From 76e7b641119d1816c5f477b5b95b0bf8fdcc3b48 Mon Sep 17 00:00:00 2001
From: Tim Daly <daly@axiom-developer.org>
Date: Tue, 20 Jan 2015 02:32:24 -0500
Subject: src/input/bitcoin.input demonstrate finite fields and graphics

---
 changelog                        |    4 +-
 patch                            |    4 +-
 src/axiom-website/patches.html   |    2 +
 src/input/bitcoin.input.pamphlet |  757 ++++++++++++++++++++
 src/input/bitcoin1.eps           | 1403 ++++++++++++++++++++++++++++++++++++
 src/input/bitcoin2.eps           | 1406 ++++++++++++++++++++++++++++++++++++
 src/input/bitcoin3.eps           | 1406 ++++++++++++++++++++++++++++++++++++
 src/input/bitcoin4.eps           | 1442 +++++++++++++++++++++++++++++++++++++
 src/input/bitcoin5.eps           | 1460 ++++++++++++++++++++++++++++++++++++++
 9 files changed, 7880 insertions(+), 4 deletions(-)
 create mode 100644 src/input/bitcoin.input.pamphlet
 create mode 100644 src/input/bitcoin1.eps
 create mode 100644 src/input/bitcoin2.eps
 create mode 100644 src/input/bitcoin3.eps
 create mode 100644 src/input/bitcoin4.eps
 create mode 100644 src/input/bitcoin5.eps

diff --git a/changelog b/changelog
index 932523d..6cbeb58 100644
--- a/changelog
+++ b/changelog
@@ -1,4 +1,6 @@
-20150118 tpd src/axiom-website/patches.html 20150119.01.tpd.patch
+20150119 tpd src/axiom-website/patches.html 20150119.02.tpd.patch
+20150119 tpd src/input/bitcoin.input demonstrate finite fields and graphics
+20150119 tpd src/axiom-website/patches.html 20150119.01.tpd.patch
 20150119 tpd bug 7296: connect from VIEW2D is not graph specific
 20150118 tpd src/axiom-website/patches.html 20150118.01.tpd.patch
 20150118 tpd src/axiom-website/download.html add Docker instructions
diff --git a/patch b/patch
index 9ab3a3a..1eb3f1d 100644
--- a/patch
+++ b/patch
@@ -1,4 +1,2 @@
-bug 7296: connect from VIEW2D is not graph specific
+src/input/bitcoin.input demonstrate finite fields and graphics
 
-The connect function in VIEW2D globally turns lines on or off.
-This doesn't allow for overlaying lines on a set of points.
diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html
index a140fa2..a1be972 100644
--- a/src/axiom-website/patches.html
+++ b/src/axiom-website/patches.html
@@ -4960,6 +4960,8 @@ books/bookvol10 add SimpleCell to the algebra regression tests
 src/axiom-website/download.html add Docker instructions<br/>
 <a href="patches/20150119.01.tpd.patch">20150119.01.tpd.patch</a>
 bug 7296: connect from VIEW2D is not graph specific<br/>
+<a href="patches/20150119.02.tpd.patch">20150119.02.tpd.patch</a>
+src/input/bitcoin.input demonstrate finite fields and graphics
  </body>
 </html>
 
diff --git a/src/input/bitcoin.input.pamphlet b/src/input/bitcoin.input.pamphlet
new file mode 100644
index 0000000..ae0b8d7
--- /dev/null
+++ b/src/input/bitcoin.input.pamphlet
@@ -0,0 +1,757 @@
+\documentclass{article}
+\usepackage{axiom}
+\usepackage{graphicx}
+\setlength{\textwidth}{400pt}
+\begin{document}
+\title{The Math Behind Bitcoin\cite{1}}
+\author{Eric Walder and Timothy Daly}
+\maketitle
+\begin{abstract}
+A review of the mathematics behind bitcoin. Also a demonstration
+of finite field arithmetic and graphics generation in Axiom.
+\end{abstract}
+\newpage
+\tableofcontents
+\newpage
+
+One reason Bitcoin can be confusing for beginners is that the
+technology behind it redefines the concept of ownership.
+
+To own something in the traditional sense, be it a house or a sum of
+money, means either having personal custody of the thing or granting
+custody to a trusted entity such as a bank.
