# Coppersmith method

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{{Short description|Factorisation algorithm}}
The '''Coppersmith method''', proposed by [Don Coppersmith](/source/Don_Coppersmith), is a method to find small integer [zeroes](/source/Zero_of_a_function) of univariate or bivariate [polynomial](/source/polynomial)s, or their small zeroes [modulo a given integer](/source/Modular_arithmetic). The method uses the [Lenstra–Lenstra–Lovász lattice basis reduction algorithm](/source/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm) (LLL) to find a polynomial that has the same zeroes as the target polynomial but smaller coefficients.

In [cryptography](/source/cryptography), the Coppersmith method is mainly used in attacks on [RSA](/source/RSA_(algorithm)) when parts of the [secret key](/source/public_key_cryptography) are known and forms a base for [Coppersmith's attack](/source/Coppersmith's_attack).

== Approach ==
Coppersmith's approach is a reduction of solving modular polynomial equations to solving polynomials over the integers.

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 \pmod{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 {{mvar|'''Q'''}} is easy using, e.g., [Newton's method](/source/Newton's_method), but such an algorithm does not work modulo a composite number {{mvar|M}}. The idea behind Coppersmith’s method is to find a different polynomial {{mvar|f}} related to {{mvar|F}} that has the same root <math>x_0</math> modulo {{mvar|M}}, but has only small coefficients. If the coefficients and <math>x_0</math> are small enough that <math>|f(x_0)| < M</math> over the integers, then we have <math>f(x_0) = 0</math>, so that <math>x_0</math> is a root of {{mvar|f}} over {{mvar|'''Q'''}} and can be found easily.  More generally, we can find a polynomial <math>f(x)</math> with the same root <math>x_0</math> modulo some power <math>M^a</math> of {{mvar|M}}, satisfying <math>|f(x_0)| < M^a</math>, and solve for <math>x_0</math> as above.

Coppersmith's algorithm uses the [Lenstra–Lenstra–Lovász lattice basis reduction algorithm](/source/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm) (LLL) to construct the polynomial {{mvar|f}} with small coefficients.
Given {{mvar|F}}, the algorithm constructs polynomials <math>p_1(x), p_2(x), \dots, p_n(x)</math> that all have the same root <math>x_0</math> modulo <math>M^a</math>, where {{mvar|a}} is some integer chosen based on the degree of {{mvar|F}} and the size of <math>x_0</math>.
Any [linear combination](/source/linear_combination) of these polynomials also has <math>x_0</math> as a root modulo <math>M^a</math>.

The next step is to use the LLL algorithm to construct a linear combination <math>f(x)=\sum c_ip_i(x)</math>
of the <math>p_i(x)</math> so that the inequality <math>|f(x_0)| < M^a</math> holds.
Now standard factorization methods can calculate the zeroes of <math>f(x)</math> over the integers.

== Implementations ==
Coppersmith's method for univariate polynomials is implemented in
* [Magma](/source/Magma_computer_algebra_system) as the function <code>SmallRoots</code>;
* [PARI/GP](/source/PARI%2FGP) as the function <code>zncoppersmith</code>;
* [SageMath](/source/SageMath) as the method <code>small_roots</code>.

==References==
* {{cite book |author=Coppersmith, D.|title=Advances in Cryptology — EUROCRYPT '96 |chapter=Finding a Small Root of a Univariate Modular Equation |series=Lecture Notes in Computer Science |volume=1070 |pages=155–165 |year=1996 |doi=10.1007/3-540-68339-9_14|isbn=978-3-540-61186-8 |doi-access=free }}
* {{cite book |author=Coppersmith, D. |title=Advances in Cryptology — EUROCRYPT '96 |chapter=Finding a Small Root of a Bivariate Integer Equation; Factoring with High Bits Known |series=Lecture Notes in Computer Science |volume=1070 |pages=178–189 |year=1996 |doi=10.1007/3-540-68339-9_16|isbn=978-3-540-61186-8 |doi-access=free }}
* {{cite book | author=Coron, J. S. |title=Advances in Cryptology - EUROCRYPT 2004 |chapter=Finding Small Roots of Bivariate Integer Polynomial Equations Revisited |series=Lecture Notes in Computer Science |year=2004 |volume=3027
|pages=492–505 |doi=10.1007/978-3-540-24676-3_29 |isbn=978-3-540-21935-4 |chapter-url=https://iacr.org/archive/eurocrypt2004/30270487/bivariate.pdf}}
* {{cite book |first1=A. |last1=Bauer |first2=A. |last2=Joux |title=Advances in Cryptology - EUROCRYPT 2007 |chapter=Toward a Rigorous Variation of Coppersmith's Algorithm on Three Variables |series=Lecture Notes in Computer Science |authorlink2=Antoine Joux |volume=4515 |year=2007 |pages=361–378 |doi=10.1007/978-3-540-72540-4_21 |isbn=978-3-540-72539-8 |doi-access=free }}
* {{cite book |author=Coron, J. S. |title=Advances in Cryptology - CRYPTO 2007 |chapter=Finding Small Roots of Bivariate Integer Polynomial Equations: A Direct Approach |series=Lecture Notes in Computer Science |year=2007 |volume=4622 |pages=379–394 |doi=10.1007/978-3-540-74143-5_21 |isbn=978-3-540-74142-8 |chapter-url=https://iacr.org/archive/crypto2007/46220372/46220372.pdf}}

{{DEFAULTSORT:Coppersmith Method}}
Category:Asymmetric-key algorithms
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Adapted from the Wikipedia article [Coppersmith method](https://en.wikipedia.org/wiki/Coppersmith_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Coppersmith_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.