1 files changed, 144 insertions, 76 deletions

diff --git a/util/plot_dsa.ipynb b/util/plot_dsa.ipynb
index 503bde8..6fa5663 100644
--- a/util/plot_dsa.ipynb
+++ b/util/plot_dsa.ipynb
@@ -12,8 +12,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:08:10.526799Z",
-     "start_time": "2019-03-18T18:08:10.073972Z"
+     "end_time": "2019-03-21T16:47:51.573214Z",
+     "start_time": "2019-03-21T16:47:50.777035Z"
     }
    },
    "outputs": [],
@@ -27,7 +27,7 @@
     "import matplotlib.pyplot as plt\n",
     "from matplotlib import ticker, colors, gridspec\n",
     "from copy import deepcopy\n",
-    "from utils import plot_hist, moving_average, hw, time_scale\n",
+    "from utils import plot_hist, moving_average, hw, time_scale, hist_size_func\n",
     "from binascii import unhexlify\n",
     "from IPython.display import display, HTML\n",
     "from ipywidgets import interact, interactive, fixed, interact_manual\n",
@@ -48,8 +48,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:15:54.067732Z",
-     "start_time": "2019-03-18T18:15:54.063679Z"
+     "end_time": "2019-03-21T16:47:51.578911Z",
+     "start_time": "2019-03-21T16:47:51.575142Z"
     }
    },
    "outputs": [],
@@ -67,7 +67,7 @@
     "scaling_factor = 1\n",
     "\n",
     "# The amount of entries skipped from the beginning of the file, as they are usually outliers.\n",
-    "skip_first = 100\n",
+    "skip_first = 10\n",
     "\n",
     "# Whether to plot things in logarithmic scale or not.\n",
     "log_scale = False\n",
@@ -103,8 +103,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:15:55.985799Z",
-     "start_time": "2019-03-18T18:15:55.495414Z"
+     "end_time": "2019-03-21T16:47:55.190977Z",
+     "start_time": "2019-03-21T16:47:52.428880Z"
     }
    },
    "outputs": [],
@@ -177,21 +177,8 @@
     "byte_size = (bit_size + 7) // 8\n",
     "bit_size = byte_size * 8\n",
     "\n",
-    "if hist_size == \"sqrt\":\n",
-    "    hist_size_func = lambda n, xmin, xmax, var, xlower, xupper: int(sqrt(n)) + 1\n",
-    "elif hist_size == \"sturges\":\n",
-    "    hist_size_func = lambda n, xmin, xmax, var, xlower, xupper: int(log(n, 2)) + 1\n",
-    "elif hist_size == \"rice\":\n",
-    "    hist_size_func = lambda n, xmin, xmax, var, xlower, xupper: int(2 * n**(1/3))\n",
-    "elif hist_size == \"scott\":\n",
-    "    hist_size_func = lambda n, xmin, xmax, var, xlower, xupper: (xmax - xmin) // int((3.5 * sqrt(var)) / (n**(1/3)))\n",
-    "elif hist_size == \"fd\":\n",
-    "    hist_size_func = lambda n, xmin, xmax, var, xlower, xupper: (xmax - xmin) // int(2 * (xupper - xlower) / (n**(1/3)))\n",
-    "else:\n",
-    "    hist_size_func = lambda n, xmin, xmax, var, xlower, xupper: hist_size\n",
-    "\n",
-    "hist_size_sign_time = hist_size_func(description_sign.nobs, min_sign_time, max_sign_time, description_sign.variance, quantiles_sign[1], quantiles_sign[3])\n",
-    "hist_size_sign_time_trim = hist_size_func(description_sign_trim.nobs, description_sign_trim.minmax[0], description_sign_trim.minmax[1], description_sign_trim.variance, quantiles_sign_trim[1], quantiles_sign_trim[3])\n",