+
+With Bitcoin the case is different. Bitcoins themselves are not stored
+either centrally or locally and so no one entity is their
+custodian. They exist as records on a distributed ledger called the
+block chain, copies of which are shared by a volunteer network of
+connected computers. To ``own'' a bitcoin simply means having the
+ability to transfer control of it to someone else by creating a record
+of the transfer in the block chain. What grants this ability? Access
+to an ECDSA private and public key pair. What does that mean and how
+does that secure Bitcoin?
+
+Let's have a look under the hood.
+
+ECDSA is short for Elliptic Curve Digital Signature Algorithm. It's a
+process that uses an elliptic curve and a finite field to ``sign'' data
+in such a way that third parties can verify the authenticity of the
+signature while the signer retains the exclusive ability to create the
+signature. With Bitcoin, the data that is signed is the transaction
+that transfers ownership.
+
+ECDSA has separate procedures for signing and verification. Each
+procedure is an algorithm composed of a few arithmetic operations. The
+signing algorithm makes use of the private key, and the verification
+process makes use of the public key. We will show an example of this
+later.
+
+But first, a crash course on elliptic curves and finite fields.
+
+\section{Elliptic Curves}
+
+An elliptic curve is represented algebraically as an equation of the form:
+
+\begin{verbatim}
+    y2 = x3 + ax + b
+\end{verbatim}
+
+For a = 0 and b = 7 (the version used by Bitcoin), it looks like this:
+
+\includegraphics[scale=0.5]{bitcoin1.eps}
+\begin{chunk}{*}
+)clear all
+bitcoin1:=draw(y^2-x^3-7=0,x,y,range==[-5..5,-5..5],title=="Bitcoin curve")
+axes(bitcoin1,1,"on")
+units(bitcoin1,1,"on")
+points(bitcoin1,1,"off")
+
+\end{chunk}
+
+Elliptic curves have useful properties. For example, a non-vertical
+line intersecting two non-tangent points on the curve will always
+intersect a third point on the curve. A further property is that a
+non-vertical line tangent to the curve at one point will intersect
+precisely one other point on the curve.
+
+We can use these properties to define two operations: point addition
+and point doubling.
+
+Point addition, P + Q = R, is defined as the reflection through the
+x-axis of the third intersecting point R’ on a line that includes P
+and Q. It's easiest to understand this using a diagram:
+
+\includegraphics[scale=0.5]{bitcoin2.eps}
+\begin{chunk}{*}
+)clear all
+t1:=draw(0.5*x+2.5,x=-5..5)
+t2:=getGraph(t1,1)
+t3:=draw(10*x-15=0,x,y,range==[-5..5,-5..5])
+t4:=getGraph(t3,1)
+bitcoin2:=draw(y^2-x^3-7=0,x,y,range==[-5..5,-5..5],title=="Point Addition")
+putGraph(bitcoin2,t2,2)
+putGraph(bitcoin2,t4,3)
+axes(bitcoin2,1,"on")
+units(bitcoin2,1,"on")
+points(bitcoin2,1,"off")
+makeViewport2D(bitcoin2)
+
+\end{chunk}
+
+Similarly, point doubling, P + P = R is defined by finding the line
+tangent to the point to be doubled, P, and taking reflection through
+the x-axis of the intersecting point R' on the curve to get R. Here's
+an example of what that would look like:
+
+\includegraphics[scale=0.5]{bitcoin3.eps}
+\begin{chunk}{*}
+)clear all
+t1:=draw(x-238/100=0,x,y,range==[-5..5,-5..5])
+t2:=getGraph(t1,1)
+-- [-1.074,2.4],[2.382,4.531]
+t3:=draw(3083/5000*(x+537/500)+12/5-y=0,x,y,range==[-5..10,-10..10])
+t4:=getGraph(t3,1)
+bitcoin3:=draw(y^2-x^3-7=0,x,y,range==[-5..5,-5..5],title=="Point Doubling")
+putGraph(bitcoin3,t2,2)
+putGraph(bitcoin3,t4,3)
+axes(bitcoin3,1,"on")
+units(bitcoin3,1,"on")
+points(bitcoin3,1,"off")
+makeViewport2D(bitcoin3)
+
+\end{chunk}
+
+Together, these two operations are used for scalar multiplication, R =
+a P, defined by adding the point P to itself a times.  For example:
+
+\begin{verbatim}
+    R = 7P
+    R = P + (P + (P + (P + (P + (P + P)))))
+\end{verbatim}
+
+The process of scalar multiplication is normally simplified by using a
+combination of point addition and point doubling operations. For
+example:
+
+\begin{verbatim}
+    R = 7P
+    R = P + 6P
+    R = P + 2 (3P)
+    R = P + 2 (P + 2P)
+\end{verbatim}
+
+Here, 7P has been broken down into two point doubling steps and two
+point addition steps.  
