Secure Watermarking using Diophantine Equations for Authentication and Recovery

2015 ◽  
Vol 3 (2) ◽  
Author(s):  
Jayashree Nair ◽  
T. Padma
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Jpeg Compression ◽  
Authentication Scheme ◽  
Original Image ◽  
Invariant Features ◽  
Jpeg2000 Compression ◽  
Frequency Properties ◽  
Visual Authentication

This paper describes an authentication scheme that uses Diophantine equations based generation of the secret locations to embed the authentication and recovery watermark in the DWT sub-bands. The security lies in the difficulty of finding a solution to the Diophantine equation. The scheme uses the content invariant features of the image as a self-authenticating watermark and a quantized down sampled approximation of the original image as a recovery watermark for visual authentication, both embedded securely using secret locations generated from solution of the Diophantine equations formed from the PQ sequences. The scheme is mildly robust to Jpeg compression and highly robust to Jpeg2000 compression. The scheme also ensures highly imperceptible watermarked images as the spatio –frequency properties of DWT are utilized to embed the dual watermarks.

Diophantine Equations for Enhanced Security in Watermarking Scheme for Image Authentication

2017 ◽  
pp. 205-229
Author(s):  
Padma T ◽  
Jayashree Nair
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Image Authentication ◽  
Authentication Scheme ◽  
Content Authentication ◽  
Image Content ◽  
Polynomial Time Algorithms ◽  
Mathematical Problems ◽  
Watermarking Scheme

Hard mathematical problems having no polynomial time algorithms to determine a solution are seemly in design of secure cryptosystems. The proposed watermarking system used number theoretic concepts of the hard higher order Diophantine equations for image content authentication scheme with three major phases such as 1) Formation of Diophantine equation; 2) Generation and embedding of dual Watermarks; and 3) Image content authentication and verification of integrity. Quality of the watermarked images, robustness to compression and security are bench-marked with two peer schemes which used dual watermarks.

NOVEL SELF-REFERENCE BASED IMAGE WATERMARKING SCHEME

2010 ◽  
Vol 19 (02) ◽  
pp. 491-502 ◽  
Author(s):  
YONG-GANG FU ◽  
RUIMIN SHEN
Keyword(s):  
Image Watermarking ◽  
Extraction Process ◽  
Jpeg Compression ◽  
Reference Image ◽  
Original Image ◽  
Quantization Step ◽  
Dct Transform ◽  
Gray Level Image ◽  
Watermarking Scheme ◽  
Self Reference

In this paper, a novel image watermarking scheme based on a self-reference technique is proposed. The meaningful watermark is embedded into a gray-level image according to the relation between the constructed reference image and the original host. In order to be robust against Jpeg compression, the reference image should be robust under Jpeg compression. Firstly, the original image is transformed into DCT domain; and then most of the high frequency coefficients are omitted; after the quantization step and inverse DCT transform, we can obtain a robust reference. By considering the relation between the original image and its reference, we can embed the watermark into the host. The watermark extraction process is oblivious. Experimental results under several attacks show good robustness of the proposed scheme. Especially under cropping and Jpeg compression attacks, the watermark can be extracted with only few errors.

scholarly journals Quadratic ideals, indefinite quadratic forms and some specific diophantine equations

2018 ◽  
Vol 36 (3) ◽  
pp. 173-192
Author(s):  
Ahmet Tekcan ◽  
Seyma Kutlu
Keyword(s):  
Diophantine Equation ◽  
Quadratic Forms ◽  
Diophantine Equations ◽  
Algebraic Properties ◽  
Integer Solutions ◽  
Indefinite Quadratic

Let $k\geq 1$ be an integer and let $P=k+2,Q=k$ and $D=k^{2}+4$. In this paper, we derived some algebraic properties of quadratic ideals $I_{\gamma}$ and indefinite quadratic forms $F_{\gamma }$ for quadratic irrationals $\gamma$, and then we determine the set of all integer solutions of the Diophantine equation $F_{\gamma }^{\pm k}(x,y)=\pm Q$.

A Reversible Watermarking Algorithm Resistant to Image Geometric Transformation

2019 ◽  
Vol 11 (1) ◽  
pp. 100-113
Author(s):  
Jian Li ◽  
Jinwei Wang ◽  
Shuang Yu ◽  
Xiangyang Luo
Keyword(s):  
High Performance ◽  
Side Information ◽  
Jpeg Compression ◽  
Reversible Watermarking ◽  
Experimental Results ◽  
Geometric Transformation ◽  
Original Image ◽  
Robust Watermark ◽  
Watermark Embedding ◽  
Watermarking Scheme

This article proposes a novel robust reversible watermarking algorithm. The proposed watermarking scheme is reversible because the original image can be recovered after extracting watermarks from the watermarked image, as long as it is not processed by an attacker. The scheme is robust because watermarks can still be extracted from watermarked images, even if it is undergone some malicious or normal operations like rotation and JPEG compression. It first selects two circles, which are centred at the centroid and the centre of image. Then, statistic quantities of these two circles are employed for robust watermark embedding by altering the pixels' value. The side information generated by above embedding process will be embedded as fragile watermarks at another stage to ensure the recovery of original image. Experimental results verify the high performance of the proposed algorithm in resisting various attacks, including JPEG compression and geometric transformation.

scholarly journals On a class of insoluble binary quadratic diophantine equations

1991 ◽  
Vol 123 ◽  
pp. 141-151 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Diophantine Equation ◽  
Class Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Quadratic Number ◽  
Number Problem ◽  
Quadratic Number Fields ◽  
Quadratic Diophantine Equation ◽  
Real Quadratic

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT

2014 ◽  
Vol 91 (1) ◽  
pp. 11-18
Author(s):  
NOBUHIRO TERAI
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Euler’S Totient Function ◽  
Exponential Diophantine Equations ◽  
Euler’S Theorem ◽  
Positive Integers

AbstractLet $a$ and $m$ be relatively prime positive integers with $a>1$ and $m>2$. Let ${\it\phi}(m)$ be Euler’s totient function. The quotient $E_{m}(a)=(a^{{\it\phi}(m)}-1)/m$ is called the Euler quotient of $m$ with base $a$. By Euler’s theorem, $E_{m}(a)$ is an integer. In this paper, we consider the Diophantine equation $E_{m}(a)=x^{l}$ in integers $x>1,l>1$. We conjecture that this equation has exactly five solutions $(a,m,x,l)$ except for $(l,m)=(2,3),(2,6)$, and show that if the equation has solutions, then $m=p^{s}$ or $m=2p^{s}$ with $p$ an odd prime and $s\geq 1$.

scholarly journals A Hypothetical Upper Bound on the Heights of the Solutions of a Diophantine Equation with a Finite Number of Solutions

2018 ◽  
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Rational Solution ◽  
Infinitely Many Solutions ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions

Let f ( 1 ) = 1 , and let f ( n + 1 ) = 2 2 f ( n ) for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove:   (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, then M is computable.

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