+    "hist_size_sign_time = hist_size_func(hist_size)(description_sign.nobs, min_sign_time, max_sign_time, description_sign.variance, quantiles_sign[1], quantiles_sign[3])\n",
+    "hist_size_sign_time_trim = hist_size_func(hist_size)(description_sign_trim.nobs, description_sign_trim.minmax[0], description_sign_trim.minmax[1], description_sign_trim.variance, quantiles_sign_trim[1], quantiles_sign_trim[3])\n",
     "\n",
     "if hist_size_sign_time < 30:\n",
     "    hist_size_sign_time = max_sign_time - min_sign_time\n",
@@ -218,8 +205,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:15:57.175564Z",
-     "start_time": "2019-03-18T18:15:57.161611Z"
+     "end_time": "2019-03-21T16:18:19.188665Z",
+     "start_time": "2019-03-21T16:18:19.164737Z"
     }
    },
    "outputs": [],
@@ -246,8 +233,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:15:58.257820Z",
-     "start_time": "2019-03-18T18:15:58.254036Z"
+     "end_time": "2019-03-21T16:18:20.110503Z",
+     "start_time": "2019-03-21T16:18:20.099333Z"
     }
    },
    "outputs": [],
@@ -269,13 +256,15 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:15:58.917927Z",
-     "start_time": "2019-03-18T18:15:58.909693Z"
+     "end_time": "2019-03-21T16:18:20.768858Z",
+     "start_time": "2019-03-21T16:18:20.758881Z"
     }
    },
    "outputs": [],
    "source": [
-    "display(\"Bitsize:\", bit_size)"
+    "display(\"Bitsize:\", bit_size)\n",
+    "display(\"Histogram time bins: {}\".format(hist_size_sign_time))\n",
+    "display(\"Histogram time bins(trimmed): {}\".format(hist_size_sign_time_trim))"
    ]
   },
   {
@@ -298,14 +287,14 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:15:59.977656Z",
-     "start_time": "2019-03-18T18:15:59.926337Z"
+     "end_time": "2019-03-21T16:18:22.549846Z",
+     "start_time": "2019-03-21T16:18:22.332677Z"
     }
    },
    "outputs": [],
    "source": [
     "fig_nonce = plt.figure(figsize=(10.5, 8), dpi=90)\n",
-    "axe_nonce = fig_nonce.add_subplot(1, 1, 1)\n",
+    "axe_nonce = fig_nonce.add_subplot(1, 1, 1, title=\"Nonce MSB vs signature time\")\n",
     "nonce_msb = np.array(list(map(lambda x: x >> (bit_size - 8), data_trimmed[\"nonce\"])), dtype=np.dtype(\"u1\"))\n",
     "max_msb = max(nonce_msb)\n",
     "min_msb = min(nonce_msb)\n",
@@ -314,10 +303,11 @@
     "extent = [min_msb, max_msb, yedges[0], yedges[-1]]\n",
     "im = axe_nonce.imshow(heatmap.T, extent=extent, aspect=\"auto\", cmap=cmap, origin=\"low\",\n",
     "                   interpolation=\"nearest\", norm=norm)\n",
-    "axe_nonce.set_xlabel(\"nonce key MSB value\")\n",
+    "axe_nonce.set_xlabel(\"nonce MSB value\")\n",
     "axe_nonce.set_ylabel(\"signature time ({})\".format(sign_disp_unit))\n",
     "fig_nonce.colorbar(im, ax=axe_nonce)\n",
     "\n",
+    "fig_nonce.tight_layout()\n",
     "del nonce_msb"
    ]
   },
@@ -336,16 +326,16 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:01.977710Z",
-     "start_time": "2019-03-18T18:16:01.717704Z"
+     "end_time": "2019-03-21T16:18:26.230002Z",
+     "start_time": "2019-03-21T16:18:24.323208Z"
     }
    },
    "outputs": [],
    "source": [
     "fig_nonce_hist = plt.figure(figsize=(10.5, 12), dpi=90)\n",
     "gs = gridspec.GridSpec(2, 1, height_ratios=[2.5, 1])\n",
