English suggestions for hidden_shift

algorithms/algebraic/hidden_shift/hidden_shift.ipynb

@@ -7,23 +7,23 @@
     "tags": []
    },
    "source": [
-    "# Hidden-Shift problem for bent functions using the classiq platform"
+    "# Hidden-Shift Problem for Bent Functions"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "450fbc3a-2837-44e7-90bc-e695a87ba220",
    "metadata": {},
    "source": [
-    "Here we implement the hidden shift algorithm for the familty of boolean bent functions."
+    "Here we implement the hidden shift algorithm for the family of Boolean bent functions using the Classiq platform."
    ]
   },
   {
    "cell_type": "markdown",
    "id": "a13a1783-065a-46aa-9967-ba2d5affd33e",
    "metadata": {},
    "source": [
-    "First, make sure we have all necessary packages:"
+    "Make sure we have all necessary packages:"
    ]
   },
   {
@@ -51,7 +51,7 @@
     "tags": []
    },
    "source": [
-    "On the first part, we assume we know how to implement the dual of $f$, and get $s$ according to the algorithm in [[1](#first)]:![Screen Shot 2023-06-27 at 18.05.48.png](attachment:663333d5-eb52-4150-a00d-8683e816d860.png)"
+    "We assume we know how to implement the dual of $f$ and get $s$ according to the algorithm in [[1](#first)]:![Screen Shot 2023-06-27 at 18.05.48.png](attachment:663333d5-eb52-4150-a00d-8683e816d860.png)"
    ]
   },
   {
@@ -63,9 +63,7 @@
     {
      "name": "stdout",
      "output_type": "stream",
-     "text": [
-      ""
-     ]
+     "text": []
     }
    ],
    "source": [
@@ -143,7 +141,7 @@
    "id": "64d54218-7d9c-420a-92ff-1f719d2aa638",
    "metadata": {},
    "source": [
-    "# More complex functions"
+    "## More Complex Functions"
    ]
   },
   {
@@ -239,7 +237,7 @@
    "id": "c27a578d-5380-460d-9833-15557d4e292a",
    "metadata": {},
    "source": [
-    "## Now create the ciruit:"
+    "## Creating the Circuit"
    ]
   },
   {
@@ -253,8 +251,7 @@
      "output_type": "stream",
      "text": [
       "f_dual: (((((((((x[5]) & (y[0])) ^ ((x[2]) & (y[1]))) ^ ((x[7]) & (y[2]))) ^ ((x[0]) & (y[3]))) ^ ((x[6]) & (y[4]))) ^ ((x[3]) & (y[5]))) ^ ((x[1]) & (y[6]))) ^ ((x[4]) & (y[7]))) ^ ((((((((x[5]) & (x[2])) & (x[7])) & (x[0])) & (x[6])) & (x[3])) & (x[1])) & (x[4]))\n",
-      "g: (((((((((x[0]) & (y[3])) ^ (((x[1]) ^ 1) & (y[6]))) ^ ((x[2]) & ((y[1]) ^ 1))) ^ (((x[3]) ^ 1) & (y[5]))) ^ ((x[4]) & (y[7]))) ^ ((x[5]) & (y[0]))) ^ ((x[6]) & (y[4]))) ^ ((x[7]) & (y[2]))) ^ ((((((((y[0]) & ((y[1]) ^ 1)) & (y[2])) & (y[3])) & (y[4])) & (y[5])) & (y[6])) & (y[7]))\n",
-      ""
+      "g: (((((((((x[0]) & (y[3])) ^ (((x[1]) ^ 1) & (y[6]))) ^ ((x[2]) & ((y[1]) ^ 1))) ^ (((x[3]) ^ 1) & (y[5]))) ^ ((x[4]) & (y[7]))) ^ ((x[5]) & (y[0]))) ^ ((x[6]) & (y[4]))) ^ ((x[7]) & (y[2]))) ^ ((((((((y[0]) & ((y[1]) ^ 1)) & (y[2])) & (y[3])) & (y[4])) & (y[5])) & (y[6])) & (y[7]))\n"
      ]
     }
    ],
@@ -359,7 +356,7 @@
     "tags": []