+
+\section{Finite Fields}
+
+A finite field, in the context of ECDSA, can be thought of as a
+predefined range of positive numbers within which every calculation
+must fall. Any number outside this range ``wraps around'' so as to fall
+within the range.
+
+The simplest way to think about this is calculating remainders, as
+represented by the modulus (mod) operator. For example, 9/7 gives 1
+with a remainder of 2:
+
+\begin{verbatim}
+    9 mod 7 = 2
+\end{verbatim}
+
+Here our finite field is modulo 7, and all mod operations over this
+field yield a result falling within a range from 0 to 6.  
+
+In Axiom, we can create a finite field and do the arithmetic there.
+\begin{chunk}{*}
+mod7:=FiniteField(7,1)
+a:mod7:=9
+
+\end{chunk}
+
+\section{Putting it Together}
+
+ECDSA uses elliptic curves in the context of a finite field, which
+greatly changes their appearance but not their underlying equations or
+special properties. The same equation plotted above, in a finite field
+of modulo 67, looks like this:
+
+\includegraphics[scale=0.5]{bitcoin4.eps}
+\begin{chunk}{*}
+)clear all
+FIN:=FiniteField(67,1)  -- declare a field p**n, p=67, n=1
+f(x:INT,y:INT):FIN == (y^2-x^3-7)::FIN -- compute a finite point
+m():LIST(POINT(DFLOAT)) ==
+  lli:LIST(LIST(INT)) := []
+  for x in 0..66 repeat
+   for y in 0..66 repeat
+    a:FIN:=f(x,y)
+    if a = 0 then lli:=concat(lli,[[x,y]])
+  ld:=lli::LIST(LIST(DFLOAT))
+  lpd:=[point(p) for p in ld]
+m()
+vfloats:=vertical::LIST(LIST(DFLOAT))
+pfloats:=[point(p) for p in vfloats]
+ldo:=title("BitCoin Modular Points")
+rng:=range([0..67,0..67])
+bitcoin4:=draw(m(),[ldo,rng])$DRAWPT
+connect(bitcoin4,1,"off")$VIEW2D -- turn off the lines
+units(bitcoin4,1,"on")$VIEW2D -- turn on the units
+axes(bitcoin4,1,"on")$VIEW2D -- turn on the axes
+makeViewport2D(bitcoin4)
+
+\end{chunk}
+
+It's now a set of points, in which all the x and y values are integers
+between 0 and 66. Note that the ``curve'' still retains its horizontal
+symmetry.
+
+In Axiom, we can declare a function that takes Integer values for
+$x$ and $y$ and returns an answer from the finite field 67.
