Dense univariate polynomials over $\ZZ/n\ZZ$ , implemented using NTL.¶

This implementation is generally slower than the FLINT implementation in polynomial_zmod_flint, so we use FLINT by default when the modulus is small enough; but NTL does not require that $n$ be int-sized, so we use it as default when $n$ is too large for FLINT.

Note that the classes Polynomial_dense_modn_ntl_zz and Polynomial_dense_modn_ntl_ZZ are different; the former is limited to moduli less than a certain bound, while the latter supports arbitrarily large moduli.

AUTHORS:

Robert Bradshaw: Split off from polynomial_element_generic.py (2007-09)
Robert Bradshaw: Major rewrite to use NTL directly (2007-09)

class sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_n¶

Bases: sage.rings.polynomial.polynomial_element.Polynomial

A dense polynomial over the integers modulo n, where n is composite, with the underlying arithmetic done using NTL.

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(16))
sage: f = x^3 - x + 17
sage: f^2
x^6 + 14*x^4 + 2*x^3 + x^2 + 14*x + 1

sage: loads(f.dumps()) == f
True

sage: R.<x> = Integers(100)[]
sage: p = 3*x
sage: q = 7*x
sage: p+q
10*x
sage: R.<x> = Integers(8)[]
sage: parent(p)
Univariate Polynomial Ring in x over Ring of integers modulo 100
sage: p + q
10*x
sage: R({10:-1})
7*x^10

degree(gen=None)¶: Return the degree of this polynomial. The zero polynomial has degree -1.

int_list()¶

lcm(singular='singular_default', have_ring=False)¶

list()¶

Return a new copy of the list of the underlying elements of self.

EXAMPLES:

sage: _.<x> = Integers(100)[]
sage: f = x^3 + 3*x - 17
sage: f.list()
[83, 3, 0, 1]

ntl_ZZ_pX()¶: Return underlying NTL representation of this polynomial. Additional ‘’bonus’’ functionality is available through this function.

Warning

You must call ntl.set_modulus(ntl.ZZ(n)) before doing arithmetic with this object!

ntl_set_directly(v)¶

Set the value of this polynomial directly from a vector or string.

Polynomials over the integers modulo n are stored internally using NTL’s ZZ_pX class. Use this function to set the value of this polynomial using the NTL constructor, which is potentially very fast. The input v is either a vector of ints or a string of the form [ n1 n2 n3 ... ] where the ni are integers and there are no commas between them. The optimal input format is the string format, since that’s what NTL uses by default.

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(100))
sage: from sage.rings.polynomial.polynomial_modn_dense_ntl import Polynomial_dense_mod_n as poly_modn_dense
sage: poly_modn_dense(R, ([1,-2,3]))
3*x^2 + 98*x + 1
sage: f = poly_modn_dense(R, 0)
sage: f.ntl_set_directly([1,-2,3])
sage: f
3*x^2 + 98*x + 1
sage: f.ntl_set_directly('[1 -2 3 4]')
sage: f
4*x^3 + 3*x^2 + 98*x + 1

quo_rem(right)¶

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

resultant(other, variable=None)¶

shift(n)¶

Returns this polynomial multiplied by the power $x^n$ . If $n$ is negative, terms below $x^n$ will be discarded. Does not change this polynomial.

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(12345678901234567890))
sage: p = x^2 + 2*x + 4
sage: p.shift(0)
 x^2 + 2*x + 4
sage: p.shift(-1)
 x + 2
sage: p.shift(-5)
 0
sage: p.shift(2)
 x^4 + 2*x^3 + 4*x^2

TESTS:

sage: p = R(0)
sage: p.shift(3).is_zero()
True
sage: p.shift(-3).is_zero()
True

AUTHOR:

David Harvey (2006-08-06)

small_roots(*args, **kwds)¶

See sage.rings.polynomial.polynomial_modn_dense_ntl.small_roots() for the documentation of this function.

EXAMPLE:

sage: N = 10001
sage: K = Zmod(10001)
sage: P.<x> = PolynomialRing(K)
sage: f = x^3 + 10*x^2 + 5000*x - 222
sage: f.small_roots()
[4]

class sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_p¶

Bases: sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_n

A dense polynomial over the integers modulo p, where p is prime.

discriminant()¶

EXAMPLES:

sage: _.<x> = PolynomialRing(GF(19))
sage: f = x^3 + 3*x - 17
sage: f.discriminant()
12

gcd(right)¶

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

resultant(other)¶

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

xgcd(right)¶

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

class sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ¶

Bases: sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_n

degree()¶

EXAMPLES:

sage: R.<x> = Integers(14^34)[]
sage: f = x^4 - x - 1
sage: f.degree()
4
sage: f = 14^43*x + 1
sage: f.degree()
0

is_gen()¶

list()¶

quo_rem(right)¶

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

reverse()¶

Reverses the coefficients of self. The reverse of $f(x)$ is $x^n f(1/x)$ .

