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solving summation polynomials

asked 2017-12-12 11:14:45 -0500

Paul Doyle gravatar image

updated 2017-12-15 10:07:38 -0500

Solving discrete logarithm problem for elliptic curves

most papers I see use Magma for solving summation polynomials, can SageMath do this?

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What do you mean precisely? Most programming languages are Turing complete (Python is one of them).

vdelecroix gravatar imagevdelecroix ( 2017-12-12 17:15:22 -0500 )edit

For reference, "summation polynomials" are introduced in section 2 of the paper "Summation polynomials and the discrete logarithm problem on elliptic curves" by Igor Semaev, available at https://eprint.iacr.org/2004/031.pdf.

See also a discussion of summation polynomials by Steven Galbraith on "The Elliptic Curve Cryptography blog" .

slelievre gravatar imageslelievre ( 2017-12-12 18:07:21 -0500 )edit

Sorry, of course any Turing complete language are equivalent, so my question is badly phrased. I am starting writing an MSc dissertation on the Elliptic Curve Discrete Logarithm Problem over Finite Fields 𝔽pFp, (pp large prime) which is considered to be computationally infeasible, but two recent papers have made some advances, they both present results using Magma, the main computation work is: generating random points on an Elliptic Curve over a finite field. R=aP+bQR=aP+bQ (a, b random integers) solving systems of summation polynomial equations (as per your reference). Sm(X1,…,Xm)=0Sm(X1,…,Xm)=0 linear algebra ( using using Gr\"obner basis algorithms ) All the papers I see use Magma, how easy it is for SageMath to do these computations?, in terms of pre-existing functions.

Paul Doyle gravatar imagePaul Doyle ( 2017-12-15 04:11:59 -0500 )edit

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answered 2017-12-15 04:11:38 -0500

Paul Doyle gravatar image

updated 2017-12-15 10:03:53 -0500

Sorry, of course any Turing complete language are equivalent, so my question is badly phrased. I am starting writing an MSc dissertation on the Elliptic Curve Discrete Logarithm Problem over Finite Fields $\mathbb{F}_p$, ($p$ large prime) which is considered to be computationally infeasible, but two recent papers have made some advances, they both present results using Magma, the main computation work is:

  1. generating random points on an Elliptic Curve over a finite field. $R = aP + bQ$ (a, b random integers)

  2. solving systems of summation polynomial equations (as per your reference). $S_m(X_1,\dots,X_m) = 0$

  3. linear algebra ( using using Gr\"obner basis algorithms )

All the papers I see use Magma, so question is better phrased, if I ask how easy it is for SageMath to do these computations?, in terms of pre-existing functions. UPDATE: I've been told by the author of one of the papers , that I should be able to do this in SageMath.

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I could be wrong, but I think you are allowed to edit your own question to provide more details--if possible you should do that rather than update your question in the form of an "answer".

Iguananaut gravatar imageIguananaut ( 2017-12-18 06:57:58 -0500 )edit

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Asked: 2017-12-12 11:14:31 -0500

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Last updated: Dec 15 '17