cipher based on vector spaces. In the classical Hill cipher, a non-singular matrix is used for encryption but it is well known that this cipher is vulnerable to the known-plaintext attack. In our proposed cryptosystem, we eradicate this problem by encrypting each plaintext block with a new invertible key matrix. This makes our scheme immune to all existing attacks in literature on this type of ciphers and so the resulting cipher can be used as other state-of-art block cipher. To generate the invertible matrices which serve as the dynamic keys, we make use of the vector spaces along with randomly generated basis and non-singular linear transformation. In addition to this, we also study the computational complexity of the proposed cryptosystem and compare this with the computational complexities of other schemes based on Hill cipher.
In this paper, the local convergence of Iteratively regularized Landweber iteration method is inv... more In this paper, the local convergence of Iteratively regularized Landweber iteration method is investigated for solving non-linear inverse problems in Banach spaces. Our analysis mainly relies on the assumption that the inverse mapping satisfies the Hölder stability estimate locally. We consider both noisy as well as non-noisy data in our analysis. Under the a-priori choice of stopping index for noisy data, we show that the iterates remain in a certain ball around exact solution and obtain the convergence rates. The convergence of the Iteratively regularized Landweber iterates to the exact solution is shown under certain assumptions in the case of non-noisy data and as a by-product, under different conditions, two different convergence rates are obtained.
In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First... more In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First one is Elliptic ElGamal type group ring public key cryptosystem whose security is greater than security of cryptosystems based on elliptic curves discrete logarithmic problem (ECDLP). Second is ElGamal type group ring public key cryptosystem, which is analogous to ElGamal public key cryptosystem but has comparatively greater security. Examples are also given for both the proposed cryptosystems.
Journal of Discrete Mathematical Sciences and Cryptography, 2021
In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First... more In this paper, we propose two cryptosystems based on group rings and existing cryptosystem. First one is Elliptic ElGamal type group ring public key cryptosystem whose security is greater than security of cryptosystems based on elliptic curves discrete logarithmic problem (ECDLP). Second is ElGamal type group ring public key cryptosystem, which is analogous to ElGamal public key cryptosystem but has comparatively greater security. Examples are also given for both the proposed cryptosystems.
In this paper we establish compactness results of multiscale and very weak multiscale type for se... more In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in L 2 (0, T ; H 1 0 (Ω)), fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation ε p ∂ t uε(x, t) − ∇ • (a(xε −1 , xε −2 , tε −q , tε −r)∇uε(x, t)) = f (x, t), where 0 < p < q < r. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by p, compared to the standard matching that gives rise to local parabolic problems.
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