-    "axe_nonce_hist = fig_nonce_hist.add_subplot(gs[0])\n",
-    "axe_nonce_hist_hw = fig_nonce_hist.add_subplot(gs[1], sharex = axe_nonce_hist)\n",
+    "axe_nonce_hist = fig_nonce_hist.add_subplot(gs[0], title=\"Nonce Hamming weight vs signature time\")\n",
+    "axe_nonce_hist_hw = fig_nonce_hist.add_subplot(gs[1], sharex=axe_nonce_hist, title=\"Nonce Hamming weight\")\n",
     "nonce_hw = np.array(list(map(hw, data_trimmed[\"nonce\"])), dtype=np.dtype(\"u2\"))\n",
     "h, xe, ye = np.histogram2d(nonce_hw, data_trimmed[\"sign_time\"], bins=[max(nonce_hw) - min(nonce_hw), hist_size_sign_time_trim])\n",
     "im = axe_nonce_hist.imshow(h.T, origin=\"low\", cmap=cmap, aspect=\"auto\", extent=[xe[0], xe[-1], ye[0], ye[-1]], norm=norm)\n",
@@ -361,11 +351,13 @@
     "norm_pdf = norm_dist.pdf(pdf_range, *param[:-2], loc=param[-2], scale=param[-1]) * description_sign_trim.nobs\n",
     "axe_nonce_hist_hw.plot(pdf_range, norm_pdf, label=\"fitted normal distribution\")\n",
     "axe_nonce_hist_hw.legend(loc=\"best\")\n",
-    "fig_nonce_hist.colorbar(im, ax=[axe_nonce_hist, axe_nonce_hist_hw])\n",
+    "\n",
     "\n",
     "display(HTML(\"<b>Nonce Hamming weight fitted with normal distribution:</b>\"))\n",
     "display(HTML(tabulate.tabulate([(\"Mean\", \"Variance\"), param], tablefmt=\"html\")))\n",
     "\n",
+    "fig_nonce_hist.tight_layout()\n",
+    "fig_nonce_hist.colorbar(im, ax=[axe_nonce_hist, axe_nonce_hist_hw])\n",
     "del nonce_hw"
    ]
   },
@@ -381,17 +373,18 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:03.232728Z",
-     "start_time": "2019-03-18T18:16:03.134237Z"
+     "end_time": "2019-03-21T16:18:48.188494Z",
+     "start_time": "2019-03-21T16:18:39.850301Z"
     }
    },
    "outputs": [],
    "source": [
     "fig_sig_hist = plt.figure(figsize=(10.5, 8), dpi=90)\n",
-    "axe_hist_full = fig_sig_hist.add_subplot(2, 1, 1)\n",
-    "axe_hist_trim = fig_sig_hist.add_subplot(2, 1, 2)\n",
+    "axe_hist_full = fig_sig_hist.add_subplot(2, 1, 1, title=\"Signature time\")\n",
+    "axe_hist_trim = fig_sig_hist.add_subplot(2, 1, 2, title=\"Signature time (trimmed)\")\n",
     "plot_hist(axe_hist_full, data[\"sign_time\"], \"signature time ({})\".format(sign_disp_unit), log_scale, hist_size_sign_time);\n",
-    "plot_hist(axe_hist_trim, data_trimmed[\"sign_time\"], \"signature time ({})\".format(sign_disp_unit), log_scale, hist_size_sign_time_trim);"
+    "plot_hist(axe_hist_trim, data_trimmed[\"sign_time\"], \"signature time ({})\".format(sign_disp_unit), log_scale, hist_size_sign_time_trim);\n",
+    "fig_sig_hist.tight_layout()"
    ]
   },
   {
@@ -406,15 +399,16 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:04.380116Z",
-     "start_time": "2019-03-18T18:16:04.227481Z"
+     "end_time": "2019-03-21T16:19:05.618320Z",
+     "start_time": "2019-03-21T16:18:53.161932Z"
     }
    },
    "outputs": [],
    "source": [
     "fig_vrfy_hist = plt.figure(figsize=(10.5, 5), dpi=90)\n",
-    "axe_hist_full = fig_vrfy_hist.add_subplot(1, 1, 1)\n",
-    "plot_hist(axe_hist_full, data[\"verify_time\"], \"verification time ({})\".format(verify_disp_unit), log_scale, hist_size_sign_time);"
