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This commit is contained in:
Landon Curt Noll
178
cal/ellip.cal
178
cal/ellip.cal
@@ -9,7 +9,7 @@
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*
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* Calc is distributed in the hope that it will be useful, but WITHOUT
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* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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* Public License for more details.
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*
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* A copy of version 2.1 of the GNU Lesser General Public License is
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@@ -17,16 +17,16 @@
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* received a copy with calc; if not, write to Free Software Foundation, Inc.
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* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*
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* Under source code control: 1990/02/15 01:50:33
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* File existed as early as: before 1990
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* Under source code control: 1990/02/15 01:50:33
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* File existed as early as: before 1990
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*
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* Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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* Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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*/
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/*
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* Attempt to factor numbers using elliptic functions:
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*
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* y^2 = x^3 + a*x + b (mod ellip_N).
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* y^2 = x^3 + a*x + b (mod ellip_N).
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*
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* Many points (x,y) (mod ellip_N) are found that solve the above equation,
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* starting from a trivial solution and 'multiplying' that point together
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@@ -47,13 +47,13 @@
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* Theory and Cryptography" by Neal Koblitz for a good explanation.
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*
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* efactor(iN, ia, B, force)
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* iN is the number to be factored.
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* ia is the initial value of a in the equation, and each successive
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* value of a is an independent attempt at factoring (default 1).
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* B is the limit of the primes that make up the high power that the
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* point is raised to for each factoring attempt (default 100).
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* force is a flag to attempt to factor numbers even if they are
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* thought to already be prime (default false).
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* iN is the number to be factored.
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* ia is the initial value of a in the equation, and each successive
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* value of a is an independent attempt at factoring (default 1).
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* B is the limit of the primes that make up the high power that the
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* point is raised to for each factoring attempt (default 100).
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* force is a flag to attempt to factor numbers even if they are
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* thought to already be prime (default false).
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*
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* Making B larger makes the power the point being raised to contain more
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* prime factors, thus increasing the chance that the order of the point
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@@ -77,114 +77,114 @@
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* of the powers so far.
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*
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* If a factor is found, it is returned and is also saved in the global
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* variable f. The number being factored is also saved in the global
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* variable f. The number being factored is also saved in the global
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* variable ellip_N.
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*/
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obj point {x, y};
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global ellip_N; /* number to factor */
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global ellip_a; /* first coefficient */
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global ellip_b; /* second coefficient */
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global ellip_f; /* found factor */
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global ellip_N; /* number to factor */
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global ellip_a; /* first coefficient */
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global ellip_b; /* second coefficient */
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global ellip_f; /* found factor */
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define efactor(iN, ia, B, force)
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{
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local C, x, p;
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local C, x, p;
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if (!force && ptest(iN, 50))
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return 1;
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if (isnull(B))
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B = 100;
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if (isnull(ia))
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ia = 1;
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obj point x;
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ellip_a = ia;
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ellip_b = -ia;
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ellip_N = iN;
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C = isqrt(ellip_N);
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C = 2 * C + 2 * isqrt(C) + 1;
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ellip_f = 0;
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while (ellip_f == 0) {
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print "A =", ellip_a;
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x.x = 1;
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x.y = 1;
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print 2, x;
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x = x ^ (2 ^ (highbit(C) + 1));
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for (p = 3; ((p < B) && (ellip_f == 0)); p += 2) {
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if (!ptest(p, 1))
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continue;
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print p, x;
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x = x ^ (p ^ ((highbit(C) // highbit(p)) + 1));
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}
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ellip_a++;
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ellip_b--;
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}
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return ellip_f;
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if (!force && ptest(iN, 50))
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return 1;
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if (isnull(B))
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B = 100;
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if (isnull(ia))
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ia = 1;
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obj point x;
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ellip_a = ia;
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ellip_b = -ia;
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ellip_N = iN;
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C = isqrt(ellip_N);
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C = 2 * C + 2 * isqrt(C) + 1;
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ellip_f = 0;
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while (ellip_f == 0) {
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print "A =", ellip_a;
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x.x = 1;
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x.y = 1;
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print 2, x;
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x = x ^ (2 ^ (highbit(C) + 1));
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for (p = 3; ((p < B) && (ellip_f == 0)); p += 2) {
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if (!ptest(p, 1))
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continue;
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print p, x;
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x = x ^ (p ^ ((highbit(C) // highbit(p)) + 1));
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}
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ellip_a++;
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ellip_b--;
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}
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return ellip_f;
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}
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define point_print(p)
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{
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print "(" : p.x : "," : p.y : ")" :;
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print "(" : p.x : "," : p.y : ")" :;
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}
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define point_mul(p1, p2)
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{
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local r, m;
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local r, m;
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if (p2 == 1)
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return p1;
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if (p1 == p2)
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return point_square(`p1);
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obj point r;
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m = (minv(p2.x - p1.x, ellip_N) * (p2.y - p1.y)) % ellip_N;
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if (m == 0) {
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if (ellip_f == 0)
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ellip_f = gcd(p2.x - p1.x, ellip_N);
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r.x = 1;
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r.y = 1;
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return r;
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}
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r.x = (m^2 - p1.x - p2.x) % ellip_N;
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r.y = ((m * (p1.x - r.x)) - p1.y) % ellip_N;
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return r;
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if (p2 == 1)
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return p1;
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if (p1 == p2)
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return point_square(`p1);
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obj point r;
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m = (minv(p2.x - p1.x, ellip_N) * (p2.y - p1.y)) % ellip_N;
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if (m == 0) {
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if (ellip_f == 0)
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ellip_f = gcd(p2.x - p1.x, ellip_N);
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r.x = 1;
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r.y = 1;
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return r;
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}
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r.x = (m^2 - p1.x - p2.x) % ellip_N;
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r.y = ((m * (p1.x - r.x)) - p1.y) % ellip_N;
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return r;
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}
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define point_square(p)
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{
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local r, m;
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local r, m;
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obj point r;
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m = ((3 * p.x^2 + ellip_a) * minv(p.y << 1, ellip_N)) % ellip_N;
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if (m == 0) {
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if (ellip_f == 0)
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ellip_f = gcd(p.y << 1, ellip_N);
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r.x = 1;
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r.y = 1;
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return r;
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}
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r.x = (m^2 - p.x - p.x) % ellip_N;
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r.y = ((m * (p.x - r.x)) - p.y) % ellip_N;
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return r;
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obj point r;
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m = ((3 * p.x^2 + ellip_a) * minv(p.y << 1, ellip_N)) % ellip_N;
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if (m == 0) {
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if (ellip_f == 0)
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ellip_f = gcd(p.y << 1, ellip_N);
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r.x = 1;
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r.y = 1;
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return r;
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}
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r.x = (m^2 - p.x - p.x) % ellip_N;
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r.y = ((m * (p.x - r.x)) - p.y) % ellip_N;
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return r;
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}
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define point_pow(p, pow)
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{
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local bit, r, t;
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local bit, r, t;
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r = 1;
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if (isodd(pow))
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r = p;
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t = p;
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for (bit = 2; ((bit <= pow) && (ellip_f == 0)); bit <<= 1) {
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t = point_square(`t);
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if (bit & pow)
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r = point_mul(`t, `r);
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}
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return r;
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r = 1;
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if (isodd(pow))
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r = p;
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t = p;
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for (bit = 2; ((bit <= pow) && (ellip_f == 0)); bit <<= 1) {
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t = point_square(`t);
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if (bit & pow)
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r = point_mul(`t, `r);
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}
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return r;
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}
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