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      "body": "@@ -800,17 +800,17 @@\n  point i\n-f\n+s\n  the res\n",
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      "author": "icostan",
      "body": "The easiest way to understand Elliptic Curve (EC), point addition, scalar multiplication and trapdoor function; explained with simple graphs and animations.\n\n\n# 1. Abstract\n\n-   What the heck is an elliptic curve?\n-   A plane algebraic curve defined by an equation of this form: ***y<sup>2</sup> = x<sup>3</sup> + a\\*x + b***\n\n-   Why are elliptic curves important in cryptography?\n-   Because elliptic curve scalar multiplication is a trapdoor function\n\n-   How does scalar multiplication works?\n-   Scalar/point multiplication is defined as repeated addition of a point to itself\n\n-   How does point addition works then?\n-   If we draw a line passing thru elliptic curve points (or draw a tangent to a single point) it will intersect another point on the curve and the inverse of this intersection point if the result of point addition\n\nSince a picture is worth a thousand words then the following elliptic curve point addition/multiplication animation has 33 frames and is worth a lot more, do the math.\n\n![ec11-animate.gif](https://cdn.steemitimages.com/DQmeaftKeUgnvojjuNFV6xHeX2mV4S33otD5rPumc78JxnU/ec11-animate.gif)\n\nGiven an elliptic curve ***E*** a point on elliptic curve ***G*** (called the generator) and a private key ***k*** we can calculate the public key ***P*** where ***P = k \\* G***.\n\nThe whole idea behind elliptic curves cryptography is that point addition (multiplication) is a trapdoor function which means that given ***G*** and ***P*** points it is infeasible to find the private key ***k***.\n\n[Read the blog post](https://blog.iuliancostan.com/post/2019-09-25-elliptic-curves/) for more details.",
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      "body": "@@ -1133,16 +1133,19 @@\n rapdoor%0A\n+---\n %0AFirst t\n@@ -2751,16 +2751,19 @@\n ryption%0A\n+---\n %0A%5C%60m%5C%60 i\n@@ -3085,16 +3085,19 @@\n ryption%0A\n+---\n %0ATo decr\n@@ -3981,16 +3981,19 @@\n tuition%0A\n+---\n %0AWhy thi\n",
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      "author": "icostan",
      "body": "@@ -1077,17 +1077,20 @@\n rapdoor%0A\n+---\n %0A\n-\n The very\n@@ -2324,17 +2324,20 @@\n ryption%0A\n+---\n %0A\n-\n Now it's\n@@ -3364,17 +3364,20 @@\n ryption%0A\n+---\n %0A\n-\n The %5C%60d%5C\n@@ -4098,16 +4098,19 @@\n tuition%0A\n+---\n %0AAt firs\n",
      "json_metadata": "{\"tags\":[\"cryptography\",\"rsa\",\"primes\",\"factorization\",\"eli5\"],\"links\":[\"https://en.wikipedia.org/wiki/RSA_(cryptosystem)\",\"https://en.wikipedia.org/wiki/Euler's_totient_function\",\"https://en.wikipedia.org/wiki/Coprime_integers\",\"https://en.wikipedia.org/wiki/Trapdoor_function\",\"https://en.wikipedia.org/wiki/Integer_factorization\",\"https://en.wikipedia.org/wiki/Greatest_common_divisor\",\"https://en.wikipedia.org/wiki/Euclidean_algorithm\",\"https://en.wikipedia.org/wiki/Multiplicative_inverse\",\"https://en.wikipedia.org/wiki/Finite_field\",\"https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
      "parent_author": "",
      "parent_permlink": "cryptography",
      "permlink": "eli5-cryptography-rsa",
      "title": "Cryptography: RSA"
    }
  ],
  "op_in_trx": 0,
  "timestamp": "2019-04-13T13:18:00",
  "trx_id": "95c0ac96e6bf66ea3f1340e49da10f6952227a98",
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{
  "block": 32008791,
  "op": [
    "comment",
    {
      "author": "icostan",
      "body": "@@ -2303,33 +2303,8 @@\n ()%0A%0A\n-!%5Bimg%5D(/assets/ec1.png)%0A%0A\n Not \n@@ -2461,34 +2461,8 @@\n ()%0A%0A\n-!%5Bimg%5D(/assets/ec2.png)%0A%0A%0A\n # El\n",
      "json_metadata": "{\"tags\":[\"ecdsa\",\"cryptography\",\"discrete\",\"logarithm\"],\"links\":[\"https://en.wikipedia.org/wiki/Elliptic-curve_cryptography\",\"https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm\",\"http://www.sagemath.org\",\"https://en.wikipedia.org/wiki/Multiplicative_inverse\",\"https://en.wikipedia.org/wiki/Trapdoor_function\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
      "parent_author": "",
      "parent_permlink": "ecdsa",
      "permlink": "cryptography-ecdsa",
      "title": "Cryptography: ECDSA"
    }
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  "op_in_trx": 0,
  "timestamp": "2019-04-13T13:16:36",
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{
  "block": 32008705,
  "op": [
    "comment",
    {
      "author": "icostan",
      "body": "# Overview\n---\n\n    \"If you can't explain it simply, you don't understand it well enough\" - Einstein\n---\nElliptic curve cryptography ([ECC](https://en.wikipedia.org/wiki/Elliptic-curve_cryptography)) and digital signature algorithm ([ECDSA](https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm)) are more complex than RSA or ElGamal but I will try my best to hide the hairy math and the implementation details.\n\nHere is the ELI5 version in 18 lines of [SageMath](http://www.sagemath.org) / Python code. I use Sage because it provides elliptic curves as first-class citizens (\\`FiniteField\\` and \\`EllipticCurve\\`) and we can take multiplication operation for granted.\n\n     1  p = 103\n     2  F = FiniteField(p)\n     3  E = EllipticCurve(F, [0, 7])\n     4  G = E([97, 10])\n     5  n = G.order()\n     6  k = randint(2, n)\n     7  P = k * G\n     8  m = 6\n     9  t = randint(2, n)\n    10  R = t * G\n    11  r = mod(R[0], n)\n    12  s = mod(inverse_mod(t, n) * (m + r * k), n)\n    13  inv_s = inverse_mod(int(s), n)\n    14  u = mod(m * inv_s, n)\n    15  v = mod(r * inv_s, n)\n    16  F = int(u) * G + int(v) * P\n    17  f = mod(F[0], n)\n    18  print \"YOU ARE A CRYPTOSTAR!\" if r == f else \"YOU SUCK!\"\n\n    YOU ARE A CRYPTOSTAR!\n\nBesides lines #7, #10 and #16 that involves multiplication between a scalar and a point on elliptic curve, everything else is just modular arithmetic and two [multiplicative inverse](https://en.wikipedia.org/wiki/Multiplicative_inverse) calculations (line #12 and #13)\n\nPlease keep reading if you want line by line explanations and a few elliptic curve graphs.\n\n\n# Basics\n\n-   What the heck is an elliptic curve?\n-   It is just a simple equation of this form \\`y2 = x3 + ax + b\\`.\n-   Why it is so special?\n-   Because it has one interesting operation: addition. Adding two points on the curve will result a third point also on the curve (aka closure). Multiplication is particular case of addition, adding the same point to itself \\`k\\` times.\n-   And how does it looks like?\n-   Well, I am afraid you won't like the graph because elliptic curves defined over finite fields get cut off and are not intelligible to humans eye\n\nBut here it is anyway: \\`a = 0; b = 7\\` and prime number is \\`103\\`.\n\n    E = EllipticCurve(FiniteField(103), [0, 7])\n    E.plot()\n\n![img](/assets/ec1.png)\n\nNot pretty huh? lets define the same elliptic curve over real numbers and the graph starts to make little sense.\n\n    E = EllipticCurve([0, 7])\n    E.plot()\n\n![img](/assets/ec2.png)\n\n\n# Elliptic curve multiplication trapdoor\n\nFirst thing first, let's define the elliptic curve parameters and notations:\n\n    1  p = 103\n    2  F = FiniteField(p)\n    3  E = EllipticCurve(F, [0, 7])\n    4  G = E([97, 10])\n    5  n = G.order()\n\nVariable notations:\n\n-   uppercase letter - elements of the elliptic curve like finite fields (F), elliptic curve itself (E) or point on elliptic curve (G)\n-   lowercase letter - scalar value like the secret key (k)\n\nDomain parameters:\n\n-   p - prime number\n-   F - finite field\n-   0,7 - coefficients for elliptic curve\n-   E - elliptic curve\n-   G - point generator\n-   n - order (the number of points) of the cyclic subgroup generated by \\`G\\`\n\nNow, it's time to generate the secret key \\`k\\` and calculate the public key \\`P\\` using the trapdoor function called Elliptic Curve Discrete Logarithm Problem (ECDLP).\n\nThis is the bread and the butter of elliptic curve cryptography, given the generator \\`G\\` and public key \\`P' it is infeasible to calculate the secret key \\`k\\`, hence the [trapdoor](https://en.wikipedia.org/wiki/Trapdoor_function)\n\n    6  k = randint(2, n)\n    7  P = k * G\n\nIf you are curious how elliptic curve point \\`P\\` looks like then it is nothing than a Cartesian (x,y) coordinate:\n\n    P\n\n    (65 : 31 : 1)\n\n\n# Signature\n\nLet's sign the message \\`m\\` and for this we need to generate a random integer \\`t\\` of order \\`n\\` and calculate the \\`R\\` point on elliptic curve.\n\n     8  m = 6\n     9  t = randint(2, n)\n    10  R = t * G\n    11  r = mod(R[0], n)\n    12  s = mod(inverse_mod(t, n) * (m + r * k), n)\n\nThe signature itself has two numbers \\`r\\` and \\`s\\` that are sent to third-party for verification:\n\n-   r - the \\`x\\` coordinate of point \\`R\\`\n-   s - is calculated with \\`s = (m + r\\*k) / t\\` but we use multiplication with inverse instead of division.\n\nAnd this is how our signature looks like:\n\n    [r, s]\n\n    [66, 51]\n\n\n# Verification\n\nGiven the signature \\`r, s\\` above and \\`G, P, r and n\\` that are public information we need to calculate the equation \\`F = u\\*G + v\\*P\\` where \\`u = m/s\\` and \\`v = r/s\\`.\n\nThe signature is valid if \\`R.x == F.x\\` is true, in other words the \\`x\\` coordinates are the same.\n\n    13  inv_s = inverse_mod(int(s), n)\n    14  u = mod(m * inv_s, n)\n    15  v = mod(r * inv_s, n)\n    16  F = int(u) * G + int(v) * P\n    17  f = mod(F[0], n)\n    18  print \"YOU ARE A CRYPTOSTAR!\" if r == f else \"YOU SUCK!\"\n\n    YOU ARE A CRYPTOSTAR!\n\n\n# Intuition\n\nRemember that uppercase letters are points on elliptic curve, lowercase are scalars:\n\n    1  R == u*G + v*P\n    2  R == u*G + v*k*P\n    3  R == m/s * G + r/s * k*G\n    4  R == m/s * G + r*k/s * G\n    5  R == (m + r*k)/s * G\n    6  R == (m + r*k)/((m + r*k)/t) * G\n\nAs always we will do the math backwards:\n\n1.  start with the equation of signature verification and substitute public key \\`P\\`\n2.  substitute \\`u\\` and \\`v\\`\n3.  multiplication is commutative and we can put \\`r\\` and \\`k\\` together\n4.  extract \\`G\\` and \\`s\\` as factors\n5.  substitute \\`s\\`\n6.  after reduction we are left with \\`t\\*G\\`\n\nAnd the intuition is valid if the \\`x\\` coordinates on both sides of the equation are equal.\n\n    print \"MAGIC\" if R[0] == (int((m + r*k)*t/(m + r*k)) * G)[0] else \"ERROR\"\n\n    MAGIC",
