Added some optimizations

erasure_code/ec65536/poly_utils.cpp

@@ -36,19 +36,21 @@ int eval_poly_at(vector<int> poly, int x) {
 }
 
 vector<int> lagrange_interp(vector<int> ys, vector<int> xs) {
-    deque<int> root;
-    root.push_back(1);
-    for (int i = 0; i < xs.size(); i++) {
+    int xs_size = xs.size();
+    vector<int> root(xs_size + 1);
+    root[xs_size] = 1;
+    for (int i = 0; i < xs_size; i++) {
         int logx = glogtable[xs[i]];
-        root.push_front(0);
-        for (int j = 0; j < root.size() - 1; j++) {
-            if (root[j + 1] and xs[i])
-                root[j] ^= gexptable[glogtable[root[j+1]] + logx];
+        int offset = xs_size - i - 1;
+        root[offset] = 0;
+        for (int j = 0; j < i + 1; j++) {
+            if (root[j + 1 + offset] and xs[i])
+                root[j + offset] ^= gexptable[glogtable[root[j+1 + offset]] + logx];
         }
     }
-    vector<vector<int> > nums;
-    for (int i = 0; i < xs.size(); i++) {
+    vector<int> b(xs_size);
     vector<int> output(root.size() - 1);
+    for (int i = 0; i < xs_size; i++) {
         output[root.size() - 2] = 1;
         int logx = glogtable[xs[i]];
         for (int j = root.size() - 2; j > 0; j--) { 
@@ -57,18 +59,11 @@ vector<int> lagrange_interp(vector<int> ys, vector<int> xs) {
             else
                 output[j - 1] = root[j];
         }
-        nums.push_back(output);
-    }
-    vector<int> denoms;
-    for (int i = 0; i < xs.size(); i++) {
-        denoms.push_back(eval_poly_at(nums[i], xs[i]));
-    }
-    vector<int> b(xs.size());
-    for (int i = 0; i < xs.size(); i++) {
-        int log_yslice = glogtable[ys[i]] - glogtable[denoms[i]] + 65535;
-        for (int j = 0; j < xs.size(); j++) {
-            if(nums[i][j] and ys[i])
-                b[j] ^= gexptable[glogtable[nums[i][j]] + log_yslice];
+        int denom = eval_poly_at(output, xs[i]);
+        int log_yslice = glogtable[ys[i]] - glogtable[denom] + 65535;
+        for (int j = 0; j < xs_size; j++) {
+            if(output[j] and ys[i])
+                b[j] ^= gexptable[glogtable[output[j]] + log_yslice];
         }
     }
     return b;

erasure_code/ec65536/poly_utils.py

@@ -70,8 +70,9 @@ def lagrange_interp(pieces, xs):
         for j in range(len(root)-1):
             if root[j+1] and x:
                 root[j] ^= gexptable[glogtable[root[j+1]] + logx]
+    #print(root)
     assert len(root) == len(pieces) + 1
-    print(root)
+    # print(root)
     # Generate per-value numerator polynomials, eg. for x=x2,
     # (x - x1) * (x - x3) * ... * (x - xn), by dividing the master
     # polynomial back by each x coordinate
@@ -86,7 +87,7 @@ def lagrange_interp(pieces, xs):
                 output[j-1] = root[j]
         assert len(output) == len(pieces)
         nums.append(output)
-    print(nums)
+    #print(nums)
     # Generate denominators by evaluating numerator polys at each x
     denoms = [eval_poly_at(nums[i], xs[i]) for i in range(len(xs))]
     # Generate output polynomial, which is the sum of the per-value numerator

