Binary Independence Model: Difference between revisions

Revision as of 01:34, 21 March 2013

The Coppersmith method, proposed by Don Coppersmith, is a method to find small integer roots of polynomial equations. These polynomials can be univariate or bivariate. In cryptography the algorithm is mainly used in attacks on RSA when parts of the secret key are known.

The method uses the LLL algorithm ^[1] to find a polynomial that has the roots of the target polynomial as roots and has small coefficients.

Coppersmith’s method is based on lattice reduction. A lattice L is a subgroup of $R^{n}$ . Also there exists a k such that $L = Z b_{1} \oplus \dots \oplus Z b_{k}$ , where $B = (b_{1}, b_{2}, \dots, b_{k})$ is a basis of L. The LLL algorithm computes a basis $(b_{1}^{*}, b_{2}^{*}, \dots, b_{k}^{*})$ of short vectors. If k=n, the determinant of the lattice is given by det(L)=det(B); in general $d e t (L) \leq \prod | | b_{i}^{*} | |$ .

For any LLL reduced basis $(b_{1}^{*}, b_{2}^{*}, \dots, b_{k}^{*})$ it holds that $| | b_{k}^{*} | | \geq (d e t (L))^{1 / k} \cdot 2^{(1 - k) / 4}$ , see.^[2]

Let $F (x) = x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$ and assume that $F (x_{0}) \equiv 0 mod M$ for some integer $| x_{0} | < M^{1 / n}$ . Coppersmith’s algorithm can be used to find this integer solution $x_{0}$ .

Finding roots over Q is easy using e.g. Newton's method but these algorithms do not work modulo a composite number M. The idea behind Coppersmith’s method is to find a different polynomial $F_{2}$ related to F that has the same $x_{0}$ as a solution and has only small coefficients. If the coefficients and $x_{0}$ are so small that $F_{2} (x_{0}) < M$ over the integers, then $x_{0}$ is a root of F over Q and can easily be found.

How to find small roots using Coppersmith's method

Coppersmith’s approach is a reduction of solving modular polynomial equations to solving polynomials over the integers. Coppersmith's algorithm uses LLL to construct the polynomial $F_{2}$ with small coefficients.

Given F, the algorithm constructs polynomials $p_{1} (x), p_{2} (x), \dots, p_{n} (x)$ that have the same $x_{0}$ as root modulo $M^{a}$ , where a is some integer chosen dependent on the degree of F and the size of $x_{0}$ . Any linear combination of these polynomials has $x_{0}$ as root modulo $M^{a}$ .

The next step is to use the LLL algorithm to construct a linear combination $F_{2} (x) = \sum c_{i} p_{i} (x)$ of the $p_{i}$ so that the inequality $| F_{2} (x) | < M^{a}$ holds. Now standard factorization methods can calculate the roots of $F_{2} (x)$ over the integers.

References

↑ Lattice Basis Reduction Algorithms (http://www.farcaster.com/papers/sm-thesis/node7.html)
↑ A. Bauer and A. Joux, Toward a Rigorous Variation of Coppersmith’s Algorithm on Three Variables, Springer, LNCS 4515, 2007

[1] Lattice Basis Reduction Algorithms (http://www.farcaster.com/papers/sm-thesis/node7.html)

[2] A. Bauer and A. Joux, Toward a Rigorous Variation of Coppersmith’s Algorithm on Three Variables, Springer, LNCS 4515, 2007

[1]

[2]

