From 20dd1cafc3047b7eb6f2c6539319ba59f6c506a6 Mon Sep 17 00:00:00 2001
From: sowle <crypto.sowle@gmail.com>
Date: Tue, 15 Dec 2020 23:57:17 +0300
Subject: [PATCH] crypto helpers improved, sc_exp and sc_invert implemented
 (using Fermat little theorem)

---
 tests/unit_tests/crypto_tests.cpp | 591 ++++++++++++++++++++++++++++--
 1 file changed, 566 insertions(+), 25 deletions(-)

diff --git a/tests/unit_tests/crypto_tests.cpp b/tests/unit_tests/crypto_tests.cpp
index ccb4e293..366a1145 100644
--- a/tests/unit_tests/crypto_tests.cpp
+++ b/tests/unit_tests/crypto_tests.cpp
@@ -8,17 +8,61 @@
 
 extern "C" {
 #include "crypto/crypto-ops.h"
+void fe_reduce(fe u, const fe h)
+{
+  /* From fe_frombytes.c */
+
+  // loading changed
+
+  int64_t h0 = h[0];
+  int64_t h1 = h[1];
+  int64_t h2 = h[2];
+  int64_t h3 = h[3];
+  int64_t h4 = h[4];
+  int64_t h5 = h[5];
+  int64_t h6 = h[6];
+  int64_t h7 = h[7];
+  int64_t h8 = h[8];
+  int64_t h9 = h[9];
+  int64_t carry0;
+  int64_t carry1;
+  int64_t carry2;
+  int64_t carry3;
+  int64_t carry4;
+  int64_t carry5;
+  int64_t carry6;
+  int64_t carry7;
+  int64_t carry8;
+  int64_t carry9;
+
+  carry9 = (h9 + (int64_t)(1 << 24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
+  carry1 = (h1 + (int64_t)(1 << 24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
+  carry3 = (h3 + (int64_t)(1 << 24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
+  carry5 = (h5 + (int64_t)(1 << 24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
+  carry7 = (h7 + (int64_t)(1 << 24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
+
+  carry0 = (h0 + (int64_t)(1 << 25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
+  carry2 = (h2 + (int64_t)(1 << 25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
+  carry4 = (h4 + (int64_t)(1 << 25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
+  carry6 = (h6 + (int64_t)(1 << 25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
+  carry8 = (h8 + (int64_t)(1 << 25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
+
+  u[0] = h0;
+  u[1] = h1;
+  u[2] = h2;
+  u[3] = h3;
+  u[4] = h4;
+  u[5] = h5;
+  u[6] = h6;
+  u[7] = h7;
+  u[8] = h8;
+  u[9] = h9;
+
+  /* End fe_frombytes.c */
+
+}
 
-/*
-Input:
-a[0]+256*a[1]+...+256^31*a[31] = a
-b[0]+256*b[1]+...+256^31*b[31] = b
-Output:
-s[0]+256*s[1]+...+256^31*s[31] = (ab) mod l
-where l = 2^252 + 27742317777372353535851937790883648493
-*/
 
-// TODO: make more efficient multiplication
 void sc_mul(unsigned char *s, const unsigned char *a, const unsigned char *b)
 {
   unsigned char c[32];
@@ -29,18 +73,312 @@ void sc_mul(unsigned char *s, const unsigned char *a, const unsigned char *b)
   sc_mulsub(s, neg_a, b, c);
 }
 
-
 } // extern "C"
 