      "json_metadata": "{\"tags\":[\"cryptography\",\"ecdsa\",\"discrete\",\"logarithm\"],\"links\":[\"https://en.wikipedia.org/wiki/Elliptic-curve_cryptography\",\"https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm\",\"http://www.sagemath.org\",\"https://en.wikipedia.org/wiki/Multiplicative_inverse\",\"https://en.wikipedia.org/wiki/Trapdoor_function\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\",\"image\":[\"/assets/ec1.png\",\"/assets/ec2.png\"]}",
      "parent_author": "",
      "parent_permlink": "ecdsa",
      "permlink": "cryptography-ecdsa",
      "title": "Cryptography: ECDSA"
    }
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  "op_in_trx": 0,
  "timestamp": "2019-04-13T13:12:15",
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    "comment",
    {
      "author": "icostan",
      "body": "@@ -4,16 +4,19 @@\n verview%0A\n+---\n %0A    %22If\n",
      "json_metadata": "{\"tags\":[\"cryptography\",\"rsa\",\"primes\",\"factorization\",\"eli5\"],\"links\":[\"https://en.wikipedia.org/wiki/RSA_(cryptosystem)\",\"https://en.wikipedia.org/wiki/Euler's_totient_function\",\"https://en.wikipedia.org/wiki/Coprime_integers\",\"https://en.wikipedia.org/wiki/Trapdoor_function\",\"https://en.wikipedia.org/wiki/Integer_factorization\",\"https://en.wikipedia.org/wiki/Greatest_common_divisor\",\"https://en.wikipedia.org/wiki/Euclidean_algorithm\",\"https://en.wikipedia.org/wiki/Multiplicative_inverse\",\"https://en.wikipedia.org/wiki/Finite_field\",\"https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
      "parent_author": "",
      "parent_permlink": "cryptography",
      "permlink": "eli5-cryptography-rsa",
      "title": "Cryptography: RSA"
    }
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  "op_in_trx": 0,
  "timestamp": "2019-04-13T13:11:24",
  "trx_id": "71b22b4eb83e9e8b4ae54278b0b4632ea272e69e",
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{
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    {
      "author": "icostan",
      "body": "@@ -4,16 +4,19 @@\n verview%0A\n+---\n %0A    %22If\n@@ -92,16 +92,20 @@\n Einstein\n+%0A---\n %0A%0AElGama\n",
      "json_metadata": "{\"tags\":[\"cryptography\",\"elgamal\",\"discrete\",\"logarithm\",\"eli5\"],\"links\":[\"https://en.wikipedia.org/wiki/ElGamal_encryption\",\"https://en.wikipedia.org/wiki/ElGamal_signature_scheme\",\"https://en.wikipedia.org/wiki/Cyclic_group\",\"https://en.wikipedia.org/wiki/Discrete_logarithm\",\"https://en.wikipedia.org/wiki/Additive_inverse\",\"https://en.wikipedia.org/wiki/Finite_field\",\"https://en.wikipedia.org/wiki/Euler%2527s_theorem\",\"https://en.wikipedia.org/wiki/Lagrange%2527s_theorem_(group_theory)\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
      "parent_author": "",
      "parent_permlink": "cryptography",
      "permlink": "eli5-cryptography-elgamal",
      "title": "Cryptography: ElGamal"
    }
  ],
  "op_in_trx": 0,
  "timestamp": "2019-04-13T13:10:48",
  "trx_id": "cc641331f751126d3f588cec2ac1ea401e6d2545",
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{
  "block": 32008654,
  "op": [
    "comment",
    {
      "author": "icostan",
      "body": "# Overview\n---\n\n    \"If you can't explain it simply, you don't understand it well enough\" - Einstein\n---\nElliptic curve cryptography ([ECC](https://en.wikipedia.org/wiki/Elliptic-curve_cryptography)) and digital signature algorithm ([ECDSA](https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm)) are more complex than RSA or ElGamal but I will try my best to hide the hairy math and the implementation details.\n\nHere is the ELI5 version in 18 lines of [SageMath](http://www.sagemath.org) / Python code. I use Sage because it provides elliptic curves as first-class citizens (\\`FiniteField\\` and \\`EllipticCurve\\`) and we can take multiplication operation for granted.\n\n     1  p = 103\n     2  F = FiniteField(p)\n     3  E = EllipticCurve(F, [0, 7])\n     4  G = E([97, 10])\n     5  n = G.order()\n     6  k = randint(2, n)\n     7  P = k * G\n     8  m = 6\n     9  t = randint(2, n)\n    10  R = t * G\n    11  r = mod(R[0], n)\n    12  s = mod(inverse_mod(t, n) * (m + r * k), n)\n    13  inv_s = inverse_mod(int(s), n)\n    14  u = mod(m * inv_s, n)\n    15  v = mod(r * inv_s, n)\n    16  F = int(u) * G + int(v) * P\n    17  f = mod(F[0], n)\n    18  print \"YOU ARE A CRYPTOSTAR!\" if r == f else \"YOU SUCK!\"\n\n    YOU ARE A CRYPTOSTAR!\n\nBesides lines #7, #10 and #16 that involves multiplication between a scalar and a point on elliptic curve, everything else is just modular arithmetic and two [multiplicative inverse](https://en.wikipedia.org/wiki/Multiplicative_inverse) calculations (line #12 and #13)\n\nPlease keep reading if you want line by line explanations and a few elliptic curve graphs.\n\n\n# Basics\n\n-   What the heck is an elliptic curve?\n-   It is just a simple equation of this form \\`y2 = x3 + ax + b\\`.\n-   Why it is so special?\n-   Because it has one interesting operation: addition. Adding two points on the curve will result a third point also on the curve (aka closure). Multiplication is particular case of addition, adding the same point to itself \\`k\\` times.\n-   And how does it looks like?\n-   Well, I am afraid you won't like the graph because elliptic curves defined over finite fields get cut off and are not intelligible to humans eye\n\nBut here it is anyway: \\`a = 0; b = 7\\` and prime number is \\`103\\`.\n\n    E = EllipticCurve(FiniteField(103), [0, 7])\n    E.plot()\n\n![img](/assets/ec1.png)\n\nNot pretty huh? lets define the same elliptic curve over real numbers and the graph starts to make little sense.\n\n    E = EllipticCurve([0, 7])\n    E.plot()\n\n![img](/assets/ec2.png)\n\n\n# Elliptic curve multiplication trapdoor\n\nFirst thing first, let's define the elliptic curve parameters and notations:\n\n    1  p = 103\n    2  F = FiniteField(p)\n    3  E = EllipticCurve(F, [0, 7])\n    4  G = E([97, 10])\n    5  n = G.order()\n\nVariable notations:\n\n-   uppercase letter - elements of the elliptic curve like finite fields (F), elliptic curve itself (E) or point on elliptic curve (G)\n-   lowercase letter - scalar value like the secret key (k)\n\nDomain parameters:\n\n-   p - prime number\n-   F - finite field\n-   0,7 - coefficients for elliptic curve\n-   E - elliptic curve\n-   G - point generator\n-   n - order (the number of points) of the cyclic subgroup generated by \\`G\\`\n\nNow, it's time to generate the secret key \\`k\\` and calculate the public key \\`P\\` using the trapdoor function called Elliptic Curve Discrete Logarithm Problem (ECDLP).\n\nThis is the bread and the butter of elliptic curve cryptography, given the generator \\`G\\` and public key \\`P' it is infeasible to calculate the secret key \\`k\\`, hence the [trapdoor](https://en.wikipedia.org/wiki/Trapdoor_function)\n\n    6  k = randint(2, n)\n    7  P = k * G\n\nIf you are curious how elliptic curve point \\`P\\` looks like then it is nothing than a Cartesian (x,y) coordinate:\n\n    P\n\n    (65 : 31 : 1)\n\n\n# Signature\n\nLet's sign the message \\`m\\` and for this we need to generate a random integer \\`t\\` of order \\`n\\` and calculate the \\`R\\` point on elliptic curve.\n\n     8  m = 6\n     9  t = randint(2, n)\n    10  R = t * G\n    11  r = mod(R[0], n)\n    12  s = mod(inverse_mod(t, n) * (m + r * k), n)\n\nThe signature itself has two numbers \\`r\\` and \\`s\\` that are sent to third-party for verification:\n\n-   r - the \\`x\\` coordinate of point \\`R\\`\n-   s - is calculated with \\`s = (m + r\\*k) / t\\` but we use multiplication with inverse instead of division.\n\nAnd this is how our signature looks like:\n\n    [r, s]\n\n    [66, 51]\n\n\n# Verification\n\nGiven the signature \\`r, s\\` above and \\`G, P, r and n\\` that are public information we need to calculate the equation \\`F = u\\*G + v\\*P\\` where \\`u = m/s\\` and \\`v = r/s\\`.\n\nThe signature is valid if \\`R.x == F.x\\` is true, in other words the \\`x\\` coordinates are the same.\n\n    13  inv_s = inverse_mod(int(s), n)\n    14  u = mod(m * inv_s, n)\n    15  v = mod(r * inv_s, n)\n    16  F = int(u) * G + int(v) * P\n    17  f = mod(F[0], n)\n    18  print \"YOU ARE A CRYPTOSTAR!\" if r == f else \"YOU SUCK!\"\n\n    YOU ARE A CRYPTOSTAR!\n\n\n# Intuition\n\nRemember that uppercase letters are points on elliptic curve, lowercase are scalars:\n\n    1  R == u*G + v*P\n    2  R == u*G + v*k*P\n    3  R == m/s * G + r/s * k*G\n    4  R == m/s * G + r*k/s * G\n    5  R == (m + r*k)/s * G\n    6  R == (m + r*k)/((m + r*k)/t) * G\n\nAs always we will do the math backwards:\n\n1.  start with the equation of signature verification and substitute public key \\`P\\`\n2.  substitute \\`u\\` and \\`v\\`\n3.  multiplication is commutative and we can put \\`r\\` and \\`k\\` together\n4.  extract \\`G\\` and \\`s\\` as factors\n5.  substitute \\`s\\`\n6.  after reduction we are left with \\`t\\*G\\`\n\nAnd the intuition is valid if the \\`x\\` coordinates on both sides of the equation are equal.\n\n    print \"MAGIC\" if R[0] == (int((m + r*k)*t/(m + r*k)) * G)[0] else \"ERROR\"\n\n    MAGIC",
