Corrigendum: On Diophantine Equations Solvable in Models of Open Induction.

1991 ◽  
Vol 56 (3) ◽  
pp. 811
Author(s):  
Margarita Otero
Keyword(s):  
Diophantine Equations ◽  
Open Induction

On Diophantine equations solvable in models of open induction

1990 ◽  
Vol 55 (2) ◽  
pp. 779-786 ◽  
Author(s):  
Margarita Otero
Keyword(s):  
Diophantine Equations ◽  
Main Lemma ◽  
Peano Arithmetic ◽  
The Real ◽  
Real Closure ◽  
Open Induction ◽  
Induction Scheme

AbstractWe consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas.We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣f(x,y) ∣ < 1 for every (x,y) Є C, where C is the curve defined on the real closure of M by C: x2 + y2 = a and a > 0 is a nonstandard element of M.

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.

Eight Diophantine Equations

1966 ◽  
Vol s3-16 (1) ◽  
pp. 153-166 ◽  
Author(s):  
J. H. E. Cohn
Keyword(s):  
Diophantine Equations
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