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 June 8th, 2018, 02:43 PM #1 Newbie   Joined: Jul 2016 From: Ames, IA Posts: 3 Thanks: 0 Two-stage decoding based on Euclidean division Define the following quantity: $$X=abs^{2\ell}+ce+Fs^{\ell}$$ where $$F=ae+cb$$ Assumptions: $a,b,c,e$ are unknown integers $s,\ell$ are known positive integers $X$ is known My goal is to find $F$. The general idea is simple. Since we know $X$, we first find the remainder of the division $X/s^{2l}$. That would be $ce+Fs^{\ell}$. Then from this quantity we subtract the remainder of the division $(ce+Fs^{\ell})/s^\ell$ which yields $Fs^{\ell}$. Then just divide by $s^\ell$. Now, if all variables were positive integers under the constraint $F0$). Definition 1 $$x=qd+r$$ in which case the $0\leq r0$, then $R=r$ and $Q=q$ - If $x<0$, then $r=R+d$ and $Q=q+1$ Matlab has an implementation for both definitions. For my initial problem if I use Definition 1 the result for $F$ is wrong if $X<0$ (it differs from the correct one by $s^\ell$ which is expected) and correct if $X>0$. If I use Definition 2, the result for $F$ is correct if $X>0$ and for $X<0$ it's sometimes correct and sometimes wrong. So my strategy is flawed but I am not sure what is wrong. Also, some assumptions need to hold a priori for the general case where some variables can be negative, to have a solution and I think that those should be $|a|,|b|,|c|,|e|,|F|<0.5(s^{\ell}-1)$.

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