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Diffstat (limited to 'buch/chapters/30-endlichekoerper/rechnungen')
3 files changed, 170 insertions, 0 deletions
diff --git a/buch/chapters/30-endlichekoerper/rechnungen/invbeispiel.maxima b/buch/chapters/30-endlichekoerper/rechnungen/invbeispiel.maxima new file mode 100644 index 0000000..4770926 --- /dev/null +++ b/buch/chapters/30-endlichekoerper/rechnungen/invbeispiel.maxima @@ -0,0 +1,97 @@ +/* + * invbeispiel.maxima + * + * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ + +m: X^3 + 2*X^2 + 2*X + 3; + +modulus:7; +factor(m); +modulus:false; + +M: matrix( + [ 0, 0, 0, -3 ], + [ 1, 0, 0, -2 ], + [ 0, 1, 0, -2 ], + [ 0, 0, 1, -1 ] +); +M: mod(M, 7); +M0: identfor(M); +M1: M; +M2: M.M1; +M3: M.M2; + +a0: 1; +a1: 2; +a2: 9; +a3: 1; + +a: a0 + a1*X + a2*X^2 + a3*X^3; + +A: a0*M0 + a1*M1 + a2*M2 + a3*M3; +A: mod(A, 7); + +T: matrix( + [ A[1,1], A[1,2], A[1,3], A[1,4], 1, 0, 0, 0 ], + [ A[2,1], A[2,2], A[2,3], A[2,4], 0, 1, 0, 0 ], + [ A[3,1], A[3,2], A[3,3], A[3,4], 0, 0, 1, 0 ], + [ A[4,1], A[4,2], A[4,3], A[4,4], 0, 0, 0, 1 ] +); + +t: inv_mod(T[1,1], 7); +T[1]: mod(t * T[1], 7); +T[2]: mod(T[2] - T[2,1]*T[1], 7); +T[3]: mod(T[3] - T[3,1]*T[1], 7); +T[4]: mod(T[4] - T[4,1]*T[1], 7); + +t: inv_mod(T[2,2], 7); +T[2]: mod(t * T[2], 7); +T[3]: mod(T[3] - T[3,2] * T[2], 7); +T[4]: mod(T[4] - T[4,2] * T[2], 7); + +t: inv_mod(T[3,3], 7); +T[3]: mod(t * T[3], 7); +T[4]: mod(T[4] - T[4,3] * T[3], 7); + +t: inv_mod(T[4,4], 7); +T[4]: mod(t * T[4], 7); +T; + +T[3]: mod(T[3] - T[3,4] * T[4], 7); +T[2]: mod(T[2] - T[2,4] * T[4], 7); +T[1]: mod(T[1] - T[1,4] * T[4], 7); + +T[2]: mod(T[2] - T[2,3] * T[3], 7); +T[1]: mod(T[1] - T[1,3] * T[3], 7); + +T[1]: mod(T[1] - T[1,2] * T[2], 7); + +T; + +C: matrix( + [ T[1,5], T[1,6], T[1,7], T[1,8] ], + [ T[2,5], T[2,6], T[2,7], T[2,8] ], + [ T[3,5], T[3,6], T[3,7], T[3,8] ], + [ T[4,5], T[4,6], T[4,7], T[4,8] ] +); + +mod(A.C, 7); + +b0: C[1,1]; +b1: C[2,1]; +b2: C[3,1]; +b3: C[4,1]; + +Cc: mod(b0*M0 + b1*M1 + b2*M2 + b3*M3, 7); +C - Cc; + +b: b0 + b1*X + b2*X^2 + b3*X^3; +p: expand(a*b); + +p: expand(p - 5 * m * X^3); +p: expand(p - 40 * m * X^2); +p: expand(p + 35 * m * X); +p: expand(p + 9 * m); + +mod(28 * M0 + 125*M1 - 18*M2,7); diff --git a/buch/chapters/30-endlichekoerper/rechnungen/inverse.maxima b/buch/chapters/30-endlichekoerper/rechnungen/inverse.maxima new file mode 100644 index 0000000..5f3682f --- /dev/null +++ b/buch/chapters/30-endlichekoerper/rechnungen/inverse.maxima @@ -0,0 +1,35 @@ +/* + * inverse.maxima + * + * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +n: 5; +m: X^5 + 15*X^3 - 30*X^2 + 45; + +M: matrix( + [ 0, 0, 0, 0, -45 ], + [ 1, 0, 0, 0, 0 ], + [ 0, 1, 0, 0, 30 ], + [ 0, 0, 1, 0, -15 ], + [ 0, 0, 0, 1, 0 ] +); +M2: M.M; +M3: M.M2; +M4: M.M3; + +y: a0 + a1*X + a2*X^2 + a3*X^3 + a4*X^4; +Y: a0*identfor(M) + a1*M + a2*M2 + a3*M3 + a4*M4; + +B: invert(Y); + +b0: B[1,1]; +b1: B[2,1]; +b2: B[3,1]; +b3: B[4,1]; +b4: B[5,1]; + +Z: b0*identfor(M) + b1*M + b2*M2 + b3*M3 + b4*M4; +z: b0 + b1*X + b2*X^2 + b3*X^3 + b4*X^4; + +w: expand(y*z); +remainder(w, m, X); diff --git a/buch/chapters/30-endlichekoerper/rechnungen/multiplikation.maxima b/buch/chapters/30-endlichekoerper/rechnungen/multiplikation.maxima new file mode 100644 index 0000000..e09f848 --- /dev/null +++ b/buch/chapters/30-endlichekoerper/rechnungen/multiplikation.maxima @@ -0,0 +1,38 @@ +/* + * multiplikation.maxima + * + * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ + +Malpha: matrix( +[ 0, 0, 0, 0, 0, 0, -m0 ], +[ 1, 0, 0, 0, 0, 0, -m1 ], +[ 0, 1, 0, 0, 0, 0, -m2 ], +[ 0, 0, 1, 0, 0, 0, -m3 ], +[ 0, 0, 0, 1, 0, 0, -m4 ], +[ 0, 0, 0, 0, 1, 0, -m5 ], +[ 0, 0, 0, 0, 0, 1, -m6 ] +); + +Malpha2: expand(Malpha . Malpha); +Malpha3: expand(Malpha . Malpha2); +Malpha4: expand(Malpha . Malpha3); +Malpha5: expand(Malpha . Malpha4); +Malpha6: expand(Malpha . Malpha5); +Malpha7: expand(Malpha . Malpha6); +Malpha8: expand(Malpha . Malpha7); + +p: m0 * identfor(Malpha) ++ m1 * Malpha ++ m2 * Malpha2 ++ m3 * Malpha3 ++ m4 * Malpha4 ++ m5 * Malpha5 ++ m6 * Malpha6 ++ Malpha7; +expand(p); + + +m(X) := m0 + m1*X + m2*X^2 + m3*X^3 + m4*X^4 + m5*X^5 + m6*X^6 + X^7; + +invert(Malpha); |