@@ Line 1: / Line 1: @@
-Chef knives, also identified as French knives, are an vital component of any kitchen. If you happen to be seeking to shop for Henckels TWIN Master eight" Chef Knife with Black Deal with on your. If you're taking a look to buy Henckels TWIN Master 8" Chef Knife with Yellow Manage on your. Very good solution online Henckels TWIN Master 11.five" Chef Knife with Red Handle & Outlet shop.  You can in acquiring the bottom rates tag on Henckels TWIN Master 11.five" Chef Knife with Red Manage.  Low good quality chef knives are [https://www.flickr.com/search/?q=troublesome troublesome] and will cost you much more cash in future.<br><br>We'll start off with your ordinary kitchen eight-inch fillet knife.  We'll take our knife and go underneath the collagen band.  We'll just get underneath that connective tissue, slide our knifeand shave off any of that unwanted tissue.  Even the best way to make a knife is debated among chefs and daily cooks.  Two words-forged or stamped-can set off heated discussions on the merits of each and every. A boning knife is slightly larger than a paring knife.<br><br>Obtaining these leading rated knife sets is excellent but it would be great realizing specifically how to use them. The Zwilling J. A.  If you loved this post and you would like to acquire much more info concerning [http://www.thebestkitchenknivesreviews.com/best-knife-set-reviews-top-kitchen-sets/ Best Kitchen Knife Set For The Money] kindly pay a visit to our own website. Henckels Comprehensive Book of Knife Skills: The Essential Guide to Use, Strategies and Care. This spiral bound book has concise step by step directions on several knife tactics and security with great is a will have to have for each and every household chef. Mastering Knife Skills: The Crucial Guide to the Most Significant Tools in Your Kitchen (with DVD).<br><br>Hold the knife manage firmly and drive the point of the knife into the wood at least 1/two to three/four inch deep if attainable.  Apply stress to the wide side of the knife blade to [http://search.huffingtonpost.com/search?q=determine&s_it=header_form_v1 determine] if the tip of the blade will rip by way of the wood or bend. Carry out any other cutting tasks to which the knife might be subjected primarily based on individual use. The 3 knives - paring knife, utility knife and chef's Santuko knife - will execute any job you need.  Cleaver knife - used to reduce bony components of meat.<br><br>A single of my buddies consider that his hunting knife substantial that it is raising once more the edge of the carbide phase Wustof edger and Phase 2. Then I was surprised to see how cut his newspaper the "new" scrap and.  Like the original product, Ginsu's International Traditions is made for the household cook who is not attuned to the upkeep essential of a more higher-finish knife.  I like the fact that there are sharp knife in a box.  Several swear blade knife, but a heavy knife forged.Knife eight-in.<br><br>Chef Rosendale received the Presidential Medallion from the American Culinary Federation and was previosuly names Chef of the Year for 2005 in San Antonio, Texas. Bob - I never consider I've been capable to hold any tools or implements for 30 entire years and I need a new set of kitchen knives mysef.  The modest set I received last Christmas was pretty great for a commence.  Thanks for your experiences and I envy you having a set of knives that extended.  I may get a set for my kitchen soon.<br><br>This 14-piece set stocks a kitchen with most cutlery demands, and packs all knives in a contemporary black hardwood storage block.  The set contains: an eight-inch chef's knife, an 8-inch slicer, an 8-inch bread knife, a five-inch boning knife, a 4-1/two-inch utility knife, a 3-1/two-inch paring knife, and six four-1/2-inch steak knives, plus a pair of shears. The second season of Top Chef began airing in October 2006.<br><br>Announcement of remedy in 2014 was a surprise and a enhance to research and info-searching for. The globe has only a single entertainer with a Paul McCartney marionette and various sets of Beatles puppets.  Tomatoes land in unexpected places among tasty globe cuisines, which includes Tomato Juice Cakes from Girl Scout pot lucks - they're quite excellent.  