The degree will go down if the constant term is zero.

EXAMPLES:

sage: R.<x> = Integers(12^29)[]
sage: f = x^4 + 2*x + 5
sage: f.reverse()
5*x^4 + 2*x^3 + 1
sage: f = x^3 + x
sage: f.reverse()
x^2 + 1

shift(n)¶

Shift self to left by $n$ , which is multiplication by $x^n$ , truncating if $n$ is negative.

EXAMPLES:

sage: R.<x> = Integers(12^30)[]
sage: f = x^7 + x + 1
sage: f.shift(1)
x^8 + x^2 + x
sage: f.shift(-1)
x^6 + 1
sage: f.shift(10).shift(-10) == f
True

TESTS:

sage: p = R(0)
sage: p.shift(3).is_zero()
True
sage: p.shift(-3).is_zero()
True

truncate(n)¶

Returns this polynomial mod $x^n$ .

EXAMPLES:

sage: R.<x> = Integers(15^30)[]
sage: f = sum(x^n for n in range(10)); f
x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
sage: f.truncate(6)
x^5 + x^4 + x^3 + x^2 + x + 1

valuation()¶

Returns the valuation of self, that is, the power of the lowest non-zero monomial of self.

EXAMPLES:

sage: R.<x> = Integers(10^50)[]
sage: x.valuation()
1
sage: f = x-3; f.valuation()
0
sage: f = x^99; f.valuation()
99
sage: f = x-x; f.valuation()
+Infinity

class sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_zz¶

Bases: sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_n

degree()¶

EXAMPLES:

sage: R.<x> = Integers(77)[]
sage: f = x^4 - x - 1
sage: f.degree()
4
sage: f = 77*x + 1
sage: f.degree()
0

int_list()¶

Returns the coefficients of self as efficiently as possible as a list of python ints.

EXAMPLES:

sage: R.<x> = Integers(100)[]
sage: from sage.rings.polynomial.polynomial_modn_dense_ntl import Polynomial_dense_mod_n as poly_modn_dense
sage: f = poly_modn_dense(R,[5,0,0,1])
sage: f.int_list()
[5, 0, 0, 1]
sage: [type(a) for a in f.int_list()]
[<type 'int'>, <type 'int'>, <type 'int'>, <type 'int'>]

is_gen()¶

ntl_set_directly(v)¶

quo_rem(right)¶

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

reverse()¶

Reverses the coefficients of self. The reverse of $f(x)$ is $x^n f(1/x)$ .

The degree will go down if the constant term is zero.

EXAMPLES:

sage: R.<x> = Integers(77)[]
sage: f = x^4 - x - 1
sage: f.reverse()
76*x^4 + 76*x^3 + 1
sage: f = x^3 - x
sage: f.reverse()
76*x^2 + 1        

shift(n)¶

Shift self to left by $n$ , which is multiplication by $x^n$ , truncating if $n$ is negative.

EXAMPLES:

sage: R.<x> = Integers(77)[]
sage: f = x^7 + x + 1
sage: f.shift(1)
x^8 + x^2 + x
sage: f.shift(-1)
x^6 + 1
sage: f.shift(10).shift(-10) == f
True

TESTS:

sage: p = R(0)
sage: p.shift(3).is_zero()
True
sage: p.shift(-3).is_zero()
True

truncate(n)¶

Returns this polynomial mod $x^n$ .

EXAMPLES:

sage: R.<x> = Integers(77)[]
sage: f = sum(x^n for n in range(10)); f
x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
sage: f.truncate(6)
x^5 + x^4 + x^3 + x^2 + x + 1

valuation()¶

Returns the valuation of self, that is, the power of the lowest non-zero monomial of self.

EXAMPLES:

sage: R.<x> = Integers(10)[]
sage: x.valuation()
1
sage: f = x-3; f.valuation()
0
sage: f = x^99; f.valuation()
99
sage: f = x-x; f.valuation()
+Infinity

sage.rings.polynomial.polynomial_modn_dense_ntl.make_element(parent, args)¶

sage.rings.polynomial.polynomial_modn_dense_ntl.small_roots(self, X, beta=None, epsilon=1.0, **kwds=None)¶

Let $N$ be the characteristic of the base ring this polynomial is defined over: N = self.base_ring().characteristic(). This method returns small roots of this polynomial modulo some factor $b$ of $N$ with the constraint that $b >= N^\beta$ . Small in this context means that if $x$ is a root of $f$ modulo $b$ then $|x| < X$ . This $X$ is either provided by the user or the maximum $X$ is chosen such that this algorithm terminates in polynomial time. If $X$ is chosen automatically it is $X = ceil(1/2 N^{\beta^2/\delta - \epsilon})$ . The algorithm may also return some roots which are larger than $X$ . ‘This algorithm’ in this context means Coppersmith’s algorithm for finding small roots using the LLL algorithm. The implementation of this algorithm follows Alexander May’s PhD thesis referenced below.