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      "parent_author": "",
      "parent_permlink": "ecdsa",
      "permlink": "cryptography-ecdsa",
      "title": "Cryptography: ECDSA"
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  "timestamp": "2019-04-13T13:09:42",
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  "block": 31521289,
  "op": [
    "comment",
    {
      "author": "icostan",
      "body": "# Overview\n\n    \"If you can't explain it simply, you don't understand it well enough\" - Einstein\n\nElGamal is a public key cryptosystem that is used in [encryption](https://en.wikipedia.org/wiki/ElGamal_encryption) , [digital signature](https://en.wikipedia.org/wiki/ElGamal_signature_scheme) and homomorphic cryptography.\n\nHere is my take in 12 lines of Python code:\n\n     1  p = 7\n     2  g = [x for x in range(1, p) if len(set([x**i % p for i in range(1, p)])) == p-1][0]\n     3  k = 4\n     4  t = g**k % p\n     5  m = 6\n     6  import random\n     7  r = random.randint(1, p-1)\n     8  c1 = g**r % p\n     9  c2 = (t**r * m) % p\n    10  inv_k = p - 1 - k\n    11  inv_c1 = c1**inv_k % p\n    12  message = c2 * inv_c1 % p\n    13  print \"YOU ARE A CRYPTOSTAR!\" if message == m else \"ERROR\"\n\n    YOU ARE A CRYPTOSTAR!\n\nBesides line #2 that involves an algorithm to find the generator \\`g\\` and line #10 that calculates the additive inverse of private key \\`k\\`, everything else is just modular multiplication and exponentiation.\n\nInterested in more details and line by line explanations? please read on.\n\n\n# Discrete logarithm trapdoor\n\nFirst thing first, lets pick a prime number:\n\n    1  p = 7\n\nFor the prime number above we need to find the generator \\`g\\` that defines a [cyclic group](https://en.wikipedia.org/wiki/Cyclic_group) which in theory is a little bit involved but in simple terms the generator is a specific number that returns unique values for each exponent between \\`1\\` and \\`p-1\\` where \\`p-1\\` is called the order of the cyclic group.\n\nLets take a few examples and see how it looks like: pick \\`2\\` as generator:\n\n    print [2**i % p for i in range(1, p)]\n\n    [2, 4, 1, 2, 4, 1]\n\nWe can easily see that returning values are not unique, lets pick \\`3\\` as generator:\n\n    print [3**i % p for i in range(1, p)]\n\n    [3, 2, 6, 4, 5, 1]\n\nBINGO! returning values are unique and we've found our generator.\n\nThe line below is just a brute force algorithm that finds multiple generators of order \\`p-1\\` but only takes the first one that is found.\n\n    2  g = [x for x in range(1, p) if len(set([x**i % p for i in range(1, p)])) == p-1][0]\n\nNow is time to pick a private key \\`k\\` between \\`1\\` and \\`p-1\\`, execute the trapdoor function and calculate \\`t\\` value:\n\n    3  k = 4\n    4  t = g**k % p\n\nThe line #4 is the secret sauce of ElGamal cryptography, the trapdoor function called [**the discrete logarithm**](https://en.wikipedia.org/wiki/Discrete_logarithm) which says that for given \\`g\\` and \\`t\\` values it is **infeasible** to calculate the private key \\`k\\` when working with very large numbers.\n\nThe \\`p, g and t\\` form the public key and will be used in encryption below, while \\`k\\` is kept secret.\n\n\n# Message encryption\n\n\\`m\\` is the message that we want to encrypt and \\`r\\` is a random number between \\`1\\` and \\`p-1\\`:\n\n    5  m = 6\n    6  import random\n    7  r = random.randint(1, p-1)\n    8  c1 = g**r % p\n    9  c2 = (t**r * m) % p\n\n\\`c1, c2\\` are the encrypted ciphers that will be used to decrypt the original message.\n\n\n# Message decryption\n\nTo decrypt the original message we need apply the following formula:\n\n\\`message = c2 / c1<sup>k</sup> mod p\\`\n\nThe problem is that division is not an operation defined for finite fields but multiplication is and we can re-write it using multiplication / exponentiation operations only:\n\n\\`message = c2 \\* c1<sup>inv</sup><sub>k</sub> mod p\\`\n\n\\`inv<sub>k</sub>\\` is called [additive inverse](https://en.wikipedia.org/wiki/Additive_inverse)  of \\`k\\` over [finite field](https://en.wikipedia.org/wiki/Finite_field) 'p' where \\`k + inv<sub>k</sub> mod p = 0\\` is valid.\nWe can brute force and find it but there is a simpler way: if \\`k < p\\` (which it is) then \\`inv<sub>k</sub> = p - 1 - k\\`.\n\n    10  inv_k = p - 1 - k\n    11  inv_c1 = c1**inv_k % p\n    12  message = c2 * inv_c1 % p\n    13  print \"YOU ARE A CRYPTOSTAR!\" if message == m else \"ERROR\"\n\n    YOU ARE A CRYPTOSTAR!\n\n\n# Intuition\n\nWhy this magic works the way it works?\n\n    1  message == (c2 * inv_c1) % p\n    2  message == (c2 * c1**(p-1-k)) % p\n    3  message == (t**r * m * (g**r)**(p-1-k)) % p\n    4  message == ((g**k)**r * m * (g**r)**(p-1-k)) % p\n    5  message == (g**(k*r) * m * (g**(r*(p-1-k)))) % p\n    6  message == (g**(k*r) * m * g**(r*(p-1)) * g**(-r*k)) % p\n\nLets do the math backwards:\n\n1.  start with last decryption formula and substitute inverses \\`inv<sub>c1</sub>\\` and \\`inv<sub>k</sub>\\`\n2.  substitute ciphers \\`c1\\` and \\`c2\\`\n3.  substitute trapdoor \\`t\\`\n4.  apply exponentiation power rule \\`(a<sup>b</sup>)<sup>c</sup> = a<sup>(b\\*c)</sup>\\`\n5.  distribute the exponent \\`r\\`\n6.  - \\`g\\*\\*(r\\*(p-1))\\` is 1 because of [Euler's theorem](https://en.wikipedia.org/wiki/Euler%2527s_theorem) and [Lagrange's theorem](https://en.wikipedia.org/wiki/Lagrange%2527s_theorem_(group_theory))\n    -   \\`g\\*\\*(k\\*r)\\` terms reduces each other\n\nCheck if the intuition is valid by substituting with numbers: generator \\`g\\` is 3, private key \\`k\\` is 4, random number \\`r\\` is 5 and message \\`m\\` is 6:\n\n    print \"MAGIC\" if 6 == (3**(4*5) * 6 * 3**(5*(7-1)) * 3**(-5*4)) % 7 else \"ERROR\"\n\n    MAGIC",
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      "author": "icostan",
      "body": "# Overview\n\n    \"If you can't explain it simply, you don't understand it well enough\" - Einstein\n---\n\n[RSA](https://en.wikipedia.org/wiki/RSA_(cryptosystem)) (Rivest-Shamir-Adleman) needs no introduction, it is well known and most used public-key cryptosystem that governs our digital lives.\n\nHere is my take, a simple implementation in 10 lines of Ruby code that is neither the best implementation nor the most efficient one but is enough for the purpose of this article.\n\n     1  p = 7\n     2  q = 11\n     3  n = p * q\n     4  phi = (p-1) * (q-1)\n     5  gcd = ->(a, b) { while(b != 0) do r, a = a % b, b; b = r end; a }\n     6  e = (2..phi).find{|i| gcd.call(i, phi) == 1 }\n     7  d = (2..phi).find{|i| (i * e) % phi == 1 }\n     8  m = 6\n     9  c = m**e % n\n    10  message = c**d % n\n    11  \"Decrypted message: #{message}\" if m == message\n\nExcept lines #5 that involves an algorithm and a while loop, everything else is mid-school math and \\`\\*\\*\\` is the exponentiation operator.\n\nThe curious minds please read simple explanations below.\n\n\n# Integer factorization trapdoor\n\nThe very first thing to do is to choose 2 prime numbers, \\`p\\` and \\`q\\`:\n\n    1  p = 7\n\n    2  q = 11\n\nNext, let's multiply the two numbers:\n\n    3  n = p * q\n\nNow it's time to define our first function \\`phi\\`:\n\n    4  phi = (p-1) * (q-1)\n\nWhich is [Euler's totient function](<https://en.wikipedia.org/wiki/Euler%27s_totient_function>) defined as number of integers less than \\`n\\` that are [coprime](https://en.wikipedia.org/wiki/Coprime_integers) with \\`n\\` and has two interesting properties:\n\n-   the function is multiplicative and \\`phi(a\\*b) = phi(a) \\* phi(b)\\`\n-   if \\`n\\` is a prime number then \\`phi(n) = n - 1\\`\n\nAnd finally we have the [cryptographic trapdoor](https://en.wikipedia.org/wiki/Trapdoor_function) which is the most important primitive to understand. Trapdoors are based on asymmetric computation that is easy to do in one direction but it is very difficult to calculate in the opposite direction.\n\nIn our case, if one (an attacker) does not know the initial \\`p\\` and \\`q\\` prime numbers it will be very hard to calculate the \\`phi\\` because [integer factorization](https://en.wikipedia.org/wiki/Integer_factorization)  of \\`n\\` is known to be infeasible to calculate for very large numbers.