Ask Your Question

# x63's profile - activity

 2024-03-04 09:55:54 +0200 received badge ● Notable Question (source) 2024-03-04 09:55:54 +0200 received badge ● Popular Question (source) 2021-12-03 09:18:14 +0200 commented answer Symbolic variables in finite-fields Thank you for the idea. I will leave my mistake with the ideal in the question to match your answer. I also tested it wi 2021-12-03 09:14:27 +0200 marked best answer Symbolic variables in finite-fields Hello. I have a finite-field k and its degree n extension E k = GF(4, 'a') n = 64 E = k.extension(n, 'X')  where I perform exponentiation v = E.random_element() res = v^123456789  My goal is to track an element in k^n, mapped to E, as the exponentiation in E occurs. Ideally, I would write it as alphas = var(["alpha_%d" % i for i in range(n)]) v = 0 for i in range(n): v += alphas[i]*E.gen()^i res = v^123456789  But this leads to "TypeError: positive characteristic not allowed in symbolic computations". I noticed that polynomials could be used instead of symbolic variables (e.g., https://ask.sagemath.org/question/59697), so another idea is to write it as k = GF(4, 'a') n = 8 R = PolynomialRing(k, ["alpha_%d" % i for i in range(n)]) P = PolynomialRing(k, 'x') E = R.extension(P.irreducible_element(n), 'X') v = 0 for i in range(n): v += R.gen(i)*E.gen()^i res = v^65536  But this leads to "OverflowError: exponent overflow (65536)". It's also slow for larger values of n. The next idea is to reduce $\alpha_i^q = \alpha_i$ using an ideal k = GF(4, 'a') n = 4 R = PolynomialRing(k, n, 'x') I = R.ideal([g^(k.order() - 1) - 1 for g in R.gens()]) QR = R.quotient(I, 'alpha') P = PolynomialRing(k, 'x') BR = PolynomialRing(QR, 'Y') E = BR.extension(P.irreducible_element(n), 'X') v = 0 for i in range(n): v += QR.gen(i)*E.gen()^i res = v^65537  It seems to work for values of n = 4, but for anything larger it gets stuck on computing primary decomposition for the ideal. See https://ask.sagemath.org/question/59987 for details. Is there a way to do this in SageMath or, perhaps, using one of the underlying systems, such as Singular, and then move back to SageMath? 2021-12-03 09:14:27 +0200 received badge ● Scholar (source) 2021-12-03 09:14:25 +0200 received badge ● Supporter (source) 2021-12-01 22:08:52 +0200 asked a question Symbolic variables in finite-fields Symbolic variables in finite-fields Hello. I have a finite-field k and its degree n extension E k = GF(4, 'a') n = 64 E 2021-11-30 02:58:18 +0200 received badge ● Good Question (source) 2021-11-29 23:20:56 +0200 received badge ● Nice Question (source) 2021-11-29 10:15:02 +0200 received badge ● Editor (source) 2021-11-29 10:15:02 +0200 edited question PolynomialRing appears to hang during construction PolynomialRing appears to hang during construction Hello. I came across a situation which hangs SageMath 9.4: k = GF(4, 2021-11-28 01:52:09 +0200 received badge ● Student (source) 2021-11-28 01:51:43 +0200 asked a question PolynomialRing appears to hang during construction PolynomialRing appears to hang during construction Hello. I came across a situation which hangs SageMath 9.4: k = GF(4,