On the Diophantine equations 
                
                  
                
                $$z^2=f(x)^2 \pm f(y)^2$$
                
                  
                    
                      
                        z
                        2
                      
                      =
                      f
                      
                        
                          (
                          x
                          )
                        
                        2
                      
                      ±
                      f
                      
                        
                          (
                          y
                          )
                        
                        2
                      
                    
                  
                
               involving quartic polynomials

Yong Zhang; Arman Shamsi Zargar

doi:10.1007/s10998-018-0259-7

On the Diophantine equations $$z^2=f(x)^2 \pm f(y)^2$$ z 2 = f ( x ) 2 ± f ( y ) 2 involving quartic polynomials

Periodica Mathematica Hungarica ◽

10.1007/s10998-018-0259-7 ◽

2018 ◽

Vol 79 (1) ◽

pp. 25-31 ◽

Cited By ~ 2

Author(s):

Yong Zhang ◽

Arman Shamsi Zargar

Keyword(s):

Diophantine Equations ◽

Quartic Polynomials

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Properties of Some Diophantine Equations

SSRN Electronic Journal ◽

10.2139/ssrn.3527801 ◽

2020 ◽

Author(s):

Viktor Grechyn

Keyword(s):

Diophantine Equations

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Secure Watermarking using Diophantine Equations for Authentication and Recovery

Journal of Network and Information Security ◽

10.21863/jnis/2015.3.2.006 ◽

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.

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