\n\n\n# Message encryption\n\nNow it's time to define our first algorithm, \\`gcd\\` which is the [greatest common divisor](<https://en.wikipedia.org/wiki/Greatest_common_divisor>) and is defined as the largest positive integer that divides both \\`a\\` and \\`b\\`.\n\nIf single-liner GCD looks scary then here is the formatted version of the [Euclid's algorithm](https://en.wikipedia.org/wiki/Euclidean_algorithm).\n\n    def gcd(a, b)\n      while(b != 0) do\n        r = a % b\n        a = b\n        b = r\n      end\n      a\n    end\n\nThe \\`e\\` key below is the encryption key and is defined as the smallest value for which \\`gcd(e, phi) == 1\\` is true.\nTo keep things simple we will just brute-force and find \\`e\\` but keep in mind that this will be very inefficient for large numbers.\n\n    5  gcd = ->(a, b) { while(b != 0) do r, a = a % b, b; b = r end; a }      (gcd)\n    6  e = (2..phi).find{|i| gcd.call(i, phi) == 1}\n\nMessage encryption is modulo exponentiation of the message \\`m\\` and the encryption key \\`e\\`.\n\n    7  m = 6\n    8  c = m**e % n\n\n\n# Message decryption\n\nThe \\`d\\` key below is the decryption key and is defined as [multiplicative inverse](https://en.wikipedia.org/wiki/Multiplicative_inverse) of \\`e\\` over [finite field](https://en.wikipedia.org/wiki/Finite_field) \\`phi\\`. Simply put \\`e \\* d mod phi = 1\\`.\n\nAgain we will just brute-force and calculate the inverse but there are better algorithms to do this, like [Extended Euclidean algorithm](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm).\n\n    9  d = (2..phi).find{|i| (e * i) % phi == 1}\n\nMessage decryption is another modulo exponentiation using the encrypted cipher \\`c\\` and decryption key \\`d\\`:\n\n    10  message = c**d % n\n\nBINGO! You can try it yourself but I can bet that \\`m = message = 6\\`\n\n\n# Intuition\n\nAt first sight it looks like magic right? but if you reason about it, it's very easy.\n\nStarting backwards, with the decryption/encryption formulas:\n\n    1  m = c^d mod n\n    2  c = m^e mod n\n\nLet's substitute #2 in #1:\n\n    m = (m^e)^d mod n\n\nExponentials Power Rule says that \\`(a<sup>b</sup>)<sup>c</sup> == a<sup>(b\\*c)</sup>\\` and we also know that \\`d\\` is the multiplicative inverse of \\`e\\`, guess what is the value of the resulting exponent? :)\n\n    m = m^(e*d mod phi) mod p\n\nIf the intuition is valid then the following expression stands true, where 7 is \\`e\\` and 43 is \\`d\\` in our little example:\n\n    'YOU ARE A CRYPTOSTAR!!!' if 6 == 6**(7 * 43 % 60) % 77",
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      "author": "icostan",
      "body": "# Bitmex-api-ruby\n\nFully featured, idiomatic Ruby library for BitMEX API. Support for both REST and Websocket APIs as described in [API Overview](https://www.bitmex.com/app/apiOverview).\n\n## Installation\n\n```shell\ngem install bitmex-api\n```\n\n## Usage\n\nListen for live trades in 3 lines of code:\n\n```ruby\n# Importing library...\n2.5.3 :001 > require 'bitmex-api'\n => true\n\n# Create client...\n2.5.3 :002 > client = Bitmex::Client.new\n => #<Bitmex::Client:0x00007fae7a569598 @host=\"www.bitmex.com\", @api_key=nil, @api_secret=nil>\n\n# Spit out live trades as they happen...\n2.5.3 :003 > client.trades.all symbol: 'XBTUSD' do |trade|\n  puts \"#{trade.side} #{trade.homeNotional} #{trade.symbol} @ #{trade.price}\"\nend\n2.5.3 :004 >\n2.5.3 :005?>\n==> {\"info\":\"Welcome to the BitMEX Realtime API.\",\"version\":\"2019-01-31T23:25:34.000Z\",\"timestamp\":\"2019-02-02T12:35:05.892Z\",\"docs\":\"https://www.bitmex.com/app/wsAPI\",\"limit\":{\"remaining\":37}}\n==> {\"success\":true,\"subscribe\":\"trade:XBTUSD\",\"request\":{\"op\":\"subscribe\",\"args\":[\"trade:XBTUSD\"]}}\nBuy 0.2924442 XBTUSD @ 3440\nSell 0.0305277 XBTUSD @ 3439.5\nBuy 0.2907 XBTUSD @ 3440\nSell 0.18810878 XBTUSD @ 3439.5\nSell 0.14537 XBTUSD @ 3439.5\nBuy 0.17442 XBTUSD @ 3440\nBuy 0.05814 XBTUSD @ 3440\nSell 0.014537 XBTUSD @ 3439.5\n```\n\nOr fetch trade history:\n\n```ruby\n# get first 10 trades of the year\n2.5.3 :006 > trades = client.trades.all symbol: 'XBTUSD', startTime: '2019-01-01', count: 10\n => [...]\n\n2.5.3 :007 > trades.size\n => 10\n\n# the very first trade in 2019\n2.5.3 :008 > trades.first\n => #<Bitmex::Mash foreignNotional=10 grossValue=270780 homeNotional=0.0027078 price=3693 side=\"Sell\" size=10 symbol=\"XBTUSD\" tickDirection=\"ZeroMinusTick\" timestamp=\"2019-01-01T00:00:03.904Z\" trdMatchID=\"2bd201ae-3b79-0908-293e-d9bd1cc489f2\">\n```\n\nThis post is just a teaser, if you want to find out more info please see [source code](https://github.com/icostan/bitmex-api-ruby) and [documentation](https://www.rubydoc.info/gems/bitmex-api).\n\nEnjoy futures trading!",
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      "author": "partiko",
      "body": "Thank you so much for sharing this amazing post with us!\n\nHave you heard about Partiko? It’s a really convenient mobile app for Steem! With Partiko, you can easily see what’s going on in the Steem community, make posts and comments (no beneficiary cut forever!), and always stayed connected with your followers via push notification!\n\nPartiko also rewards you with Partiko Points (3000 Partiko Point bonus when you first use it!), and Partiko Points can be converted into Steem tokens. You can earn Partiko Points easily by making posts and comments using Partiko.\n\nWe also noticed that your Steem Power is low. We will be very happy to delegate 15 Steem Power to you once you have made a post using Partiko! With more Steem Power, you can make more posts and comments, and earn more rewards!\n\nIf that all sounds interesting, you can: \n\n- Download Partiko Android at [Google Play](http://bit.ly/2SRFIta)\n- Or Download Partiko iOS on the [App Store](https://apple.co/2PcXkSd)\n\nThank you so much for reading this message!",
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      "parent_author": "icostan",
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      "author": "icostan",
      "body": "I just want to announce the very first release [v0.0.3] of Cryptos-ruby project, a simple and very easy to use Ruby API to  manipulate multiple crypto coins.\n\nFor more information please check [Github project page](https://github.com/icostan/cryptos-ruby).\n\n## Features\n  * Generate private/public keys\n  * Generate Bitcoin/Litecoin addresses (more to come)\n  * Create Bitcoin/Litecoin transactions\n  * Execute atomic swaps between Bitcoin and Litecoin\n\n## Installation\n```shell\ngem install cryptos\n```\n## Usage\n\n```ruby\n# Importing library...\n2.5.1 :001 > require 'cryptos'\n\n# Generating private key...\n2.5.1 :003 > private_key = Cryptos::PrivateKey.generate\n => #<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>\n\n# Generating public key...\n2.5.1 :004 > public_key = Cryptos::PublicKey.new private_key\n => #<Cryptos::PublicKey:0x00007fc9d122fc48 @private_key=#<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>, @x=37935518911551901189910488779135983333966395037400135768523765072809885233888, @y=60841756395196742058661917858407687798753194961782162321894682162865330386541>\n\n# Generating Bitcoin address for testnet\n2.5.1 :005 > address = Cryptos::Bitcoin::Address.new public_key\n => #<Cryptos::Bitcoin::Address:0x00007fc9d18a29b8 @public_key=#<Cryptos::PublicKey:0x00007fc9d122fc48 @private_key=#<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>, @x=37935518911551901189910488779135983333966395037400135768523765072809885233888, @y=60841756395196742058661917858407687798753194961782162321894682162865330386541>, @testnet=true>\n2.5.1 :006 > address.to_s\n => \"n37t517bJymxg3YftzqyVTAnjg4wKb3Tth\"\n\n# Generating Litecoin address for mainnet\n2.5.1 :007 > address = Cryptos::Litecoin::Address.new public_key, testnet: false\n => #<Cryptos::Litecoin::Address:0x00007fc9d119a1e8 @public_key=#<Cryptos::PublicKey:0x00007fc9d122fc48 @private_key=#<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>, @x=37935518911551901189910488779135983333966395037400135768523765072809885233888, @y=60841756395196742058661917858407687798753194961782162321894682162865330386541>, @testnet=false>\n2.5.1 :008 > address.to_s\n => \"Lgpt3ALSacam9jmDMZrtwZ2E5tqWVYzEkH\"\n```\n\n## Todos\n  * Work in progress: Zcash and Ethereum support\n  * Add more and more coins, cool crypto tech",
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      "title": "Introducing  Cryptos-ruby: The easiest way to craft your own transactions"
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      "author": "icostan",
      "body": "@@ -1,37 +1,4 @@\n-# Release: Cryptos-ruby project%0A%0A\n I ju\n",
      "json_metadata": "{\"tags\":[\"crypto\",\"bitcoin\",\"litecoin\",\"transaction\",\"atomicswaps\"],\"links\":[\"https://github.com/icostan/cryptos-ruby\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
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      "title": "Release: Cryptos-ruby project"
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      "author": "icostan",