+\begin{chunk}{*}
+FIN:=FiniteField(67,1)  -- declare a field p**n, p=67, n=1
+f(x:INT,y:INT):FIN == (y^2-x^3-7)::FIN -- compute a finite point
+k():LIST(LIST(INT)) ==
+  b:LIST(LIST(INT)) := []
+  for x in 0..66 repeat
+   for y in 0..66 repeat
+    a:FIN:=f(x,y)
+    if a = 0 then b:=concat(b,[[x,y]])
+  b
+k()
+
+\end{chunk}
+
+Point addition and doubling are now slightly different visually. Lines
+drawn on this graph will wrap around the horizontal and vertical
+directions, just like in a game of Asteroids, maintaining the same
+slope. So adding points (2, 22) and (6, 25) looks like this:
+
+\includegraphics[scale=0.5]{bitcoin5.eps}
+\begin{chunk}{*}
+)clear all
+FIN:=FiniteField(67,1)  -- declare a field p**n, p=67, n=1
+f(x:INT,y:INT):FIN == (y^2-x^3-7)::FIN -- compute a finite point
+m():LIST(POINT(DFLOAT)) ==
+  lli:LIST(LIST(INT)) := []
+  for x in 0..66 repeat
+   for y in 0..66 repeat
+    a:FIN:=f(x,y)
+    if a = 0 then lli:=concat(lli,[[x,y]])
+  ld:=lli::LIST(LIST(DFLOAT))
+  lpd:=[point(p) for p in ld]
+m()
+line(x1:INT,y1:INT,x2:INT,y2:INT):EXPR(INT) ==
+    -- y-y1 = m(x-x1) where m = (y2-y1)/x2-x1)
+  (y2-y1)/(x2-x1)*(x-x1) + y1
+
+line1:=draw(line(2,22,6,25),x=0..67)
+line1graph:=getGraph(line1,1)
+
+line2:=draw(line(30,26,47,39),x=0..67)
+line2graph:=getGraph(line2,1)
+
+-- this never draws for some reason
+--line3:=draw(x-47=0,x,y,range==[0..67,0..67])
+--line3graph:=getGraph(line3,1)
+
+vertical:LIST(LIST(INT)):=_
+[[47,0],[47,1],[47,2],[47,3],[47,4],[47,5],[47,6],[47,7],[47,8],_
+[47,9],[47,10],[47,11],[47,12],[47,13],[47,14],[47,15],[47,16],[47,17],_
+[47,18],[47,19],[47,20],[47,21],[47,22],[47,23],[47,24],[47,25],[47,26],_
+[47,27],[47,28],[47,29],[47,30],[47,31],[47,32],[47,33],[47,34],[47,35],_
+[47,36],[47,37],[47,38],[47,39],[47,40],[47,41],[47,42],[47,43],[47,44],_
+[47,45],[47,46],[47,47],[47,48],[47,49],[47,50],[47,51],[47,52],[47,53],_
+[47,54],[47,55],[47,56],[47,57],[47,58],[47,59],[47,60],[47,61],[47,62],_
+[47,63],[47,64],[47,65],[47,66]]
+vfloats:=vertical::LIST(LIST(DFLOAT))
+pfloats:=[point(p) for p in vfloats]
+
+ldo:=title("BitCoin Modular Points")
+rng:=range([0..67,0..67])
+bitcoin5:=draw(append(m(),pfloats),[ldo,rng])$DRAWPT
+
+putGraph(bitcoin5,line1graph,2)
+putGraph(bitcoin5,line2graph,3)
+putGraph(bitcoin5,line3graph,4)
+axes(bitcoin5,1,"on")$VIEW2D -- turn on the axes
+units(bitcoin5,1,"on")$VIEW2D -- turn on the units
+connect(bitcoin5,1,"off")$VIEW2D -- turn off the lines
+makeViewport2D(bitcoin5)
+
+\end{chunk}
+
+The third intersecting point is (47, 39) and its reflection point is
+(47, 28). We can see this reflection point by looking that the list
+of points which make up the modular graph
+\begin{verbatim}
+   [[ 2,22], [ 2,45], [ 4, 2], [4,65],  [ 5,20], [ 5,47], [ 6,25], [ 6,42], 
+    [ 7,22], [ 7,45], [11,20], [11,47], [12,23], [12,44], [13,23], [13,44], 
+    [14, 2], [14,65], [16, 4], [16,63], [17,30], [17,37], [18,12], [18,55],
+    [21,25], [21,42], [23,28], [23,39], [24,30], [24,37], [25,17], [25,50],
+    [26,30], [26,37], [27,27], [27,40], [30,26], [30,41], [34, 7], [34,60],
+    [38,26], [38,41], [40,25], [40,42], [42,23], [42,44], [46,27], [46,40],
+    [47,28], [47,39], [48, 7], [48,60], [49, 2], [49,65], [51,20], [51,47],
+    [52, 7], [52,60], [53,12], [53,55], [54,17], [54,50], [55,17], [55,50],
+    [56, 4], [56,63], [58,22], [58,45], [61,27], [61,40], [62, 4], [62,63],
+    [63,12], [63,55], [64,28], [64,39], [66,26], [66,41]]
+\end{verbatim}
+
+Notice that the points come in pairs on the vertical axis. So we can
+look for the two points with first component of 47, which are
+[47,28] and [47,39]. Thus the reflection point of [47,39] is 
+[47,28].
