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: 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), implementation='NTL')
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> = PolynomialRing(Integers(100), implementation='NTL')
sage: p = 3*x
sage: q = 7*x
sage: p+q
10*x
sage: R.<x> = PolynomialRing(Integers(8), implementation='NTL')
sage: parent(p)
Univariate Polynomial Ring in x over Ring of integers modulo 100 (using NTL)
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.

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(100), implementation='NTL')
sage: (x^3 + 3*x - 17).degree()
3
sage: R.zero().degree()
-1

int_list()

list(copy=True)

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

EXAMPLES:

sage: _.<x> = PolynomialRing(Integers(100), implementation='NTL')
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), implementation='NTL')
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)

Returns a tuple (quotient, remainder) where self = quotient*other + remainder.

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), implementation='NTL')
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

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.

EXAMPLES:

sage: N = 10001
sage: K = Zmod(10001)
sage: P.<x> = PolynomialRing(K, implementation='NTL')
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: Polynomial_dense_mod_n

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

discriminant()

EXAMPLES:

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

gcd(right)

Return the greatest common divisor of this polynomial and other, as a monic polynomial.

INPUT:

• other – a polynomial defined over the same ring as self

EXAMPLES:

sage: R.<x> = PolynomialRing(GF(3),implementation="NTL")
sage: f,g = x + 2, x^2 - 1
sage: f.gcd(g)
x + 2

resultant(other)

Returns the resultant of self and other, which must lie in the same polynomial ring.

INPUT:

• other – a polynomial

OUTPUT: an element of the base ring of the polynomial ring

EXAMPLES:

sage: R.<x> = PolynomialRing(GF(19),implementation='NTL')
sage: f = x^3 + x + 1;  g = x^3 - x - 1
sage: r = f.resultant(g); r
11
sage: r.parent() is GF(19)
True

xgcd(other)

Compute the extended gcd of this element and other.

INPUT:

• other – an element in the same polynomial ring

OUTPUT:

A tuple (r,s,t) of elements in the polynomial ring such that r = s*self + t*other.

EXAMPLES:

sage: R.<x> = PolynomialRing(GF(3),implementation='NTL')
sage: x.xgcd(x)
(x, 0, 1)
sage: (x^2 - 1).xgcd(x - 1)
(x + 2, 0, 1)
sage: R.zero().xgcd(R.one())
(1, 0, 1)
sage: (x^3 - 1).xgcd((x - 1)^2)
(x^2 + x + 1, 0, 1)
sage: ((x - 1)*(x + 1)).xgcd(x*(x - 1))
(x + 2, 1, 2)

class sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ

Bases: Polynomial_dense_mod_n

degree()

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(14^34), implementation='NTL')
sage: f = x^4 - x - 1
sage: f.degree()
4
sage: f = 14^43*x + 1
sage: f.degree()
0

is_gen()

list(copy=True)

quo_rem(right)

Returns \(q\) and \(r\), with the degree of \(r\) less than the degree of \(right\), such that \(q * right + r = self\).

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(10^30), implementation='NTL')
sage: f = x^5+1; g = (x+1)^2
sage: q, r = f.quo_rem(g)
sage: q
x^3 + 999999999999999999999999999998*x^2 + 3*x + 999999999999999999999999999996
sage: r
5*x + 5
sage: q*g + r
x^5 + 1

reverse(degree=None)

Return the reverse of the input polynomial thought as a polynomial of degree degree.

If \(f\) is a degree-\(d\) polynomial, its reverse is \(x^d f(1/x)\).

INPUT:

• degree (None or an integer) - if specified, truncate or zero pad the list of coefficients to this degree before reversing it.

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(12^29), implementation='NTL')
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
sage: f.reverse(1)
1
sage: f.reverse(5)
x^4 + x^2

shift(n)

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

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(12^30), implementation='NTL')
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

truncate(n)

Returns this polynomial mod \(x^n\).

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(15^30), implementation='NTL')
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> = PolynomialRing(Integers(10^50), implementation='NTL')
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: Polynomial_dense_mod_n

Polynomial on \(\ZZ/n\ZZ\) implemented via NTL.

_add_(_right)

_sub_(_right)

_lmul_(c)

_rmul_(c)

_mul_(_right)

_mul_trunc_(right, n)

Return the product of self and right truncated to the given length \(n\)

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(100), implementation="NTL")
sage: f = x - 2
sage: g = x^2 - 8*x + 16
sage: f*g
x^3 + 90*x^2 + 32*x + 68
sage: f._mul_trunc_(g, 42)
x^3 + 90*x^2 + 32*x + 68
sage: f._mul_trunc_(g, 3)
90*x^2 + 32*x + 68
sage: f._mul_trunc_(g, 2)
32*x + 68
sage: f._mul_trunc_(g, 1)
68
sage: f._mul_trunc_(g, 0)
0
sage: f = x^2 - 8*x + 16
sage: f._mul_trunc_(f, 2)
44*x + 56

degree()

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(77), implementation='NTL')
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> = PolynomialRing(Integers(100), implementation='NTL')
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()]
[<... 'int'>, <... 'int'>, <... 'int'>, <... 'int'>]

is_gen()

ntl_set_directly(v)

quo_rem(right)

Returns \(q\) and \(r\), with the degree of \(r\) less than the degree of \(right\), such that \(q * right + r = self\).

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(125), implementation='NTL')
sage: f = x^5+1; g = (x+1)^2
sage: q, r = f.quo_rem(g)
sage: q
x^3 + 123*x^2 + 3*x + 121
sage: r
5*x + 5
sage: q*g + r
x^5 + 1

reverse(degree=None)

Return the reverse of the input polynomial thought as a polynomial of degree degree.

If \(f\) is a degree-\(d\) polynomial, its reverse is \(x^d f(1/x)\).

INPUT:

• degree (None or an integer) - if specified, truncate or zero pad the list of coefficients to this degree before reversing it.

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(77), implementation='NTL')
sage: f = x^4 - x - 1
sage: f.reverse()
76*x^4 + 76*x^3 + 1
sage: f.reverse(2)
76*x^2 + 76*x
sage: f.reverse(5)
76*x^5 + 76*x^4 + x
sage: g = x^3 - x
sage: g.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> = PolynomialRing(Integers(77), implementation='NTL')
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

truncate(n)

Returns this polynomial mod \(x^n\).

EXAMPLES:

sage: R.<x> = PolynomialRing(Integers(77), implementation='NTL')
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> = PolynomialRing(Integers(10), implementation='NTL')
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=None, beta=1.0, epsilon=None, **kwds)

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, implementation='NTL')
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, implementation='NTL')
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), implementation='NTL')
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: from sage.misc.verbose import set_verbose
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.cs.uni-paderborn.de/uploads/tx_sibibtex/bp.pdf