Division of ideals in a Dedekind Domain
$IS \mid JS$ implies that $I\mid J$ where $I$ and $J$ are ideals of a number ring contained in another number ring $S$
Changes to parameter not showing on Report Server after deployment We denote the s value of an ECDSA signature $(r, s)$ on a message $m$ as: $s=\frac{H(m)+xr}{k}$  Assume two ECDSA signatures sharing the same nonce $(r, s_1) , (r, s_2)$ on two messages $m_1, m_2$, that verify under two pubkeys $x_1G, x_2G$.    If the two public keys are equal then the secret keys should be equal $x_1 = x_2$ and we can easily recover the $k$ using the standard attack on nonce reuse. Once we know $k$ we can recover the secret key.  $\frac{H(m_1)-H(m_2)}{(s_1 - s_2)} =\frac{k(H(m_1)-H(m_2))}{H(m_1)-H(m_2)+x_1r - x_2r}$  $x_1 = x_2 \rightarrow x_1r - x_2r = 0$  $\frac{H(m_1)-H(m_2)}{(s_1 - s_2)} =\frac{k(H(m_1)-H(m_2))}{H(m_1)-H(m_2)} = k$  My question is can this attack be made to work if the secret keys are not equal i.e. $x_1 \ne x_2$:  $\frac{H(m_1)-H(m_2)}{(s_1 - s_2)} =\frac{k(H(m_1)-H(m_2))}{H(m_1)-H(m_2)+x_1r - x_2r} = \frac{k(H(m_1)-H(m_2))}{H(m_1)-H(m_2)+ (x_1 - x_2)r}$  If you know either $x_1 - x2$ or $\frac{x_1}{x_2}$  you should be able to compute $k$ as long as $s_1 \ne s_2$.  You can calculate $x_1 - x_2 = \frac{H(m_2) - H(m_1)}{r}$ in case where $s_1 - s_2 = 0$. However this case seems to reduce to the hardness of ECDSA since anyone can compute the pubkey for a new message $m_2$ that verifies under first signature $(s, r)$ using public key recovery.  If $s_1 \ne s_2$ you can compute $\frac{x_1 - x_2}{k}$ which allows you to convert $s_1$ into $s_2$ and vice versa. Can Spirit Guardians affect enemies through walls? I have Xubuntu 18.04.3, and whenever I try to install a package through apt-get I get the following message.    Removing linux-image-5.0.0-31-generic (5.0.0-31.33~18.04.1) ... /etc/kernel/postrm.d/initramfs-tools: update-initramfs: Deleting /boot/initrd.img-5.0.0-31-generic /etc/kernel/postrm.d/zz-update-grub: Sourcing file `/etc/default/grub' /usr/sbin/grub-mkconfig: 35: /etc/default/grub: FROM:: not found run-parts: /etc/kernel/postrm.d/zz-update-grub exited with return code 127 dpkg: error processing package linux-image-5.0.0-31-generic (--remove):  installed linux-image-5.0.0-31-generic package post-removal script subprocess returned error exit status 1 Errors were encountered while processing:  linux-image-5.0.0-31-generic libdvd-pkg: Package libdvdcss2-1.4.2-1 was removed, stop processing... E: Sub-process /usr/bin/dpkg returned an error code (1)   Contents of /etc/default/grub:  # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 GRUB_TIMEOUT_STYLE=hidden GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="persistent"  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM="0x01234567,0xfefefefe,0x89abcdef,0xefefefef"  # Uncomment to disable graphical terminal (grub-pc only) #GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass "root=UUID=xxx" parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY="true"  # Uncomment to get a beep at grub start #GRUB_INIT_TUNE="480 440 1"  FROM: GRUB_CMDLINE_LINUX_DEFAULT="" GRUB_CMDLINE_LINUX="" TO: GRUB_CMDLINE_LINUX_DEFAULT="ipv6.disable=1" GRUB_CMDLINE_LINUX="ipv6.disable=1" How can I rank observations in-group faster? Cannot inherit from multiple classes defining __slots__? I want to prove, that $GL(n,K)$ is open in $M(n,K)$, where ($K=\{\mathbb{R},\mathbb{C},\mathbb{H}\}$.  For the prove I don't want to use that the determinant is continous. Alternativly I assume a submultiplicative norm.  First step: Define $U_A:=\{B\in M(n,K):\Vert A-B\Vert&lt;\frac{1}{\Vert A^{-1}\Vert}\}$ and show that it is a neighbourhod of $A$ in $M(n,K)$.  Second step: I show that the series $\sum_{k\ge0}(1-A^{-1}B)^kA^{-1}$ converges absolutely and the limit is inverse to $B$.  For the second step I should use the Neumann series, right? But I have some problems with the first step. Any help? Accused of cheating on online exams Using an NSString in a switch statement Default currency symbol How to check if a thread is sleeping? 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