      "body": "# Release: Cryptos-ruby project\n\nI just want to announce the very first release [v0.0.3] of Cryptos-ruby project, a simple and very easy to use Ruby API to  manipulate multiple crypto coins.\n\nFor more information please check [Github project page](https://github.com/icostan/cryptos-ruby).\n\n## Features\n  * Generate private/public keys\n  * Generate Bitcoin/Litecoin addresses (more to come)\n  * Create Bitcoin/Litecoin transactions\n  * Execute atomic swaps between Bitcoin and Litecoin\n\n## Installation\n```shell\ngem install cryptos\n```\n## Usage\n\n```ruby\n# Importing library...\n2.5.1 :001 > require 'cryptos'\n\n# Generating private key...\n2.5.1 :003 > private_key = Cryptos::PrivateKey.generate\n => #<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>\n\n# Generating public key...\n2.5.1 :004 > public_key = Cryptos::PublicKey.new private_key\n => #<Cryptos::PublicKey:0x00007fc9d122fc48 @private_key=#<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>, @x=37935518911551901189910488779135983333966395037400135768523765072809885233888, @y=60841756395196742058661917858407687798753194961782162321894682162865330386541>\n\n# Generating Bitcoin address for testnet\n2.5.1 :005 > address = Cryptos::Bitcoin::Address.new public_key\n => #<Cryptos::Bitcoin::Address:0x00007fc9d18a29b8 @public_key=#<Cryptos::PublicKey:0x00007fc9d122fc48 @private_key=#<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>, @x=37935518911551901189910488779135983333966395037400135768523765072809885233888, @y=60841756395196742058661917858407687798753194961782162321894682162865330386541>, @testnet=true>\n2.5.1 :006 > address.to_s\n => \"n37t517bJymxg3YftzqyVTAnjg4wKb3Tth\"\n\n# Generating Litecoin address for mainnet\n2.5.1 :007 > address = Cryptos::Litecoin::Address.new public_key, testnet: false\n => #<Cryptos::Litecoin::Address:0x00007fc9d119a1e8 @public_key=#<Cryptos::PublicKey:0x00007fc9d122fc48 @private_key=#<Cryptos::PrivateKey:0x00007fc9d123e2e8 @value=47930083789607287790662857866073624449854924554643360243140359905082181414216, @order=115792089237316195423570985008687907852837564279074904382605163141518161494337>, @x=37935518911551901189910488779135983333966395037400135768523765072809885233888, @y=60841756395196742058661917858407687798753194961782162321894682162865330386541>, @testnet=false>\n2.5.1 :008 > address.to_s\n => \"Lgpt3ALSacam9jmDMZrtwZ2E5tqWVYzEkH\"\n```\n\n## Todos\n  * Work in progress: Zcash and Ethereum support\n  * Add more and more coins, cool crypto tech",
      "json_metadata": "{\"tags\":[\"crypto\",\"bitcoin\",\"litecoin\",\"transaction\",\"atomicswaps\"],\"links\":[\"https://github.com/icostan/cryptos-ruby\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
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      "permlink": "release-cryptos-ruby-project",
      "title": "Release: Cryptos-ruby project"
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      "author": "icostan",
      "body": "# The hard way - Bitcoin: Transaction\n\n\nThis is Part 2 of 'The hard way - Bitcoin' series and I will start with 'the easy way' section first because even this gets a bit complex, then will continue with the hard stuff, crafting a Bitcoin transaction from scratch using basic math and cryptography.\n\n## A. The easy way\n\n### Create private key, public key and address\n\nWe are going to generate private key, public key and Bitcoin testnet address to be used in this article.\n\n```shell\nshell> bx seed # generate seed\n27634686aa8838acaa68e27aea6c072534394ce411cb7d75\n\nshell> echo 27634686aa8838acaa68e27aea6c072534394ce411cb7d75 | bx ec-new # generate private key\n79020296790075fc8e36835e045c513df8b20d3b3b9dbff4d043be84ae488f8d\n\nshell> echo 79020296790075fc8e36835e045c513df8b20d3b3b9dbff4d043be84ae488f8d | bx ec-to-public # generate public key\n03996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1\n\nshell> echo 03996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1 | bx ec-to-address # generate address\nn1C8nsmi4sc4hMBGgVZrnhxeFtk1sTbMZ4\n```\n### Previous transaction to be spent\n\nIn Bitcoin we have this concept called UTXO (Unspent Transaction Output) which is the unspent output of a previous transaction.\n\n```shell\nshell> bx fetch-tx d30de2a476060e08f4761ad99993ea1f7387bfcb3385f0d604a36a04676cdf93\ntransaction\n{\n    hash d30de2a476060e08f4761ad99993ea1f7387bfcb3385f0d604a36a04676cdf93\n    inputs\n    {\n        input\n        {\n            address_hash 10924d4f0aefd6dc652ce1fe906694eba4dfc8ae\n            previous_output\n            {\n                hash 264d0f5be4ebdd1102e23c4b440955b04b291c654323bdd33cd51d8989e3d54f\n                index 1\n            }\n            script \"[30450221009553c91c4aaf5ab137d5ed77cbc94a2c0511461f0de51b83fc2f00202f6a6beb022029a691221e1f664fa268524565d44f9e8a5387c5c091ec55f98ad12705720e7201] [02a46fc215bb6899814dc5bd026d815017784a2c4ceb78e7befdab1d5dfd51ef1a]\"\n            sequence 4294967294\n        }\n    }\n    lock_time 1384777\n    outputs\n    {\n        output\n        {\n            address_hash de69e4fc748732103653236dcc31c79e87422925\n            script \"dup hash160 [de69e4fc748732103653236dcc31c79e87422925] equalverify checksig\"\n            value 3625875580\n        }\n        output\n        {\n            address_hash d7d35ff2ed9cbc95e689338af8cd1db133be6a4a\n            script \"dup hash160 [d7d35ff2ed9cbc95e689338af8cd1db133be6a4a] equalverify checksig\"\n            value 65000000\n        }\n    }\n    version 2\n}\n```\n\n### Create transaction\n\nCreate transaction with two parameters: input (previous transaction hash and previous transaction output index) and output (receiving address and amount to send in Satoshi).\n\n```shell\nshell> bx tx-encode -i d30de2a476060e08f4761ad99993ea1f7387bfcb3385f0d604a36a04676cdf93:1 -o 2NFrxEjw5v2i7L8pm9dWjWSFpDRXmj8dBTn:64000000\n010000000193df6c67046aa304d6f08533cbbf87731fea9399d91a76f4080e0676a4e20dd30100000000ffffffff010090d0030000000017a914f81498040e79014455a5e8f7bd39bce5428121d38700000000\n```\n\n### Sign transaction\n\nSign the input to be spent using private key and lock script.\n\n```shell\nshell> bx input-sign 79020296790075fc8e36835e045c513df8b20d3b3b9dbff4d043be84ae488f8d \"dup hash160 [d7d35ff2ed9cbc95e689338af8cd1db133be6a4a] equalverify checksig\" 010000000193df6c67046aa304d6f08533cbbf87731fea9399d91a76f4080e0676a4e20dd30100000000ffffffff010090d0030000000017a914f81498040e79014455a5e8f7bd39bce5428121d38700000000\n3045022100b290086350a59ce28dd80cc89eac80eac097c20a50ed8c4f35b1ecbed789b65c02200129f4c34a9b05705d4f5e55acff0ce44b5565ab4a8c7faa4a74cf5e1367451101\n```\n\nCreate unlock script using signed endorsement and the public key then set it as transaction's input.\n\n```shell\nshell> bx input-set \"[3045022100b290086350a59ce28dd80cc89eac80eac097c20a50ed8c4f35b1ecbed789b65c02200129f4c34a9b05705d4f5e55acff0ce44b5565ab4a8c7faa4a74cf5e1367451101] [03996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1]\" 010000000193df6c67046aa304d6f08533cbbf87731fea9399d91a76f4080e0676a4e20dd30100000000ffffffff010090d0030000000017a914f81498040e79014455a5e8f7bd39bce5428121d38700000000\n010000000193df6c67046aa304d6f08533cbbf87731fea9399d91a76f4080e0676a4e20dd3010000006b483045022100b290086350a59ce28dd80cc89eac80eac097c20a50ed8c4f35b1ecbed789b65c02200129f4c34a9b05705d4f5e55acff0ce44b5565ab4a8c7faa4a74cf5e13674511012103996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1ffffffff010090d0030000000017a914f81498040e79014455a5e8f7bd39bce5428121d38700000000\n```\n\n### Validate transaction\n\nValidate input with public key, lock script and signed endorsement.\n\n```shell\nshell> bx input-validate 03996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1 \"dup hash160 [d7d35ff2ed9cbc95e689338af8cd1db133be6a4a] equalverify checksig\" 3045022100b290086350a59ce28dd80cc89eac80eac097c20a50ed8c4f35b1ecbed789b65c02200129f4c34a9b05705d4f5e55acff0ce44b5565ab4a8c7faa4a74cf5e1367451101 010000000193df6c67046aa304d6f08533cbbf87731fea9399d91a76f4080e0676a4e20dd3010000006b483045022100b290086350a59ce28dd80cc89eac80eac097c20a50ed8c4f35b1ecbed789b65c02200129f4c34a9b05705d4f5e55acff0ce44b5565ab4a8c7faa4a74cf5e13674511012103996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1ffffffff010090d0030000000017a914f81498040e79014455a5e8f7bd39bce5428121d38700000000\nThe endorsement is valid.\n```\n\nValidate encoded transaction as a whole.\n\n```shell\nshell> bx validate-tx 010000000193df6c67046aa304d6f08533cbbf87731fea9399d91a76f4080e0676a4e20dd3010000006b483045022100b290086350a59ce28dd80cc89eac80eac097c20a50ed8c4f35b1ecbed789b65c02200129f4c34a9b05705d4f5e55acff0ce44b5565ab4a8c7faa4a74cf5e13674511012103996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1ffffffff010090d0030000000017a914f81498040e79014455a5e8f7bd39bce5428121d38700000000\nThe transaction is valid.\n```\n\n### Broadcast transaction\n\nBroadcast the transaction to network.\n\n```shell\nbx send-tx 010000000193df6c67046aa304d6f08533cbbf87731fea9399d91a76f4080e0676a4e20dd3010000006b483045022100b290086350a59ce28dd80cc89eac80eac097c20a50ed8c4f35b1ecbed789b65c02200129f4c34a9b05705d4f5e55acff0ce44b5565ab4a8c7faa4a74cf5e13674511012103996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1ffffffff010090d0030000000017a914f81498040e79014455a5e8f7bd39bce5428121d38700000000\nSent transaction.\n```\n\nCheck address and transaction history.