 
+unsigned char Lm2[32] = { 0xeb, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x0, 0x10 };
+
+// out = z ^ s (mod l)
+void sc_exp(unsigned char* out, const unsigned char* z, const unsigned char* s)
+{
+  sc_0(out);
+  out[0] = 1;
+  if (sc_isnonzero(s) == 0)
+  {
+    return;
+  }
+
+  // calc position of the most significant bit of s
+  size_t msb_s = 0;
+  for (size_t i = 31; i != SIZE_MAX; --i)
+  {
+    if (s[i] != 0)
+    {
+      msb_s = 8 * i;
+      uint8_t t = s[i] >> 1;
+      while (t)
+      {
+        t >>= 1;
+        ++msb_s;
+      }
+      break;
+    }
+  }
+
+  //memcpy(out, z, sizeof(crypto::ec_scalar));
+
+  for (size_t i = msb_s; i != SIZE_MAX; --i)
+  {
+    sc_mul(out, out, out);
+    std::cout << "sc_mul(out, out, out);" << std::endl;
+    uint8_t bit = (s[i / 8] >> (i % 8)) & 1;
+    if (bit)
+    {
+      sc_mul(out, out, z);
+      std::cout << "sc_mul(out, out, z);" << std::endl;
+    }
+  }
+}
+
+void sc_invert(unsigned char* out, const unsigned char* z)
+{
+  memcpy(out, z, sizeof(crypto::ec_scalar));
+  for(size_t i = 0; i < 128; ++i)
+    sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+  sc_mul(out, out, out);
+  sc_mul(out, out, z);
+}
+
+
+
+
+template<class pod_t>
+std::string pod_to_hex_big_endian(const pod_t &h)
+{
+  constexpr char hexmap[] = "0123456789abcdef";
+  const char* data = reinterpret_cast<const char*>(&h);
+  size_t len = sizeof h;
+
+  std::string s(len * 2, ' ');
+  for (size_t i = 0; i < len; ++i) {
+    s[2 * i]     = hexmap[(data[len - 1 - i] & 0xF0) >> 4];
+    s[2 * i + 1] = hexmap[(data[len - 1 - i] & 0x0F)];
+  }
+
+  return s;
+}
+
+
+int fe_cmp(const fe a, const fe b)
+{
+  for (size_t i = 9; i != SIZE_MAX; --i)
+  {
+    if (reinterpret_cast<const uint32_t&>(a[i]) < reinterpret_cast<const uint32_t&>(b[i])) return -1;
+    if (reinterpret_cast<const uint32_t&>(a[i]) > reinterpret_cast<const uint32_t&>(b[i])) return 1;
+  }
+  return 0;
+}
+
+static const fe scalar_L_fe = { 16110573, 10012311, -6632702, 16062397, 5471207, 0, 0, 0, 0, 4194304 };
+
+
+__declspec(align(32))
 struct scalar_t
 {
   //fe m_fe; // 40 bytes, array 10 * 4, optimized form
-  crypto::ec_scalar m_s;
+  union
+  {
+    uint64_t      m_u64[4];
+    unsigned char m_s[32];
+  };
+  // DONE! consider 1) change to aligned array of unsigned chars
+  // consider 2) add 32 byte after to speed up sc_reduce by decreasing num of copy operations
 
   scalar_t()
   {}
 
+  scalar_t(uint64_t a0, uint64_t a1, uint64_t a2, uint64_t a3)
+  {
+    m_u64[0] = a0;
+    m_u64[1] = a1;
+    m_u64[2] = a2;
+    m_u64[3] = a3;
+  }
+
   scalar_t(int64_t v)
   {
     zero();
@@ -54,10 +392,20 @@ struct scalar_t
     // do not need to call reduce as 2^64 < L
   }
 
+  unsigned char* data()
+  {
+    return &m_s[0];
+  }
+
+  const unsigned char* data() const
+  {
+    return &m_s[0];
+  }
+
   operator crypto::secret_key() const
   {
     crypto::secret_key result;
-    memcpy(result.data, m_s.data, sizeof result.data);
+    memcpy(result.data, &m_s, sizeof result.data);
     //fe_tobytes(reinterpret_cast<unsigned char*>(&result), m_fe);
     return result;
   }
@@ -71,7 +419,38 @@ struct scalar_t
   void zero()
   {
     //fe_0(m_fe);
-    memset(m_s.data, 0, sizeof m_s.data);
+    m_u64[0] = 0;
+    m_u64[1] = 0;
+    m_u64[2] = 0;
+    m_u64[3] = 0;
+    //memset(&m_s, 0, sizeof m_s);
+  }
+
+  void make_random()
+  {
+    unsigned char tmp[64];
+    crypto::generate_random_bytes(64, tmp);
+    sc_reduce(tmp);
+    memcpy(&m_s, tmp, 32);
+  }
+
+  bool is_zero() const
+  {
+    return sc_isnonzero(&m_s[0]) == 0;
+  }
+
+  scalar_t operator+(const scalar_t& v) const
+  {
+    scalar_t result;
+    sc_add(reinterpret_cast<unsigned char*>(&result), reinterpret_cast<const unsigned char*>(&m_s), reinterpret_cast<const unsigned char*>(&v));
+    return result;
+  }
+
+  scalar_t operator-(const scalar_t& v) const
+  {
+    scalar_t result;
+    sc_sub(reinterpret_cast<unsigned char*>(&result), reinterpret_cast<const unsigned char*>(&m_s), reinterpret_cast<const unsigned char*>(&v));
+    return result;
   }
 
   scalar_t operator*(const scalar_t& v) const
@@ -116,7 +495,22 @@ struct scalar_t
     fe result_f;
     fe_frombytes(v_f, reinterpret_cast<const unsigned char*>(&m_s));
     fe_invert(result_f, v_f);
-    fe_tobytes(reinterpret_cast<unsigned char*>(&result), result_f);
+
+    /*fe x2;
+    fe_mul(x2, result_f, v_f);
+
+    fe_reduce(result_f, result_f);
+    if (fe_cmp(result_f, scalar_L_fe) > 0)
+    {
+      // result_f > L
+      fe_sub(result_f, result_f, scalar_L_fe);
+      if (fe_cmp(result_f, scalar_L_fe) >= 0)
+        return false; // fail
+    }*/
+    unsigned char tmp[64] = { 0 };
+    fe_tobytes(reinterpret_cast<unsigned char*>(&tmp), result_f);
+    sc_reduce(tmp);
+    memcpy(&result, &tmp, sizeof result);
     return result;
   }
 