+    "axe_hist_full = fig_vrfy_hist.add_subplot(1, 1, 1, title=\"Verification time\")\n",
+    "plot_hist(axe_hist_full, data[\"verify_time\"], \"verification time ({})\".format(verify_disp_unit), log_scale, hist_size_sign_time);\n",
+    "fig_vrfy_hist.tight_layout()"
    ]
   },
   {
@@ -429,15 +423,15 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:05.236199Z",
-     "start_time": "2019-03-18T18:16:05.123540Z"
+     "end_time": "2019-03-21T16:19:56.783127Z",
+     "start_time": "2019-03-21T16:19:56.375647Z"
     }
    },
    "outputs": [],
    "source": [
     "fig_avg = plt.figure(figsize=(10.5, 8), dpi=90)\n",
-    "axe_sign_avg = fig_avg.add_subplot(2, 1, 1)\n",
-    "axe_vrfy_avg = fig_avg.add_subplot(2, 1, 2, sharex=axe_sign_avg)\n",
+    "axe_sign_avg = fig_avg.add_subplot(2, 1, 1, title=\"Moving average of signature time\")\n",
+    "axe_vrfy_avg = fig_avg.add_subplot(2, 1, 2, sharex=axe_sign_avg, title=\"Moving average of verification time\")\n",
     "avg_sign_100 = moving_average(data[\"sign_time\"], 100)\n",
     "avg_sign_1000 = moving_average(data[\"sign_time\"], 1000)\n",
     "axe_sign_avg.plot(avg_sign_100, label=\"window = 100\")\n",
@@ -458,6 +452,8 @@
     "axe_vrfy_avg.set_xlabel(\"index\")\n",
     "axe_vrfy_avg.legend(loc=\"best\")\n",
     "\n",
+    "fig_avg.tight_layout()\n",
+    "\n",
     "del avg_sign_100, avg_sign_1000, avg_vrfy_100, avg_vrfy_1000"
    ]
   },
@@ -474,8 +470,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:06.352067Z",
-     "start_time": "2019-03-18T18:16:06.059476Z"
+     "end_time": "2019-03-21T16:20:00.354254Z",
+     "start_time": "2019-03-21T16:19:59.289152Z"
     }
    },
    "outputs": [],
@@ -483,11 +479,12 @@
     "fig_nonce_hists = plt.figure(figsize=(10.5, 8), dpi=90)\n",
     "nonce_msb = np.array(list(map(lambda x: x >> (bit_size - 8), data[\"nonce\"])), dtype=np.dtype(\"u1\"))\n",
     "nonce_lsb = np.array(list(map(lambda x: x & 0xff, data[\"nonce\"])), dtype=np.dtype(\"u1\"))\n",
-    "axe_msb_n_hist = fig_nonce_hists.add_subplot(2, 1, 1)\n",
-    "axe_lsb_n_hist = fig_nonce_hists.add_subplot(2, 1, 2)\n",
+    "axe_msb_n_hist = fig_nonce_hists.add_subplot(2, 1, 1, title=\"Nonce MSB\")\n",
+    "axe_lsb_n_hist = fig_nonce_hists.add_subplot(2, 1, 2, title=\"Nonce LSB\")\n",
     "plot_hist(axe_msb_n_hist, nonce_msb, \"nonce MSB\", log_scale, False, False)\n",
     "plot_hist(axe_lsb_n_hist, nonce_lsb, \"nonce LSB\", log_scale, False, False)\n",
     "\n",
+    "fig_nonce_hists.tight_layout()\n",
     "del nonce_msb, nonce_lsb"
    ]
   },
@@ -504,16 +501,16 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:07.625289Z",
-     "start_time": "2019-03-18T18:16:07.544334Z"
+     "end_time": "2019-03-21T16:20:02.737798Z",
+     "start_time": "2019-03-21T16:20:02.481429Z"
     }
    },
    "outputs": [],
    "source": [
     "fig_bl = plt.figure(figsize=(10.5, 12), dpi=90)\n",
     "gs = gridspec.GridSpec(2, 1, height_ratios=[2.5, 1])\n",
-    "axe_bl_heat = fig_bl.add_subplot(gs[0])\n",
-    "axe_bl_hist = fig_bl.add_subplot(gs[1], sharex=axe_bl_heat)\n",
+    "axe_bl_heat = fig_bl.add_subplot(gs[0], title=\"Nonce bit length vs signature time\")\n",
