Sage 2.11 Release Tour

Sage 2.10.1 was released on XXX 2008. For the official, comprehensive release notes, see the HISTORY.txt file that comes with the release. For the latest changes see [http://sagemath.org/announce/sage-2.11.txt sage-2.11.txt].

This is Work-in-Progress! Hence the links on this page won't work and information will be added as we got along!

Bugfixes/Upgrades

• ATLAS:
  • #2260 Upgraded to 3.8.1
  • #2108 ATLAS tuning info for Intel Prescott CPUs
  • #2368 ATLAS tuning info for Powerbook G4
• misc:

  • #2148 PolyBoRi monomial orders are wrong

  • #2437 Update eclib.spkg to eclib-20080304
  • #2468 inverting a non-invertible matrix over RDF returns weird results
  • #2517 ignore bad values in plot

  • #2545 FractionFieldElement lacks derivative method

  • #2566 fix all known bugs in graph_isom and binary_code

  • #2571 problem with copy() on sage.rings.integer_mod.IntegerMod_gmp

  • #2574 problem with Abelian groups and trivial elements
  • #2576 preserve docstrings of decorated methods in multi_polynomial_ideal.py
  • #2579 Inconsistency in integer quotient
  • #2581 extend solve_right to all cases; implement solve_left

  • #2582 fix bug in PermutationGroupElement

  • #2585 padic bugfix - check=False in constructor
  • #2587 subgroup of a permutation group is so slow it's silly
  • #2588 documentation and tests for sage.schemes.hyperelliptic_curves.jacobian_morphism
  • #2593 Sage chokes on utf-8 in .sage files

  • #2594 MPolynomial_polydict floordiv wrong arithmetic fixed

  • #2602 plot_vector_field docs are unnecessarily complicated (and use the slow lambda functions!)
  • #2584 printing bug with list_function()

zn_poly

David Harvey's zn_poly library is now a standard package for Sage. zn_poly is a new C library for polynomial arithmetic in (Z/nZ)[x] where 3 \le n \le ULONG\_MAX (i.e. any machine-word-sized modulus). The main benefit is speed. Three examples on sage.math, from my current development code (this code is not yet in the spkg):

• Multiplying length 200 polynomials over Z/nZ where n has 10 bits:

  • NTL (zz_pX): 113 µs
  • Magma: 44 µs

  • zn_poly: 13 µs

• Multiplying length 10^6 polynomials over Z/nZ where n has 40 bits and is odd:

  • NTL (zz_pX): 9.1s
  • Magma: 8.3s

  • zn_poly: 2.06s

• Reciprocal of a length 10^6 power series over Z/nZ where n has 40 bits and is odd:

  • NTL (zz_pX): 25.4s
  • Magma: ludicrously slow, maybe I'm doing something wrong

  • zn_poly: 3.62s

The library is used so far only to compute the zeta function for hyperelliptic curves.

small roots method for polynomials mod N (N composite)

Coppersmith's method for finding small roots of univariate polynomials modulo N where N is composite was implemented. 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.

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

We choose two primes of size 256-bit each.

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

We choose a random key

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

and pad it with 512-56=456 1s

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

Now we encrypt the resulting message:

   1 sage: C = ZmodN(M)^e

To recover K we consider the following polynomial modulo N:

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

and recover its small roots:

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

Generic Multivariate Polynomial Arithmetic

Joel Mohler improved the efficiency of the generic multivariate polynomial arithmetic in Sage. Before his patch was applied:

   1 sage: R.<x,y,z,a,b>=ZZ[]
   2 sage: f=prod([2*g^2-4*g+8 for g in R.gens()])
   3 sage: %time _=f*f
   4 CPU times: user 2.23 s, sys: 0.00 s, total: 2.23 s
   5 Wall time: 2.24

and after:

   1 sage: R.<x,y,z,a,b>=ZZ[]
   2 sage: f=prod([2*g^2-4*g+8 for g in R.gens()])
   3 sage: %time _=f*f
   4 CPU times: user 0.22 s, sys: 0.00 s, total: 0.22 s
   5 Wall time: 0.22