\n\n```shell\nshell> bx fetch-history n1C8nsmi4sc4hMBGgVZrnhxeFtk1sTbMZ4\ntransfers\n{\n    transfer\n    {\n        received\n        {\n            hash d30de2a476060e08f4761ad99993ea1f7387bfcb3385f0d604a36a04676cdf93\n            height 1384780\n            index 1\n        }\n        spent\n        {\n            hash fb8c52b201eb474ea3b2cb1f482cf4f774f710c711267818fbdd4763a5ca38b9\n            height 1384802\n            index 0\n        }\n        value 65000000\n    }\n}\n```\n\n## B. The hard way\n\nNow lets get down to real business and craft our own transaction starting with a few utility methods that we are going to use along the way.\n\nTwo methods to convert big numbers to bytes string and vice-versa, padding with `0x00` to satisfy the required length if any.\n\n```ruby\ndef bytes_to_bignum(bytes_string)\n  bytes_string.bytes.reduce { |n, b| (n << 8) + b }\nend\ndef bignum_to_bytes(n, length=nil)\n  a = []\n  while n > 0\n    a << (n & 0xFF)\n    n >>= 8\n  end\n  a.fill 0x00, a.length, length - a.length if length\n  a.reverse.pack('C*')\nend\n```\n\nBase58 encoding / decoding methods, Bitcoin address decoding to Hash160 (hashing twice with sha256 then ripe160) and build Bitcoin script based on destination address that can be Pay-to-PublicKey-Hash or Pay-to-Script-hash.\n\n```ruby\ndef bitcoin_base58_encode(ripe160_hash)\n  alphabet = '123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz'\n  value = ripe160_hash.to_i 16\n  output = ''\n  while value > 0\n    remainder = value % 58\n    value /= 58\n    output += alphabet[remainder]\n  end\n  output += alphabet[0] * [ripe160_hash].pack('H*').bytes.find_index{|b| b != 0}\n  output.reverse\nend\ndef bitcoin_base58_decode(address)\n  alphabet = '123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz'\n  int_val = 0\n  address.reverse.chars.each_with_index do |char, index|\n    char_index = alphabet.index(char)\n    int_val += char_index * 58**index\n  end\n  bignum_to_bytes(int_val, 25).unpack('H*').first\nend\ndef bitcoin_address_decode(address)\n  wrap_encode = bitcoin_base58_decode address\n  wrap_encode[2, 40]\nend\ndef bitcoin_script(address)\n  hash160 = bitcoin_address_decode address\n  if address.start_with? '2'\n    \"OP_HASH160 #{hash160} OP_EQUAL\"\n  else\n    \"OP_DUP OP_HASH160 #{hash160} OP_EQUALVERIFY OP_CHECKSIG\"\n  end\nend\n```\n\nMultiple utility methods to transform different data types into hex string.\n\n```ruby\nclass Struct\n  OPCODES = {\n    'OP_DUP' =>  0x76,\n    'OP_HASH160' =>  0xA9,\n    'OP_EQUAL' =>  0x87,\n    'OP_EQUALVERIFY' =>  0x88,\n    'OP_CHECKSIG' =>  0xAC\n  }.freeze\n  def opcode(token)\n    raise \"opcode #{token} not found\" unless OPCODES.include?(token)\n    OPCODES[token].to_s 16\n  end\n  def data(token)\n    bin_size = hex_size token\n    byte_to_hex(bin_size) + token\n  end\n  def hex_size(hex)\n    [hex].pack('H*').size\n  end\n  def to_hex(binary_bytes)\n    binary_bytes.unpack('H*').first\n  end\n  def hash_to_hex(value)\n    to_hex [value].pack('H*').reverse\n  end\n  def int_to_hex(value)\n    to_hex [value].pack('V')\n  end\n  def byte_to_hex(value)\n    to_hex [value].pack('C')\n  end\n  def long_to_hex(value)\n    to_hex [value].pack('Q<')\n  end\n  def script_to_hex(script_string)\n    script_string.split.map { |token| token.start_with?('OP') ? opcode(token) : data(token) }.join\n  end\n  def sha256(hex)\n    Digest::SHA256.hexdigest([hex].pack('H*'))\n  end\nend\n```\n\nNow, lets check the previous transaction that we want to spend.\n\n```shell\nshell> bx fetch-transaction ea8a64e122304637e8da016836e9afd59fc1179dfa15affc40d63b34e7db0cec\n\ntransaction\n{\n    hash ea8a64e122304637e8da016836e9afd59fc1179dfa15affc40d63b34e7db0cec\n    inputs\n    {\n        input\n        {\n            address_hash d7d35ff2ed9cbc95e689338af8cd1db133be6a4a\n            previous_output\n            {\n                hash 6b22d8a69432774ebbc00566755ff8da241be83c1b364ca177e241e23a0e111c\n                index 0\n            }\n            script \"[3045022100e4c449be7643140b9a62635afe8f99406311372f6ad4215d19a6d0b0ea306c7902203c6966502dba43b86d34d9252adb7632632bce1cf83ed721b5a1ac00e9e0f48a01] [03996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1]\"\n            sequence 4294967295\n        }\n    }\n    lock_time 0\n    outputs\n    {\n        output\n        {\n            address_hash ad99d5964d9180d501c8863f4bcd7b04f0766787\n            script \"dup hash160 [ad99d5964d9180d501c8863f4bcd7b04f0766787] equalverify checksig\"\n            value 1000000\n        }\n        output\n        {\n            address_hash d7d35ff2ed9cbc95e689338af8cd1db133be6a4a\n            script \"dup hash160 [d7d35ff2ed9cbc95e689338af8cd1db133be6a4a] equalverify checksig\"\n            value 23990000\n        }\n    }\n    version 1\n}\n```\n\nBuild transaction input from previous UTXO (Unspent Transaction Output): value, tx hash and output index.\n\n```ruby\nInput = Struct.new :value, :tx_hash, :index, :unlock_script, :sequence do\n  def initialize(value, tx_hash, index, unlock_script: '', sequence: 0xfffffffff)\n    super value, tx_hash, index, unlock_script, sequence\n  end\n  def serialize\n    script_hex = script_to_hex(unlock_script)\n    hash_to_hex(tx_hash) + int_to_hex(index) +\n      byte_to_hex(hex_size(script_hex)) + script_hex + int_to_hex(sequence)\n  end\nend\ninput = Input.new 23_990_000, 'ea8a64e122304637e8da016836e9afd59fc1179dfa15affc40d63b34e7db0cec', 1\n```\n\nBuild transaction output with output value and lock script.\n\n```ruby\nOutput = Struct.new :value, :lock_script do\n  def serialize\n    script_hex = script_to_hex(lock_script)\n    long_to_hex(value) + byte_to_hex(hex_size(script_hex)) + script_hex\n  end\nend\noutput_script = bitcoin_script '2N959B4qEPkce8jbzQC7EQaS6uaEBB9YTgQ'\noutput = Output.new 1_000_000, output_script\n```\n\nBuild the second output which is the 'change' transaction.\n\n```ruby\nchange_value = input.value - output.value - 10_000\nchange_script = bitcoin_script 'n1C8nsmi4sc4hMBGgVZrnhxeFtk1sTbMZ4'\nchange = Output.new change_value, change_script\n```\n\nBefore creating the actual transction lets introduce a few elliptic curve methods methods and parameters that we all know and love already.\n\n```ruby\nEC_Gx = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798\nEC_Gy = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8\nEC_p = 2**256 - 2**32 - 2**9 - 2**8 - 2**7 - 2**6 - 2**4 - 1\nEC_n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141\n\ndef extended_euclidean_algorithm(a, b)\n  s, old_s = 0, 1\n  t, old_t = 1, 0\n  r, old_r = b, a\n  while r != 0\n    quotient = old_r / r\n    old_r, r = r, old_r - quotient * r\n    old_s, s = s, old_s - quotient * s\n    old_t, t = t, old_t - quotient * t\n  end\n  [old_r, old_s, old_t]\nend\ndef inverse(n, p)\n  gcd, x, y = extended_euclidean_algorithm(n, p)\n  (n * x + p * y) % p == gcd || raise('invalid gcd')\n  gcd == 1 || raise('no multiplicative inverse')\n  x % p\nend\ndef ec_double(px, py, pn)\n  i_2y = inverse(2 * py, pn)\n  slope = (3 * px**2 * i_2y) % pn\n  x = (slope**2 - 2 * px) % pn\n  y = (slope*(px - x) - py) % pn\n  [x, y]\nend\ndef ec_add(ax, ay, bx, by, pn)\n  return [ax, ay] if bx == 0 && by == 0\n  return [bx, by] if ax == 0 && ay == 0\n  return ec_double(ax, ay, pn) if ax == bx && ay == by\n\n  i_bax = inverse(ax - bx, pn)\n  slope = ((ay - by) * i_bax) % pn\n  x = (slope**2 - ax - bx) % pn\n  y = (slope*(ax - x) - ay) % pn\n  [x, y]\nend\ndef ec_multiply(m, px, py, pn)\n  nx, ny = px, py\n  qx, qy = 0, 0\n  while m > 0\n    qx, qy = ec_add qx, qy, nx, ny, pn if m&1 == 1\n    nx, ny = ec_double nx, ny, pn\n    m >>= 1\n  end\n  [qx, qy]\nend\n```\n\nAnd these are the core methods of Elliptic Curve Digital Signature Algorithm (ECDSA), the sign/verify methods with DER signature encoding.\n\n```ruby\ndef ecdsa_sign(private_key, digest, temp_key = nil)\n  temp_key ||= 1 + SecureRandom.random_number(EC_n - 1)\n  rx, _ry = ec_multiply(temp_key, EC_Gx, EC_Gy, EC_p)\n  r = rx % EC_n\n  r > 0 || raise('r is zero, try again new temp key')\n  i_tk = inverse temp_key, EC_n\n  m = bytes_to_bignum digest\n  s = (i_tk * (m + r * private_key)) % EC_n\n  s > 0 || raise('s is zero, try again new temp key')\n  [r, s]\nend\ndef ecdsa_verify?(px, py, digest, signature)\n  r, s = signature\n  i_s = inverse s, EC_n\n  m = bytes_to_bignum digest\n  u1 = i_s * m % EC_n\n  u2 = i_s * r % EC_n\n  u1Gx, u1Gy = ec_multiply u1, EC_Gx, EC_Gy, EC_p\n  u2Px, u2Py = ec_multiply u2, px, py, EC_p\n  rx, _ry = ec_add u1Gx, u1Gy, u2Px, u2Py, EC_p\n  r == rx\nend\nDer = Struct.new :der, :length, :ri, :rl, :r, :si, :sl, :s, :sighash_type do\n  def initialize(der: 0x30, length: 0x45, ri: 0x02, rl: 0x21, r: nil, si: 0x02, sl: 0x20, s: nil, sighash_type: 0x01)\n    super der, length, ri, rl, r, si, sl, s, sighash_type\n  end\n  def serialize\n    byte_to_hex(der) + byte_to_hex(length) +\n      byte_to_hex(ri) + byte_to_hex(rl) + to_hex(bignum_to_bytes(r, 33)) +\n      byte_to_hex(si) + byte_to_hex(sl) + to_hex(bignum_to_bytes(s, 32)) +\n      byte_to_hex(sighash_type)\n  end\n  def self.parse(signature)\n    fields = *[signature].pack('H*').unpack('CCCCH66CCH64C')\n    Der.new r: fields[4], s: fields[7], sighash_type: fields[8]\n  end\nend\n```\n\nWe are getting there, build Bitcoin transaction passing in the input and the two outputs.