+    "axe_bl_hist = fig_bl.add_subplot(gs[1], sharex=axe_bl_heat, title=\"Nonce bit length\")\n",
     "bl_data = np.array(list(map(lambda x: x.bit_length(), data_trimmed[\"nonce\"])), dtype=np.dtype(\"u2\"))\n",
     "\n",
     "h, xe, ye = np.histogram2d(bl_data, data_trimmed[\"sign_time\"], bins=[max(bl_data) - min(bl_data), hist_size_sign_time_trim])\n",
@@ -522,6 +519,8 @@
     "axe_bl_heat.set_ylabel(\"signature time ({})\".format(sign_disp_unit))\n",
     "\n",
     "plot_hist(axe_bl_hist, bl_data, \"nonce bit length\", log_scale, align=\"right\")\n",
+    "\n",
+    "fig_bl.tight_layout()\n",
     "fig_bl.colorbar(im, ax=[axe_bl_heat, axe_bl_hist])\n",
     "\n",
     "del bl_data"
@@ -531,9 +530,58 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "### Nonce bit length histogram given time\n",
+    "Interactively shows the histogram of nonce bit length given a selected time range centered around `center` of width `width`. Ideally, the means of these conditional distributions are equal, while the variances can vary."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2019-03-21T16:20:10.306709Z",
+     "start_time": "2019-03-21T16:20:10.102626Z"
+    },
+    "scrolled": false
+   },
+   "outputs": [],
+   "source": [
+    "fig_bl_time = plt.figure(figsize=(10.5, 5), dpi=90)\n",
+    "axe_bl_time = fig_bl_time.add_subplot(111)\n",
+    "axe_bl_time.set_autoscalex_on(False)\n",
+    "def f(center, width):\n",
+    "    lower_bnd = center - width/2\n",
+    "    upper_bnd = center + width/2\n",
+    "    values = data_trimmed[np.logical_and(data_trimmed[\"sign_time\"] <= upper_bnd,\n",
+    "                                         data_trimmed[\"sign_time\"] >= lower_bnd)]\n",
+    "    axe_bl_time.clear()\n",
+    "    axe_bl_time.set_title(\"Nonce bit length, given signature time $\\in ({}, {})$ {}\".format(int(lower_bnd), int(upper_bnd), sign_disp_unit))\n",
+    "    bl_data = np.array(list(map(lambda x: x.bit_length(), values[\"nonce\"])), dtype=np.dtype(\"u2\"))\n",
+    "    plot_hist(axe_bl_time, bl_data, \"nonce bit length\", bins=11, range=(bit_size-10, bit_size+1), align=\"left\")\n",
+    "    axe_bl_time.set_xlim((bit_size-10, bit_size))\n",
+    "    fig_bl_time.tight_layout()\n",
+    "\n",
+    "center_w = widgets.IntSlider(min=min(data_trimmed[\"sign_time\"]),\n",
+    "                             max=max(data_trimmed[\"sign_time\"]),\n",
+    "                             step=1,\n",
+    "                             value=description_sign_trim.mean,\n",
+    "                             continuous_update=False,\n",
+    "                             description=\"center {}\".format(sign_disp_unit))\n",
+    "width_w = widgets.IntSlider(min=1, max=100, continuous_update=False,\n",
+    "                             description=\"width {}\".format(sign_disp_unit))\n",
+    "w = interactive(f, center=center_w,\n",
+    "                width=width_w)\n",
+    "display(w)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "## Validation\n",
     "Perform some tests on the produced data and compare to expected results.\n",
     "\n",
+    "\n",
     "This requires some information about the used curve, enter it below."