INPUT:

X – an absolute bound for the root (default: see above)
beta – compute a root mod $b$ where $b$ is a factor of $N$ and $b \ge N^\beta$ . (Default: 1.0, so $b = N$ .)
epsilon – the parameter $\epsilon$ described above. (Default: $\beta/8$ )
**kwds – passed through to method Matrix_integer_dense.LLL().

EXAMPLES:

First consider a small example:

sage: N = 10001
sage: K = Zmod(10001)
sage: P.<x> = PolynomialRing(K)
sage: f = x^3 + 10*x^2 + 5000*x - 222

This polynomial has no roots without modular reduction (i.e. over $\ZZ$ ):

sage: f.change_ring(ZZ).roots()
[]

To compute its roots we need to factor the modulus $N$ and use the Chinese remainder theorem:

sage: p,q = N.prime_divisors()
sage: f.change_ring(GF(p)).roots()
[(4, 1)]
sage: f.change_ring(GF(q)).roots()
[(4, 1)]

sage: crt(4, 4, p, q) 
4

This root is quite small compared to $N$ , so we can attempt to recover it without factoring $N$ using Coppersmith’s small root method:

sage: f.small_roots()
[4]

An application of this method is to consider RSA. We are using 512-bit RSA with public exponent $e=3$ to encrypt a 56-bit DES key. Because it would be easy to attack this setting if no padding was used we pad the key $K$ with 1s to get a large number:

sage: Nbits, Kbits = 512, 56
sage: e = 3

We choose two primes of size 256-bit each:

sage: p = 2^256 + 2^8 + 2^5 + 2^3 + 1
sage: q = 2^256 + 2^8 + 2^5 + 2^3 + 2^2 + 1
sage: N = p*q
sage: ZmodN = Zmod( N )

We choose a random key:

sage: K = ZZ.random_element(0, 2^Kbits)

and pad it with 512-56=456 1s:

sage: Kdigits = K.digits(2)
sage: M = [0]*Kbits + [1]*(Nbits-Kbits)
sage: for i in range(len(Kdigits)): M[i] = Kdigits[i]

sage: M = ZZ(M, 2)

Now we encrypt the resulting message:

sage: C = ZmodN(M)^e

To recover $K$ we consider the following polynomial modulo $N$ :

sage: P.<x> = PolynomialRing(ZmodN)
sage: f = (2^Nbits - 2^Kbits + x)^e - C

and recover its small roots:

sage: Kbar = f.small_roots()[0]
sage: K == Kbar
True

The same algorithm can be used to factor $N = pq$ if partial knowledge about $q$ is available. This example is from the Magma handbook:

First, we set up $p$ , $q$ and $N$ :

sage: length = 512
sage: hidden = 110
sage: p = next_prime(2^int(round(length/2)))
sage: q = next_prime( round(pi.n()*p) )
sage: N = p*q

Now we disturb the low 110 bits of $q$ :

sage: qbar = q + ZZ.random_element(0,2^hidden-1)

And try to recover $q$ from it:

sage: F.<x> = PolynomialRing(Zmod(N))
sage: f = x - qbar

We know that the error is $\le 2^{\text{hidden}}-1$ and that the modulus we are looking for is $\ge \sqrt{N}$ :

sage: set_verbose(2)
sage: d = f.small_roots(X=2^hidden-1, beta=0.5)[0] # time random
verbose 2 (<module>) m = 4
verbose 2 (<module>) t = 4
verbose 2 (<module>) X = 1298074214633706907132624082305023
verbose 1 (<module>) LLL of 8x8 matrix (algorithm fpLLL:wrapper)
verbose 1 (<module>) LLL finished (time = 0.006998)
sage: q == qbar - d
True

REFERENCES:

Don Coppersmith. Finding a small root of a univariate modular equation. In Advances in Cryptology, EuroCrypt 1996, volume 1070 of Lecture Notes in Computer Science, p. 155–165. Springer, 1996. http://cr.yp.to/bib/2001/coppersmith.pdf

Alexander May. New RSA Vulnerabilities Using Lattice Reduction Methods. PhD thesis, University of Paderborn, 2003. http://www.informatik.tu-darmstadt.de/KP/publications/03/bp.ps

Dense univariate polynomials over $\ZZ/n\ZZ$ , implemented using NTL.¶

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Dense univariate polynomials over $\ZZ/n\ZZ$ , implemented using NTL.¶