\n\n```ruby\nTransaction = Struct.new :version, :inputs, :outputs, :locktime do\n  def serialize\n    inputs_hex = inputs.map(&:serialize).join\n    outputs_hex = outputs.map(&:serialize).join\n    int_to_hex(version) + byte_to_hex(inputs.size) + inputs_hex +\n      byte_to_hex(outputs.size) + outputs_hex + int_to_hex(locktime)\n  end\n  def hash\n    hash_to_hex sha256(sha256(serialize))\n  end\n  def signature_hash(lock_script = nil, sighash_type = 0x1)\n    inputs.first.unlock_script = lock_script if lock_script\n    hash = sha256(sha256(serialize + int_to_hex(sighash_type)))\n    [hash].pack('H*')\n  end\n  def sign(private_key, public_key, lock_script, sighash_type = 0x01)\n    bytes_string = signature_hash lock_script, sighash_type\n    r, s = ecdsa_sign private_key, bytes_string\n    der = Der.new r: r, s: s\n    inputs.first.unlock_script = \"#{der.serialize} #{public_key}\"\n    serialize\n  end\nend\ntransaction = Transaction.new 1, [input], [output, change], 0\n```\n\nFinally, to sign the transaction we need 3 things: private key, public key and the lock script of previous UTXO to spend.\n\n```ruby\nprivate_key = 0x79020296790075fc8e36835e045c513df8b20d3b3b9dbff4d043be84ae488f8d\npublic_key = '03996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1'\nlock_script = bitcoin_script 'n1C8nsmi4sc4hMBGgVZrnhxeFtk1sTbMZ4'\n\ntx_hex = transaction.sign private_key, public_key, lock_script\nputs \"TX hash: #{transaction.hash}\"\nputs \"TX hex: #{tx_hex}\"\n```\n\nAnd VOILA! take the returned transaction hex and decode/validate it:\n\n```shell\nshell> bx tx-decode 0100000001ec0cdbe7343bd640fcaf15fa9d17c19fd5afe9366801dae837463022e1648aea010000006b483045022100c1c85b5865f64f8f18c54f028a50942cd21cbda9c4a5f9296132defe51017c640220304793e361e71b3bae812923ca4aa5481a2fead05a73583ad5d930e7229f562f012103996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1ffffffff0240420f000000000017a914ad99d5964d9180d501c8863f4bcd7b04f076678787a0a55e01000000001976a914d7d35ff2ed9cbc95e689338af8cd1db133be6a4a88ac00000000\n\ntransaction\n{\n    hash c218dc394fef23ab9652ed6fd49539aa5dfb4f0153127b5c433032918948e60b\n    inputs\n    {\n        input\n        {\n            address_hash d7d35ff2ed9cbc95e689338af8cd1db133be6a4a\n            previous_output\n            {\n                hash ea8a64e122304637e8da016836e9afd59fc1179dfa15affc40d63b34e7db0cec\n                index 1\n            }\n            script \"[3045022100c1c85b5865f64f8f18c54f028a50942cd21cbda9c4a5f9296132defe51017c640220304793e361e71b3bae812923ca4aa5481a2fead05a73583ad5d930e7229f562f01] [03996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1]\"\n            sequence 4294967295\n        }\n    }\n    lock_time 0\n    outputs\n    {\n        output\n        {\n            address_hash ad99d5964d9180d501c8863f4bcd7b04f0766787\n            script \"hash160 [ad99d5964d9180d501c8863f4bcd7b04f0766787] equal\"\n            value 1000000\n        }\n        output\n        {\n            address_hash d7d35ff2ed9cbc95e689338af8cd1db133be6a4a\n            script \"dup hash160 [d7d35ff2ed9cbc95e689338af8cd1db133be6a4a] equalverify checksig\"\n            value 22980000\n        }\n    }\n    version 1\n}\n\n```\n\nthen submit it to Bitcoin testnet network via:\n\n- [Broadcast TX](https://testnet.blockchain.info/pushtx)\n- or using `bitcoin-cli` command line tool\n\n```shell\nshell> bitcoin-cli -testnet sendrawtransaction 0100000001ec0cdbe7343bd640fcaf15fa9d17c19fd5afe9366801dae837463022e1648aea010000006b483045022100c1c85b5865f64f8f18c54f028a50942cd21cbda9c4a5f9296132defe51017c640220304793e361e71b3bae812923ca4aa5481a2fead05a73583ad5d930e7229f562f012103996c918f74f0a6f1aeed99ebd81ab8eed8df99bc96fc082b20839259d332bad1ffffffff0240420f000000000017a914ad99d5964d9180d501c8863f4bcd7b04f076678787a0a55e01000000001976a914d7d35ff2ed9cbc95e689338af8cd1db133be6a4a88ac00000000\n```\n\n## C. Conclusions\n\nCreating transaction is a bit more difficult than generating a Bitcoin address that we saw in Part 1 but it is all possible once you understand the basics of Bitcoin like UTXOs, script and transactions.\n\n## D. References\n\nHere are the most important references but feel free to get in touch for more information.\n\n  * [OP_CHECKSIG](https://en.bitcoin.it/wiki/OP_CHECKSIG)\n  * [Bitcoin Script](https://en.bitcoin.it/wiki/Script)\n  * [Address prefixes](https://en.bitcoin.it/wiki/List_of_address_prefixes)\n  * [Libbitcoin-explorer](https://github.com/libbitcoin/libbitcoin-explorer/wiki)\n\nThe very end.",
      "json_metadata": "{\"tags\":[\"bitcoin\",\"cryptography\",\"ecdsa\",\"utxo\",\"transaction\"],\"links\":[\"https://testnet.blockchain.info/pushtx\",\"https://en.bitcoin.it/wiki/OP_CHECKSIG\",\"https://en.bitcoin.it/wiki/Script\",\"https://en.bitcoin.it/wiki/List_of_address_prefixes\",\"https://github.com/libbitcoin/libbitcoin-explorer/wiki\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
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      "author": "icostan",
      "body": "@@ -1,64 +1,4 @@\n-# The hard way - Bitcoin: private key, public key, address%0A%0A\n This\n",
      "json_metadata": "{\"tags\":[\"bitcoin\",\"cryptography\",\"elliptic-curve\",\"math\",\"crypto\"],\"links\":[\"https://gist.github.com/icostan/231b459f610e025d31e99b0bcbd8f90e\",\"https://en.bitcoin.it/wiki/Secp256k1\",\"https://it.slashdot.org/story/13/09/11/1224252/are-the-nist-standard-elliptic-curves-back-doored\",\"https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication\",\"https://en.bitcoin.it/wiki/List_of_address_prefixes\",\"https://en.bitcoin.it/wiki/Base58Check_encoding\",\"https://github.com/libbitcoin/libbitcoin-explorer/wiki/Command-Equivalence\",\"https://en.bitcoin.it/wiki/Technical_background_of_version_1_Bitcoin_addresses\",\"https://www.blockchain.com/btc/address/1PMycacnJaSqwwJqjawXBErnLsZ7RkXUAs\",\"http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/\",\"https://blockgeeks.com/guides/cryptocurrencies-cryptography/\",\"https://www.coindesk.com/math-behind-bitcoin/\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
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      "body": "@@ -9665,17 +9665,18 @@\n fo on th\n-e\n+is\n  subject\n",
      "json_metadata": "{\"tags\":[\"bitcoin\",\"cryptography\",\"elliptic-curve\",\"math\",\"crypto\"],\"links\":[\"https://gist.github.com/icostan/231b459f610e025d31e99b0bcbd8f90e\",\"https://en.bitcoin.it/wiki/Secp256k1\",\"https://it.slashdot.org/story/13/09/11/1224252/are-the-nist-standard-elliptic-curves-back-doored\",\"https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication\",\"https://en.bitcoin.it/wiki/List_of_address_prefixes\",\"https://en.bitcoin.it/wiki/Base58Check_encoding\",\"https://github.com/libbitcoin/libbitcoin-explorer/wiki/Command-Equivalence\",\"https://en.bitcoin.it/wiki/Technical_background_of_version_1_Bitcoin_addresses\",\"https://www.blockchain.com/btc/address/1PMycacnJaSqwwJqjawXBErnLsZ7RkXUAs\",\"http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/\",\"https://blockgeeks.com/guides/cryptocurrencies-cryptography/\",\"https://www.coindesk.com/math-behind-bitcoin/\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
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      "body": "@@ -9570,16 +9570,123 @@\n rences%0A%0A\n+See the most important references below and feel free to contact me if you need more info on the subject.%0A%0A\n   * %5BBit\n",
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      "body": "@@ -233,16 +233,22 @@\n In this \n+first \n section \n",
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      "body": "@@ -57,26 +57,427 @@\n ss%0A%0A\n-## A. The hard way\n+This article is all about two ways of generating a Bitcoin address: the hard way using simple math and the easy way using an existing Bitcoin library.%0A%0A## A. The hard way%0A%0AIn this section I am going to use simple math functions like addition and multiplication to generate a valid Bitcoin address starting from a number, the private key and calculating everything all the way up to public key and final Bitcoin address.\n %0A%0A##\n",
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      "body": "# The hard way - Bitcoin: private key, public key, address\n\n## A. The hard way\n\n### TL;DR;\n\n  * private key is just a number (like a PIN code) but a very very large one, `10^77` order of magnitude (put this in context of number of atoms in the known, observable universe which is `10^80`)\n  * public key is the result of scalar multiplication of private key and a point on Elliptic curve\n  * Bitcoin address is derived from public key by doing SHA256 / RIPE160 hashing and adding network prefix and checksum suffix\n  * we will generate valid Bitcoin address using two hash functions and simple math operations like addition, multiplication\n  * copy / paste Ruby code snippets below or get the[gist](https://gist.github.com/icostan/231b459f610e025d31e99b0bcbd8f90e)\n\n### Elliptic Curve domain parameters\n\nFirst of all let's talk about the 6 domain parameters that are defined in [secp256k1](https://en.bitcoin.it/wiki/Secp256k1): p,a,b,G,n,h. Read [this Slashdot thread](https://it.slashdot.org/story/13/09/11/1224252/are-the-nist-standard-elliptic-curves-back-doored) if you want to dive into a little bit of conspiracy theory.\n\n#### Prime number - p\n\nThe migthy prime number itself.\n\n``` ruby\nruby> p = 2**256 - 2**32 - 2**9 - 2**8 - 2**7 - 2**6 - 2**4 - 1\n => 115792089237316195423570985008687907853269984665640564039457584007908834671663\n```\n\n#### Elliptic Curve (EC) - a, b\n\nIn general elliptic curves are defined as `y^2 = x^3 + ax + b` equation but since `a = 0` and `b = 7` it becomes `y^2 = x^3 + 7` which is the elliptic curve used in Bitcoin.