+
+\section{Back to ECDSA and Bitcoin}
+
+A protocol such as Bitcoin selects a set of parameters for the
+elliptic curve and its finite field representation that is fixed for
+all users of the protocol. The parameters include the equation used,
+the prime modulo of the field, and a base point that falls on the
+curve. The order of the base point, which is not independently
+selected but is a function of the other parameters, can be thought of
+graphically as the number of times the point can be added to itself
+until its slope is infinite, or a vertical line. The base point is
+selected such that the order is a large prime number.
+
+Bitcoin uses very large numbers for its base point, prime modulo, and
+order. In fact, all practical applications of ECDSA use enormous
+values. The security of the algorithm relies on these values being
+large, and therefore impractical to brute force or reverse engineer.
+
+In the case of Bitcoin:
+
+\begin{verbatim}
+    Elliptic curve equation: y2 = x3 + 7
+
+    Prime modulo = 2256 – 232 – 29 – 28 – 27 – 26 – 24 - 1 = FFFFFFFF
+    FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F
+
+    Base point = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB
+    2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8
+    FD17B448 A6855419 9C47D08F FB10D4B8
+
+    Order = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B
+    BFD25E8C D0364141
+\end{verbatim}
+
+Who chose these numbers, and why? A great deal of research, and a fair
+amount of intrigue, surrounds the selection of appropriate
+parameters. After all, a large, seemingly random number could hide a
+backdoor method of reconstructing the private key. In brief, this
+particular realization goes by the name of secp256k1 and is part of a
+family of elliptic curve solutions over finite fields proposed for use
+in cryptography.
+
+\section{Private Keys and Public Keys}
+
+With these formalities out of the way, we are now in a position to
+understand private and public keys and how they are related. Here it
+is in a nutshell: In ECDSA, the private key is an unpredictably chosen
+number between 1 and the order. The public key is derived from the
+private key by scalar multiplication of the base point a number of
+times equal to the value of the private key. Expressed as an equation:
+
+\begin{verbatim}
+    public key = private key * base point
+\end{verbatim}
+
+This shows that the maximum possible number of private keys (and thus
+Bitcoin addresses) is equal to the order.
+
+In a continuous field we could plot the tangent line and pinpoint the
+public key on the graph, but there are some equations that accomplish
+the same thing in the context of finite fields. Point addition of p +
+q to find r is defined component-wise as follows:
+
+\begin{verbatim}
+    c = (qy - py) / (qx - px)
+    rx = c2 - px - qx
+    ry = c (px - rx) - py
+\end{verbatim}
+
+And point doubling of p to find r is as follows:
+
+\begin{verbatim}
+    c = (3px2 + a) / 2py
+    rx = c2 - 2px
+    ry = c (px - rx) - py 
+\end{verbatim}
+
+In practice, computation of the public key is broken down into a
+number of point doubling and point addition operations starting from
+the base point.
+
+Let's run a back of the envelope example using small numbers, to get
+an intuition about how the keys are constructed and used in signing
+and verifying. The parameters we will use are:
+
+\begin{verbatim}
+    Equation: y2 = x3 + 7  (which is to say, a = 0 and b = 7)
+    Prime Modulo: 67
+    Base Point: (2, 22)
+    Order: 79
+    Private key:  2
+\end{verbatim}
+
+First, let's find the public key. Since we have selected the simplest
+possible private key with value = 2, it will require only a single
+point doubling operation from the base point. The calculation looks
+like this:
+
+\begin{verbatim}
+    c = (3 * 22 + 0) / (2 * 22) mod 67
+    c = (3 * 4) / (44) mod 67
+    c = 12 / 44 mod 67
+\end{verbatim}
+
+Here we have to pause for a bit of sleight-of-hand: how do we perform
+division in the context of a finite field, where the result must
+always be an integer? We have to multiply by the inverse, which space
+does not permit us to define here (we refer you to here and here if
+interested). In the case at hand, you will have to trust us for the
+moment that:
+
+\begin{verbatim}
+      -1
+    44  = 32
+\end{verbatim}
+\begin{chunk}{*}
+)clear all 
+FIN:=FiniteField(67,1)  -- declare a field p**n, p=67, n=1
+inv(44)$FIN
+
+\end{chunk}
+
+Moving right along:
+
+\begin{verbatim}
+    c = 12 * 32 mod 67
+    c = 384 mod 67
+    c = 49
+\end{verbatim}
+\begin{chunk}{*}
+)clear all 
+FIN:=FiniteField(67,1)  -- declare a field p**n, p=67, n=1
+inv(12*32)::FIN
+
+\end{chunk}
+
+\begin{verbatim}
+    rx = (49^2 - 2 * 2) mod 67
+    rx = (2401 - 4) mod 67
+    rx = 2397 mod 67
+    rx = 52
+\end{verbatim}
+\begin{chunk}{*}
+)clear all 
+FIN:=FiniteField(67,1)  -- declare a field p**n, p=67, n=1
+(49^2 - 2 * 2)::FIN
+
+\end{chunk}
+
+\begin{verbatim}
+    ry = (49 * (2 - 52) - 22) mod 67
+    ry = (49 * (-50) - 22) mod 67
+    ry = (-2450 - 22) mod 67
+    ry = -2472 mod 67
+    ry = 7
+\end{verbatim}
+\begin{chunk}{*}
+)clear all 
+FIN:=FiniteField(67,1)  -- declare a field p**n, p=67, n=1
+(49 * (2 - 52) - 22)::FIN
+
+\end{chunk}
+
+Our public key thus corresponds to the point (52, 7). All that work
+for a private key of 2!