erasure_code/ec65536/subquadratic_poly_utils.py

@@ -0,0 +1,193 @@
+modulus_poly = [1, 0, 0, 0, 0, 0, 0, 0,
+                0, 0, 1, 0, 1, 0, 0, 1,
+                1]
+modulus_poly_as_int = sum([(v << i) for i, v in enumerate(modulus_poly)])
+degree = len(modulus_poly) - 1
+
+two_to_the_degree = 2**degree
+two_to_the_degree_m1 = 2**degree - 1
+
+def galoistpl(a):
+    # 2 is not a primitive root, so we have to use 3 as our logarithm base
+    if a * 2 < two_to_the_degree:
+        return (a * 2) ^ a
+    else:
+        return (a * 2) ^ a ^ modulus_poly_as_int
+
+# Precomputing a log table for increased speed of addition and multiplication
+glogtable = [0] * (two_to_the_degree)
+gexptable = []
+v = 1
+for i in range(two_to_the_degree_m1):
+    glogtable[v] = i
+    gexptable.append(v)
+    v = galoistpl(v)
+
+gexptable += gexptable + gexptable
+
+# Add two values in the Galois field
+def galois_add(x, y):
+    return x ^ y
+
+# In binary fields, addition and subtraction are the same thing
+galois_sub = galois_add
+
+# Multiply two values in the Galois field
+def galois_mul(x, y):
+    return 0 if x*y == 0 else gexptable[glogtable[x] + glogtable[y]]
+
+# Divide two values in the Galois field
+def galois_div(x, y):
+    return 0 if x == 0 else gexptable[(glogtable[x] - glogtable[y]) % two_to_the_degree_m1]
+
+# Evaluate a polynomial at a point
+def eval_poly_at(p, x):
+    if x == 0:
+        return p[0]
+    y = 0
+    logx = glogtable[x]
+    for i, p_coeff in enumerate(p):
+        if p_coeff:
+            # Add x**i * coeff
+            y ^= gexptable[(logx * i + glogtable[p_coeff]) % two_to_the_degree_m1]
+    return y
+
+
+# Given p+1 y values and x values with no errors, recovers the original
+# p+1 degree polynomial.
+# Lagrange interpolation works roughly in the following way.
+# 1. Suppose you have a set of points, eg. x = [1, 2, 3], y = [2, 5, 10]
+# 2. For each x, generate a polynomial which equals its corresponding
+#    y coordinate at that point and 0 at all other points provided.
+# 3. Add these polynomials together.
+
+def lagrange_interp(pieces, xs):
+    # Generate master numerator polynomial, eg. (x - x1) * (x - x2) * ... * (x - xn)
+    root = mk_root_2(xs)
+    #print(root)
+    assert len(root) == len(pieces) + 1
+    # print(root)
+    # Generate the derivative
+    d = derivative(root)
+    # Generate denominators by evaluating numerator polys at each x
+    denoms = multi_eval_2(d, xs)
+    # denoms = [eval_poly_at(d, xs[i]) for i in range(len(xs))]
+    # Generate output polynomial, which is the sum of the per-value numerator
+    # polynomials rescaled to have the right y values
+    return multi_root_derive(xs, [galois_div(p, d) for p, d in zip(pieces, denoms)])
+
+def multi_root_derive(xs, muls):
+    if len(xs) == 1:
+        return [muls[0]]
+    R1 = mk_root_2(xs[:len(xs) // 2])
+    R2 = mk_root_2(xs[len(xs) // 2:])
+    x1 = karatsuba_mul(R1, multi_root_derive(xs[len(xs) // 2:], muls[len(muls) // 2:]) + [0])
+    x2 = karatsuba_mul(R2, multi_root_derive(xs[:len(xs) // 2], muls[:len(muls) // 2]) + [0])
+    o = [v1 ^ v2 for v1, v2 in zip(x1, x2)][:len(xs)]
+    # print(len(R1), len(x1), len(xs), len(o))
+    return o
+
+def multi_root_derive_1(xs, muls):
+    o = [0] * len(xs)
+    for i in range(len(xs)):
+        _xs = xs[:i] + xs[(i+1):]
+        root = mk_root_2(_xs)
+        for j in range(len(root)):
+            o[j] ^= galois_mul(root[j], muls[i])
+    return o
+
+a = 124
+b = 8932
+c = 12415
+
+assert galois_mul(galois_add(a, b), c) == galois_add(galois_mul(a, c), galois_mul(b, c))