    },
    "source": [
-    "# Hidden Shift without the dual function"
+    "## Hidden Shift Without the Dual Function"
    ]
   },
   {
@@ -372,7 +369,7 @@
    "id": "0bf70e08-3ea7-43b8-afc2-0271f124323f",
    "metadata": {},
    "source": [
-    "We now use the second algorithm described in [[2](#second)]. This algorithm only requires to implement $f$ and not its dual, however requires $O(n)$ samples from the circuit.\n",
+    "We now use the second algorithm described in [[2](#second)]. This algorithm only requires implementing $f$ and not its dual; however, it requires $O(n)$ samples from the circuit.\n",
     "![Screen Shot 2023-06-27 at 18.08.23.png](attachment:e8a93a2f-8965-4181-9083-78e12dc0f48b.png)"
    ]
   },
@@ -389,8 +386,7 @@
      "output_type": "stream",
      "text": [
       "f: (((((((((x[0]) & (y[3])) ^ ((x[1]) & (y[6]))) ^ ((x[2]) & (y[1]))) ^ ((x[3]) & (y[5]))) ^ ((x[4]) & (y[7]))) ^ ((x[5]) & (y[0]))) ^ ((x[6]) & (y[4]))) ^ ((x[7]) & (y[2]))) ^ ((((((((y[0]) & (y[1])) & (y[2])) & (y[3])) & (y[4])) & (y[5])) & (y[6])) & (y[7]))\n",
-      "g: (((((((((x[0]) & (y[3])) ^ (((x[1]) ^ 1) & (y[6]))) ^ ((x[2]) & ((y[1]) ^ 1))) ^ (((x[3]) ^ 1) & (y[5]))) ^ ((x[4]) & (y[7]))) ^ ((x[5]) & (y[0]))) ^ ((x[6]) & (y[4]))) ^ ((x[7]) & (y[2]))) ^ ((((((((y[0]) & ((y[1]) ^ 1)) & (y[2])) & (y[3])) & (y[4])) & (y[5])) & (y[6])) & (y[7]))\n",
-      ""
+      "g: (((((((((x[0]) & (y[3])) ^ (((x[1]) ^ 1) & (y[6]))) ^ ((x[2]) & ((y[1]) ^ 1))) ^ (((x[3]) ^ 1) & (y[5]))) ^ ((x[4]) & (y[7]))) ^ ((x[5]) & (y[0]))) ^ ((x[6]) & (y[4]))) ^ ((x[7]) & (y[2]))) ^ ((((((((y[0]) & ((y[1]) ^ 1)) & (y[2])) & (y[3])) & (y[4])) & (y[5])) & (y[6])) & (y[7]))\n"
      ]
     }
    ],
@@ -451,7 +447,7 @@
    "metadata": {},
    "source": [
     "Out of the sampled results, we look for $n$ independent samples, from which we can extract s.\n",
-    "1000 samples should be enough with a very high probability."
+    "One thousand samples should be enough with a very high probability."
    ]
   },
   {
@@ -464,10 +460,10 @@
    "outputs": [],
    "source": [
     "# The galois library is a package that extends NumPy arrays to operate over finite fields.\n",
-    "# we wlll use it as our equations are binary equations\n",
+    "# we will use it as our equations are binary equations\n",
     "import galois\n",
     "\n",
-    "# here we work over boolean arithmetics - F(2)\n",
+    "# here we work over Boolean arithmetics - F(2)\n",
     "GF = galois.GF(2)\n",
     "\n",
     "\n",
@@ -507,7 +503,7 @@
     "tags": []
    },
    "source": [
-    "We now left with solving the equation and extracting $s$:"
+    "We now solve the equation and extract $s$:"
    ]
   },
   {
@@ -543,7 +539,7 @@
    "id": "2e47bb31-04e8-4698-b42e-13ca9722018e",
    "metadata": {},
    "source": [
-    "And we got successfully the same shift."
+    "And we successfully received the same shift."
    ]
   },
   {