+
+This operation - going from private to public key - is computationally
+easy in comparison to trying to work backwards to deduce the private
+key from the public key, which while theoretically possible is
+computationally infeasible due to the large parameters used in actual
+elliptic cryptography.
+
+Therefore, going from the private key to the public key is by design a
+one-way trip.
+
+As with the private key, the public key is normally represented by a
+hexadecimal string. But wait, how do we get from a point on a plane,
+described by two numbers, to a single number? In an uncompressed
+public key the two 256-bit numbers representing the x and y
+coordinates are just stuck together in one long string. We can also
+take advantage of the symmetry of the elliptic curve to produce a
+compressed public key, by keeping just the x value and noting which
+half of the curve the point is on. From this partial information we
+can recover both coordinates.
+
+\section{Signing Data with the Private Key}
+
+Now that we have a private and public key pair, let's sign some data!
+
+The data can be of any length. The usual first step is to hash the
+data to generate a number containing the same number of bits (256) as
+the order of the curve. Here, for the sake of simplicity, we’ll skip
+the hashing step and just sign the raw data z. We’ll call G the base
+point, n the order, and d the private key. The recipe for signing is
+as follows:
+
+\begin{enumerate}
+\item Choose some integer k between 1 and n - 1.
+\item Calculate the point (x, y) = k * G, using scalar multiplication.
+\item Find r = x mod n. If r = 0, return to step 1.
+\item Find s = (z + r * d) / k mod n. If s = 0, return to step 1.
+\item The signature is the pair (r, s)
+\end{enumerate}
+
+As a reminder, in step 4, if the numbers result in a fraction (which
+in real life they almost always will), the numerator should be
+multiplied by the inverse of the denominator. In step 1, it is
+important that k not be repeated in different signatures and that it
+not be guessable by a third party. That is, k should either be random
+or generated by deterministic means that are kept secret from third
+parties. Otherwise it would be possible to extract the private key
+from step 4, since s, z, r, k and n are all known. You can read about
+a past exploit of this type\cite{2}.
+
+Let's pick our data to be the number 17, and follow the recipe. Our variables:
+
+\begin{verbatim}
+    z = 17 (data)
+    n = 79 (order)
+    G = (2, 22) (base point)
+    d = 2 (private key)
+\end{verbatim}
+
+1. Pick a random number:
+
+\begin{verbatim}
+    k = rand(1, n - 1)
+    k = rand(1, 79 - 1)
+    k = 3  (is this really random?  OK you got us, but it will make 
+            our example simpler!)
+\end{verbatim}
+
+2. Calculate the point. This is done in the same manner as determining
+the public key, but for brevity let's omit the arithmetic for point
+addition and point doubling.