    ]
   },
@@ -542,8 +590,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:48.791656Z",
-     "start_time": "2019-03-18T18:16:45.435426Z"
+     "end_time": "2019-03-21T15:24:57.397880Z",
+     "start_time": "2019-03-21T15:24:37.395614Z"
     }
    },
    "outputs": [],
@@ -557,8 +605,8 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:55.343989Z",
-     "start_time": "2019-03-18T18:16:49.543154Z"
+     "end_time": "2019-03-21T15:25:05.137250Z",
+     "start_time": "2019-03-21T15:24:59.218945Z"
     }
    },
    "outputs": [],
@@ -571,7 +619,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "All of the following tests should pass (e.g. be true):"
+    "All of the following tests should pass (e.g. be true), given a large enough sample and run without the `--fixed` or `-priv/-npriv` flags:"
    ]
   },
   {
@@ -579,20 +627,40 @@
    "execution_count": null,
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2019-03-18T18:16:56.289305Z",
-     "start_time": "2019-03-18T18:16:56.278296Z"
-    }
+     "end_time": "2019-03-21T16:23:08.618543Z",
+     "start_time": "2019-03-21T16:23:08.451827Z"
+    },
+    "scrolled": true
    },
    "outputs": [],
    "source": [
     "max_priv = max(data[\"priv\"])\n",
     "max_nonce = max(data[\"nonce\"])\n",
-    "display(max_priv < r)\n",
-    "display(r <= p or max_priv > p)\n",
-    "display(max_nonce < r)\n",
-    "display(r <= p or max_nonce > p)\n",
-    "display(max_priv.bit_length() == r.bit_length())\n",
-    "display(max_nonce.bit_length() == r.bit_length())"
+    "un = len(np.unique(data[\"priv\"])) != 1\n",
+    "if un:\n",
+    "    print(\"Private keys are smaller than order:\\t\\t\\t\" + str(max_priv < r))\n",
+    "    print(\"Private keys are larger than prime(if order > prime):\\t\" + str(r <= p or max_priv > p))\n",
+    "print(\"Nonces are smaller than order:\\t\\t\\t\\t\" + str(max_nonce < r))\n",
+    "print(\"Nonces are larger than prime(if order > prime):\\t\\t\" + str(r <= p or max_nonce > p))\n",
+    "if un:\n",
+    "    print(\"Private keys reach full bit length of order:\\t\\t\" + str(max_priv.bit_length() == r.bit_length()))\n",
+    "print(\"Nonces reach full bit length of order:\\t\\t\\t\" + str(max_nonce.bit_length() == r.bit_length()))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2019-03-21T16:23:09.355514Z",
+     "start_time": "2019-03-21T16:23:09.315702Z"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "if un:\n",
+    "    print(\"Private key bit length (min, max):\" + str(min(data[\"priv\"]).bit_length()) + \", \" + str(max(data[\"priv\"]).bit_length()))\n",
+    "print(\"Nonce bit length (min, max):\" + str(min(data[\"nonce\"]).bit_length()) + \", \" + str(max(data[\"nonce\"]).bit_length()))"
    ]
   },
   {
@@ -605,8 +673,8 @@
  ],
  "metadata": {
   "@webio": {
-   "lastCommId": "7c4c5d836a8d43e5846df95890bbafa3",
-   "lastKernelId": "b01f6c07-c08b-4348-a503-dc2c9cf1db89"
+   "lastCommId": "9597ec08f0744c0c9aff275fad95d237",
+   "lastKernelId": "e6055c6c-6313-439b-909b-d90592f67d26"
   },
   "hide_input": false,
   "kernelspec": {