+
+def karatsuba_mul(p1, p2):
+    L = len(p1)
+    # assert L == len(p2)
+    if L <= 16:
+        o = [0] * (L * 2)
+        for i, v1 in enumerate(p1):
+            for j, v2 in enumerate(p2):
+                if v1 and v2:
+                    o[i + j] ^= gexptable[glogtable[v1] + glogtable[v2]]
+        return o
+    if L % 2:
+        p1 = p1 + [0]
+        p2 = p2 + [0]
+        L += 1
+    halflen = L // 2
+    low1 = p1[:halflen]
+    high1 = p1[halflen:]
+    sum1 = [l ^ h for l, h in zip(low1, high1)]
+    low2 = p2[:halflen]
+    high2 = p2[halflen:]
+    sum2 = [l ^ h for l, h in zip(low2, high2)]
+    z2 = karatsuba_mul(high1, high2)
+    z0 = karatsuba_mul(low1, low2)
+    z1 = [m ^ _z0 ^ _z2 for m, _z0, _z2 in zip(karatsuba_mul(sum1, sum2), z0, z2)]
+    o = z0[:halflen] + \
+        [a ^ b for a, b in zip(z0[halflen:], z1[:halflen])] + \
+        [a ^ b for a, b in zip(z2[:halflen], z1[halflen:])] + \
+        z2[halflen:]
+    return o
+
+def mk_root_1(xs):
+    root = [1]
+    for x in xs:
+        logx = glogtable[x]
+        root.insert(0, 0)
+        for j in range(len(root)-1):
+            if root[j+1] and x:
+                root[j] ^= gexptable[glogtable[root[j+1]] + logx]
+    return root
+
+def mk_root_2(xs):
+    if len(xs) >= 128:
+        return karatsuba_mul(mk_root_2(xs[:len(xs) // 2]), mk_root_2(xs[len(xs) // 2:]))[:len(xs) + 1]
+    root = [1]
+    for x in xs:
+        logx = glogtable[x]
+        root.insert(0, 0)
+        for j in range(len(root)-1):
+            if root[j+1] and x:
+                root[j] ^= gexptable[glogtable[root[j+1]] + logx]
+    return root
+
+def derivative(root):
+    return [0 if i % 2 else r for i, r in enumerate(root[1:])]
+
+# Credit to http://people.csail.mit.edu/madhu/ST12/scribe/lect06.pdf for the algorithm
+def xn_mod_poly(p):
+    if len(p) == 1:
+        return [galois_div(1, p[0])]
+    halflen = len(p) // 2
+    lowinv = xn_mod_poly(p[:halflen])
+    submod_high = karatsuba_mul(lowinv, p[:halflen])[halflen:]
+    med = karatsuba_mul(p[halflen:], lowinv)[:halflen]
+    med_plus_high = [x ^ y for x, y in zip(med, submod_high)]
+    highinv = karatsuba_mul(med_plus_high, lowinv)
+    o = (lowinv + highinv)[:len(p)]
+    # assert karatsuba_mul(o, p)[:len(p)] == [1] + [0] * (len(p) - 1)
+    return o
+
+def mod(a, b):
+    assert len(a) == 2 * (len(b) - 1)
+    L = len(b)
+    inv_rev_b = xn_mod_poly(b[::-1] + [0] * (len(a) - L))[:L]
+    quot = karatsuba_mul(inv_rev_b, a[::-1][:L])[:L-1][::-1]
+    subt = karatsuba_mul(b, quot + [0])[:-1]
+    o = [x ^ y for x, y in zip(a[:L-1], subt[:L-1])]
+    # assert [x^y for x, y in zip(karatsuba_mul(quot + [0], b), o)] == a
+    return o
+
+def multi_eval_1(poly, xs):
+    return [eval_poly_at(poly, x) for x in xs]
+
+def multi_eval_2(poly, xs):
+    if len(xs) <= 1024:
+        return [eval_poly_at(poly, x) for x in xs]
+    halflen = len(xs) // 2
+    return multi_eval_2(mod(poly, mk_root_2(xs[:halflen])), xs[:halflen]) + \
+           multi_eval_2(mod(poly, mk_root_2(xs[halflen:])), xs[halflen:])
+           # [eval_poly_at(poly, xs[-2]), eval_poly_at(poly, xs[-1])]

iceage.py

@@ -1,12 +1,12 @@
 import random
 import datetime
 
-diffs = [257.74 * 10**12]
-hashpower = diffs[0] / 14.7
-times = [1491920186]
+diffs = [286.86 * 10**12]
+hashpower = diffs[0] / 15.0
+times = [1492650109]
 
 
-for i in range(3517029, 6010000):
+for i in range(3566076, 6010000):
     blocktime = random.expovariate(hashpower / diffs[-1])
     adjfac = max(1 - int(blocktime / 10), -99) / 2048.
     newdiff = diffs[-1] * (1 + adjfac)