\n\n#### Generator point, order and cofactor - G, n and h\n\nThese 3 are all related and together they define a cyclic subgroup of elliptic curve where:\n\n  * `G` is a point on EC, also called the generator or base point, `(x,y)` coordinates represented in uncompressed format as a concatenation of `x` and `y` prefixed with `04` or in compressed format as `x` coordinate prefix with `02`.\n  * `n` is the order of subgroup generated by point `G`; in other words it is the number of points that belong to this subgroup\n  * `h` is called cofactor and can be calculated as `N/n` where `N` is the order of the entire elliptic curve group. In this case cofactor is 1 so the order of the subgroup is equal to the order of the entire elliptic curve.\n\n``` ruby\nruby> G = '0479BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8'\n => \"0479BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8\"\nruby> Gx = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798\n => 55066263022277343669578718895168534326250603453777594175500187360389116729240\nruby> Gy = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8\n => 32670510020758816978083085130507043184471273380659243275938904335757337482424\nruby> n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141\nruby> h = 1\n```\n\n### 0. Private key\n\nWe all know that Bitcoin's private key is 256 bits long that has to be an integer between `1` and `n`, the order of subgroup. Here is my take in binary representation.\n\n``` ruby\nruby> k = 0b1100011100001010010100111101101101010001100000111111101000010011010101001010011111000000100010100011100000001111001111100100011100111011101001110011111111001101001000111111000101100001000000011010111011011001010011010001000000110001100100001011100100101\n => 11253563012059685825953619222107823549092147699031672238385790369351542642469\nruby> k.to_s 16\n => \"18e14a7b6a307f426a94f8114701e7c8e774e7f9a47e2c2035db29a206321725\"\n```\n\n### 1. Public key\n\nNow, public key is just a scalar multiplication of private key (k) and elliptic curve generator point (G) as `P = k * G`. The result is another point `(x, y)` on elliptic curve, we take that `x` coordinate, prefix it with `02` if `y` is positive or `03` if `y` is negative and there it is, the public key in compressed format.\n\nA little bit of math and a few [algorithms](https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication) that we need to use for this calculation. For scalar multiplication you can add `G` to itself `k` number of times but this will be very inefficient so we are going to use \"double and add\" algorithm implemented in `ec_multiply` that in turn uses `ec_add` and `ec_double`, addition and doubling operations on elliptic curves. And last, since division is not defined over finite fields we need to multiply by the `multiplicative inverse` and use `extended_euclidean_algorithm` to find the greatest common divisor.\n\n```ruby\ndef ec_multiply(m, px, py, pn)\n  nx, ny = px, py\n  qx, qy = 0, 0\n  while m > 0\n    qx, qy = ec_add qx, qy, nx, ny, pn if m&1 == 1\n    nx, ny = ec_double nx, ny, pn\n    m >>= 1\n  end\n  [qx, qy]\nend\n\ndef ec_add(ax, ay, bx, by, pn)\n  return [ax, ay] if bx == 0 && by == 0\n  return [bx, by] if ax == 0 && ay == 0\n  return ec_double(ax, ay, pn) if ax == bx && ay == by\n\n  i_bax = inverse(ax - bx, pn)\n  slope = ((ay - by) * i_bax) % pn\n  x = (slope**2 - ax - bx) % pn\n  y = (slope*(ax - x) - ay) % pn\n  [x, y]\nend\n\ndef ec_double(px, py, pn)\n  i_2y = inverse(2 * py, pn)\n  slope = (3 * px**2 * i_2y) % pn\n  x = (slope**2 - 2 * px) % pn\n  y = (slope*(px - x) - py) % pn\n  [x, y]\nend\n\ndef inverse(n, p)\n  gcd, x, y = extended_euclidean_algorithm(n, p)\n  (n * x + p * y) % p == gcd || raise('invalid gcd')\n  gcd == 1 || raise('no multiplicative inverse')\n  x % p\nend\n\ndef extended_euclidean_algorithm(a, b)\n  s, old_s = 0, 1\n  t, old_t = 1, 0\n  r, old_r = b, a\n  while r != 0\n    quotient = old_r / r\n    old_r, r = r, old_r - quotient * r\n    old_s, s = s, old_s - quotient * s\n    old_t, t = t, old_t - quotient * t\n  end\n  [old_r, old_s, old_t]\nend\n```\n\n And here it is, point on elliptic curve and scalar multiplication.\n\n```ruby\nruby> Px, Py = ec_multiply(k, Gx, Gy, p)\n => [36422191471907241029883925342251831624200921388586025344128047678873736520530, 20277110887056303803699431755396003735040374760118964734768299847012543114150]\nruby> P = \"#{Py > 0 ? '02' : '03'}#{Px.to_s(16)}\"\n => \"0250863ad64a87ae8a2fe83c1af1a8403cb53f53e486d8511dad8a04887e5b2352\"\n```\n\nWe can easily check if resulting point is still on elliptic curve.\n\n```ruby\n((Px**3 + 7 - Py**2) % p == 0) || raise('public key point is not on the curve')\n```\n\n### 2. SHA256 hashing\n\nStandard SHA256 hashing here, I am going to use the implementation provided in Ruby.\n\n``` ruby\nruby> require 'digest'\nruby> sha256 = Digest::SHA256.hexdigest([P].pack('H*'))\n => \"0b7c28c9b7290c98d7438e70b3d3f7c848fbd7d1dc194ff83f4f7cc9b1378e98\"\n```\n\n### 3. RIPEMD160 hashing\n\nSame here, nothing new, standard RIPEMD160 hashing, no point reinventing the wheel.\n\n``` ruby\nruby> ripemd160 = Digest::RMD160.hexdigest([sha256].pack('H*'))\n => \"f54a5851e9372b87810a8e60cdd2e7cfd80b6e31\"\n```\n\n### 4. Add version byte\n\nHere are a few but see [Bitcoin address prefixes](https://en.bitcoin.it/wiki/List_of_address_prefixes) for a complete list of all supported prefixes and resulting Base58 encodings:\n  * 0x00 - Bitcoin address - result prefix is 1\n  * 0x05 - Pay-to-Script-Hash address - result prefix is 3\n  * 0x6F - Testnet address - resulting prefix is either m or n\n\n``` ruby\nruby> with_version = \"00#{ripemd160}\"\n => \"00f54a5851e9372b87810a8e60cdd2e7cfd80b6e31\"\n```\n\n### 5.6.7. Calculate checksum\n\nDouble SHA256 of previously calculated RIPEMD160 prefixed with version byte:\n\n``` ruby\nruby> checksum = Digest::SHA256.hexdigest(Digest::SHA256.digest([with_version].pack('H*')))[0, 8]\n => \"c7f18fe8\"\n```\n\n### 8. Wrap encoding\n\n``` ruby\nruby> wrap_encode = \"#{with_version}#{checksum}\"\n => \"00f54a5851e9372b87810a8e60cdd2e7cfd80b6e31c7f18fe8\"\n```\n\n### 9. Bitcoin address\n\nThe [Base58](https://en.bitcoin.it/wiki/Base58Check_encoding) algorithm removes confusing characters like \"oOiI\" and also \"+/\" signs to make entire address selectable on double click.\n\n```ruby\ndef base58(binary_hash)\n  alphabet = '123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz'\n  value = binary_hash.unpack('H*')[0].to_i 16\n  output = ''\n  while value > 0\n    remainder = value % 58\n    value /= 58\n    output += alphabet[remainder]\n  end\n  output += alphabet[0] * binary_hash.bytes.find_index{|b| b != 0}\n  output.reverse\nend\n```\n\nBase58 encoding and the final Bitcoin address:\n\n``` ruby\nruby> base58([wrap_encode].pack('H*'))\n => \"1PMycacnJaSqwwJqjawXBErnLsZ7RkXUAs\"\n```\n\n## B. The easy way\n\nAll the steps above can be done 'the easy way', as one-liner, using excellent [Libbitcoin Explorer](https://github.com/libbitcoin/libbitcoin-explorer/wiki/Command-Equivalence) tool:\n\n```shell\nshell> echo 18e14a7b6a307f426a94f8114701e7c8e774e7f9a47e2c2035db29a206321725 | bx ec-to-public | bx sha256 | bx ripemd160 | bx wrap-encode | bx base58-encode\n1PMycacnJaSqwwJqjawXBErnLsZ7RkXUAs\n```\n\n## C. Conclusions\n\nAnd here we are, the exact same Bitcoin address used in [Bitcoin address generation](https://en.bitcoin.it/wiki/Technical_background_of_version_1_Bitcoin_addresses) tutorial is generated the hard way and the easy way. You can [check the address](https://www.blockchain.com/btc/address/1PMycacnJaSqwwJqjawXBErnLsZ7RkXUAs).\n\n# References\n\n  * [Bitcoin address generation](https://en.bitcoin.it/wiki/Technical_background_of_version_1_Bitcoin_addresses)\n  * [Elliptic curve point multiplication](https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication)\n  * [Elliptic Curve Cryptography](http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/)\n  * [The Science Behind Cryptocurrencies Cryptography](https://blockgeeks.com/guides/cryptocurrencies-cryptography/)\n  * [Math behind Bitcoin](https://www.coindesk.com/math-behind-bitcoin/)",
      "json_metadata": "{\"tags\":[\"bitcoin\",\"cryptography\",\"elliptic-curve\",\"math\",\"crypto\"],\"links\":[\"https://gist.github.com/icostan/231b459f610e025d31e99b0bcbd8f90e\",\"https://en.bitcoin.it/wiki/Secp256k1\",\"https://it.slashdot.org/story/13/09/11/1224252/are-the-nist-standard-elliptic-curves-back-doored\",\"https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication\",\"https://en.bitcoin.it/wiki/List_of_address_prefixes\",\"https://en.bitcoin.it/wiki/Base58Check_encoding\",\"https://github.com/libbitcoin/libbitcoin-explorer/wiki/Command-Equivalence\",\"https://en.bitcoin.it/wiki/Technical_background_of_version_1_Bitcoin_addresses\",\"https://www.blockchain.com/btc/address/1PMycacnJaSqwwJqjawXBErnLsZ7RkXUAs\",\"http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/\",\"https://blockgeeks.com/guides/cryptocurrencies-cryptography/\",\"https://www.coindesk.com/math-behind-bitcoin/\"],\"app\":\"steemit/0.1\",\"format\":\"markdown\"}",
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