The DIA practically lost its one hundred art collections to settle the bankruptcy of the City of Detroit.  A serrated knife has a lengthy jagged edge.<br><br>Use controlled chopping and cutting strokes with the knife blade and stay away from striking it forcefully against any surface because this will harm the knife blade. Rub the dishcloth along the blade of the knife to clean any food residue from the knife blade and then wash the handle briefly.  Rinse the knife completely under warm water and dry it right away with the dish towel. Shop the chef knife in a  Best Kitchen Knife Set Amazon knife rack or butcher's block.  A good butcher knife is very best.
+{{Multiple issues|cleanup =January 2010|refimprove =January 2010|
+{{expert-subject|Mathematics|date=January 2010}}
+}}
+The '''Coppersmith method''', proposed by [[Don Coppersmith]], is a method to find small integer [[Root of a function|roots]] of [[polynomial]] equations. These polynomials can be univariate or bivariate. In [[cryptography]] the algorithm is mainly used in attacks on [[RSA (algorithm)|RSA]] when parts of the [[public key cryptography|secret key]] are known.
+The method uses the [[LLL algorithm]] <ref>Lattice Basis Reduction Algorithms (http://www.farcaster.com/papers/sm-thesis/node7.html)</ref> to find a
+polynomial that has the roots of the target polynomial as roots and has small coefficients.
+Coppersmith’s method is based on lattice reduction. A [[lattice (group)|lattice]] ''L'' is a subgroup of <math>\mathbf{R}^n</math>.
+Also there exists a ''k'' such that <math>L = \mathbf{Z}b_1\oplus \ldots \oplus \mathbf{Z}b_k</math>, where
+<math>B=(b_1,b_2,\ldots ,b_k)</math> is a basis of ''L''. The LLL algorithm computes a basis
+<math>(b_1^*,b_2^*,\dots ,b_k^*)</math> of short vectors.
+If ''k=n'', the determinant of the lattice is given by det(''L'')=det(''B''); in general <math>\mathrm{det}(L)\le \prod||b_i^*||</math>.
+For any LLL reduced basis <math>(b_1^*,b_2^*,\dots ,b_k^*)</math> it holds that
+<math>||b_k^*||\ge (\mathrm{det}(L))^{1/k}\cdot 2^{(1-k)/4}</math>, see.<ref>A. Bauer and A. Joux, Toward a Rigorous Variation of Coppersmith’s Algorithm on Three Variables, Springer, LNCS 4515, 2007</ref>
+Let <math>F(x) = x^n+a_{n-1}x^{n-1}+\ldots +a_1x+a_0</math> and assume that <math>F(x_0)\equiv 0 \mod M</math> for some
+integer <math>|x_0|< M^{1/n}</math>.
+Coppersmith’s algorithm can be used to find this integer solution <math>x_0</math>.
+Finding  roots over '''Q''' is easy using e.g. [[Newton's method]] but these algorithms do not work modulo a composite number ''M''. The idea behind Coppersmith’s method is to find a different polynomial <math>F_2</math> related to ''F'' that has the same  <math>x_0</math>  as a solution and has only small coefficients. If the coefficients and <math>x_0</math> are so small that <math>F_2(x_0) < M</math> over the integers, then
+<math>x_0</math> is a root of ''F'' over '''Q''' and can easily be found.
+==How to find small roots using Coppersmith's method==
+Coppersmith’s approach is a reduction of solving modular polynomial equations to solving polynomials over the integers.
+Coppersmith's algorithm uses LLL to construct the polynomial <math>F_2</math> with small coefficients.
+Given ''F'', the algorithm constructs polynomials <math>p_1(x),p_2(x),\dots ,p_n(x)</math> that have the same <math>x_0</math> as root modulo <math>M^a</math>, where ''a'' is some integer chosen dependent on the degree of ''F'' and the size of <math>x_0</math>.
+Any linear combination of these polynomials has <math>x_0</math> as root modulo <math>M^a</math>.
+The next step is to use the LLL algorithm to construct a linear combination <math>F_2(x)=\sum c_ip_i(x)</math>
+of the <math>p_i</math> so that the inequality <math>|F_2(x)| < M^a</math>  holds.
+Now standard factorization methods can calculate the roots of <math>F_2(x)</math> over the integers.
+==See also==
+[[Coppersmith's Attack]]
+==References==
+  <references/>
+{{DEFAULTSORT:Coppersmith Method}}
+[[Category:Asymmetric-key algorithms]]

Binary Independence Model: Difference between revisions

Revision as of 01:34, 21 March 2013

How to find small roots using Coppersmith's method

See also

References

Navigation menu

Search