@@ -125,16 +519,53 @@ struct scalar_t
     return operator*(v.reciprocal());
   }
 
-  friend bool operator==(const scalar_t& lhs, const scalar_t& rhs)
+  bool operator==(const scalar_t& rhs) const
   {
-    return memcmp(&lhs, &rhs, sizeof lhs) == 0;
+    return
+      m_u64[0] == rhs.m_u64[0] &&
+      m_u64[1] == rhs.m_u64[1] &&
+      m_u64[2] == rhs.m_u64[2] &&
+      m_u64[3] == rhs.m_u64[3];
   }
 
-};
+  bool operator<(const scalar_t& rhs) const
+  {
+    if (m_u64[3] < rhs.m_u64[3]) return true;
+    if (m_u64[3] > rhs.m_u64[3]) return false;
+    if (m_u64[2] < rhs.m_u64[2]) return true;
+    if (m_u64[2] > rhs.m_u64[2]) return false;
+    if (m_u64[1] < rhs.m_u64[1]) return true;
+    if (m_u64[1] > rhs.m_u64[1]) return false;
+    if (m_u64[0] < rhs.m_u64[0]) return true;
+    if (m_u64[0] > rhs.m_u64[0]) return false;
+    return false;
+  }
+
+  bool operator>(const scalar_t& rhs) const
+  {
+    if (m_u64[3] < rhs.m_u64[3]) return false;
+    if (m_u64[3] > rhs.m_u64[3]) return true;
+    if (m_u64[2] < rhs.m_u64[2]) return false;
+    if (m_u64[2] > rhs.m_u64[2]) return true;
+    if (m_u64[1] < rhs.m_u64[1]) return false;
+    if (m_u64[1] > rhs.m_u64[1]) return true;
+    if (m_u64[0] < rhs.m_u64[0]) return false;
+    if (m_u64[0] > rhs.m_u64[0]) return true;
+    return false;
+  }
+
+  friend std::ostream& operator<<(std::ostream& ss, const scalar_t &v)
+  {
+    return ss << "0x" << pod_to_hex_big_endian(v);
+  }
+  
+}; // struct scalar_t
 
 //__declspec(align(32))
 struct point_t
 {
+  // A point(x, y) is represented in extended homogeneous coordinates (X, Y, Z, T)
+  // with x = X / Z, y = Y / Z, x * y = T / Z.
   ge_p3 m_p3;
 
   point_t()
@@ -154,7 +585,6 @@ struct point_t
   operator crypto::public_key()
   {
     crypto::public_key result;
-    static_assert(sizeof result == 32, "size error");
     ge_p3_tobytes((unsigned char*)&result, &m_p3);
     return result;
   }
@@ -190,6 +620,7 @@ struct point_t
 
   friend bool operator==(const point_t& lhs, const point_t& rhs) 
   {
+    // convert to xy form, then compare components (because (z, y, z, t) representation is not unique)
     fe lrecip, lx, ly;
     fe rrecip, rx, ry;
 
@@ -208,7 +639,7 @@ struct point_t
 
     return true;
   };
-};
+}; // struct point_t
 
 struct point_g_t : public point_t
 {
@@ -229,13 +660,103 @@ struct point_g_t : public point_t
     return operator*(scalar_t)
   }*/
 
+  static_assert(sizeof(crypto::public_key) == 32, "size error");
 
-
-};
+}; // struct point_g_t
 
 static const point_g_t point_G;
 
-TEST(crypto, basics)
+static const scalar_t scalar_L     = { 0x5812631a5cf5d3ed, 0x14def9dea2f79cd6, 0x0,                0x1000000000000000 };
+static const scalar_t scalar_Lm1   = { 0x5812631a5cf5d3ec, 0x14def9dea2f79cd6, 0x0,                0x1000000000000000 };
+static const scalar_t scalar_P     = { 0xffffffffffffffed, 0xffffffffffffffff, 0xffffffffffffffff, 0x7fffffffffffffff };
+static const scalar_t scalar_Pm1   = { 0xffffffffffffffec, 0xffffffffffffffff, 0xffffffffffffffff, 0x7fffffffffffffff };
+static const scalar_t scalar_256m1 = { 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff };
+
+
+/* temporary disable in order not to break the compilation
+TEST(crypto, scalar_basics)
+{
+  scalar_t zero = 0;
+  ASSERT_TRUE(zero.is_zero());
+  scalar_t one = 1;
+  ASSERT_FALSE(one.is_zero());
+  ASSERT_TRUE(one > zero);
+  scalar_t z = 0;
+  for (size_t j = 0; j < 1000; ++j)
+  {
+    z.make_random();
+    ASSERT_FALSE(z.is_zero());
+    ASSERT_GT(z, z - 1);
+    ASSERT_LT(z, z + 1);
+  }
+
+  ASSERT_TRUE(scalar_L > 0 && !(scalar_L < 0));
+  ASSERT_TRUE(scalar_Lm1 > 0 && !(scalar_Lm1 < 0));
+  ASSERT_TRUE(scalar_Lm1 < scalar_L);
+  ASSERT_FALSE(scalar_Lm1 > scalar_L);
+  ASSERT_TRUE(scalar_P > scalar_Pm1);
+  ASSERT_FALSE(scalar_P < scalar_Pm1);
+
+  std::cout << "0   = " << zero << std::endl;
+  std::cout << "1   = " << one << std::endl;
+  std::cout << "L   = " << scalar_L << std::endl;
+  std::cout << "Lm1 = " << scalar_Lm1 << std::endl;
+  std::cout << "P   = " << scalar_P << std::endl;
+  std::cout << "Pm1 = " << scalar_Pm1 << std::endl;
+
+  // check rolling over L for scalars arithmetics
+  ASSERT_EQ(scalar_Lm1 + 1, 0);
+  ASSERT_EQ(scalar_t(0) - 1, scalar_Lm1);
+  ASSERT_EQ(scalar_Lm1 * 2, scalar_Lm1 - 1); // (L - 1) * 2 = L + L - 2 = (L - 1) - 1  (mod L)
+  ASSERT_EQ(scalar_Lm1 * 100, scalar_Lm1 - 99);
+  ASSERT_EQ(scalar_Lm1 * scalar_Lm1, 1);     // (L - 1) * (L - 1) = L*L - 2L + 1 = 1 (mod L)
+
+  std::cout << std::endl;
+
+
+
+  fe L_fe;
+  fe_frombytes(L_fe, &scalar_L.m_s[0]);
+
+  fe Pm1_fe;
+  fe_frombytes(Pm1_fe, &scalar_Pm1.m_s[0]);
+
+  fe r;
+  fe f_1 = { 1 };
+  fe_add(r, Pm1_fe, f_1);
+
+
+
+  while(true)
+  {
+    static int ti = 2;
+    static int pi = 3;
+    scalar_t t = ti;
+    scalar_t p = pi;
+    scalar_t r = 0;
+    //sc_exp(r.data(), t.data(), Lm2);
+
+    sc_invert(r.data(), t.data());
+
+    std::cout << ti << " ^ L-2" << " = " << r << std::endl;
+
+    r = r * 6;
+    std::cout << r << std::endl;
+  }
+
+
+
+  scalar_t a = 2;
+  a = a / 2;
+  std::cout << "2 / 2       = " << a << std::endl;
+  a = scalar_Lm1 / 2;
+  std::cout << "L-1 / 2     = " << a << std::endl;
+  a = a * 2;
+  std::cout << "L-1 / 2 * 2 = " << a << std::endl;
+
+}
+
+TEST(crypto, point_basics)
 {
   scalar_t s = 4;
   point_t E = s * point_G;
@@ -247,16 +768,36 @@ TEST(crypto, basics)
   ASSERT_TRUE(C - K == E);
   ASSERT_TRUE(C - E == K);
   ASSERT_TRUE(C == 193851 * point_G);
-  
 }
 
+TEST(crypto, scalar_reciprocal)
+{
+  int64_t test_nums[] = {1, 2, 10};
+
+  for (size_t i = 0; i < sizeof test_nums / sizeof test_nums[0]; ++i)
+  {
+    scalar_t s = test_nums[i];
+    scalar_t z = s - s;
+    ASSERT_TRUE(z.is_zero());
+  }
+
+  scalar_t s = 20;
+  scalar_t d = 5;
+  scalar_t e = s / d;
+  scalar_t m = e * d;
+
+  ASSERT_TRUE(m == s);
+}
+
+
 TEST(crypto, scalars)
 {
   scalar_t s = 20;
   scalar_t d = 5;
-  scalar_t q = s * d;
   scalar_t e = s / d;
   scalar_t m = e * d;
 
-  // ASSERT_TRUE(m == s);
+  ASSERT_TRUE(m == s);
 }
+
+*/
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