+
+\begin{verbatim}
+    (x, y) = 3G
+    (x, y) = G + 2G
+    (x, y) = (2, 22) + (52, 7)
+    (x, y) = (62, 63)
+    x = 62
+    y = 63
+\end{verbatim}
+
+3. Find r:
+
+\begin{verbatim}
+    r = x mod n
+    r = 62 mod 79
+    r = 62
+\end{verbatim}
+
+4. Find s:
+
+\begin{verbatim}
+    s = (z + r * d) / k mod n
+    s = (17 + 62 * 2) / 3 mod 79
+    s = (17 + 124) / 3 mod 79
+    s = 141 / 3 mod 79
+    s = 47 mod 79
+    s = 47
+\end{verbatim}
+
+Note that above we were able to divide by 3 since the result was an
+integer. In real-life cases we would use the inverse of k (like
+before, we have hidden some gory details by computing it elsewhere):
+
+\begin{verbatim}
+    s = (z + r * d) / k mod n
+    s = (17 + 62 * 2) / 3 mod 79
+    s = (17 + 124) / 3 mod 79
+    s = 141 / 3 mod 79
+    s = 141 * 3-1 mod 79
+    s = 141 * 53 mod 79
+    s = 7473 mod 79
+    s = 47
+\end{verbatim}
+
+5. Our signature is the pair (r, s) = (62, 47).
+
+As with the private and public keys, this signature is normally
+represented by a hexadecimal string.  
+
+\section{Verifying the Signature with the Public Key}
+
+We now have some data and a signature for that data. A third party who
+has our public key can receive our data and signature, and verify that
+we are the senders. Let's see how this works.
+
+With Q being the public key and the other variables defined as before,
+the steps for verifying a signature are as follows:
+
+\begin{enumerate}
+\item Verify that r and s are between 1 and n - 1.
+\item Calculate w = s-1 mod n
+\item Calculate u = z * w mod n
+\item Calculate v = r * w mod n
+\item Calculate the point (x, y) = uG + vQ
+\item Verify that r = x mod n. The signature is invalid if it is not.
+\end{enumerate}
+
+Why do these steps work? We are skipping the proof, but you can read
+the details here. Let's follow the recipe and see how it works. Our
+variables, once again:
+
+\begin{verbatim}
+    z = 17 (data)
+    (r, s) = (62, 47) (signature)
+    n = 79 (order)
+    G = (2, 22) (base point)
+    Q = (52, 7) (public key)
+\end{verbatim}
+
+1. Verify that r and s are between 1 and n - 1. Check and check.
+
+\begin{verbatim}
+    r: 1 <= 62 < 79
+    s: 1 <= 47 < 79
+\end{verbatim}
+
+2. Calculate w:
+
+\begin{verbatim}
+    w = s-1 mod n
+    w = 47-1 mod 79
+    w = 37
+\end{verbatim}
+
+3. Calculate u:
+
+\begin{verbatim}
+    u = zw mod n
+    u = 17 * 37 mod 79
+    u = 629 mod 79
+    u = 76
+\end{verbatim}
+
+4. Calculate v:
+
+\begin{verbatim}
+    v = rw mod n
+    v = 62 * 37 mod 79
+    v = 2294 mod 79
+    v = 3
+\end{verbatim}
+
+5. Calculate the point (x, y):
+
+\begin{verbatim}
+    (x, y) = uG + vQ
+\end{verbatim}
+
+Let's break down the point doubling and addition in uG and vQ separately.
+
+\begin{verbatim}
+    uG = 76G
+    uG = 2(38G)
+    uG = 2( 2(19G) )
+    uG = 2( 2(G + 18G) )
+    uG = 2( 2(G + 2(9G) ) )
+    uG = 2( 2(G + 2(G + 8G) ) )
+    uG = 2( 2(G + 2(G + 2(4G) ) ) )
+    uG = 2( 2(G + 2(G + 2( 2(2G) ) ) ) )
+\end{verbatim}
+
+Sit back for a moment to appreciate that by using the grouping trick
+we reduce 75 successive addition operations to just six operations of
+point doubling and two operations of point addition. These tricks will
+come in handy when the numbers get really large.
+
+Working our way from the inside out:
+
+\begin{verbatim}
+    uG = 2( 2(G + 2(G + 2( 2( 2(2, 22) ) ) ) ) )
+    uG = 2( 2(G + 2(G + 2( 2(52, 7) ) ) ) )
+    uG = 2( 2(G + 2(G + 2(25, 17)  ) ) )
+    uG = 2( 2(G + 2( (2, 22) + (21, 42) ) ) )
+    uG = 2( 2(G + 2(13, 44) ) )
+    uG = 2( 2( (2, 22) + (66, 26) ) )
+    uG = 2( 2(38, 26) )
+    uG = 2(27, 40)
+    uG = (62, 4)
+\end{verbatim}
+
+And now for vQ:
+
+\begin{verbatim}
+    vQ = 3Q
+    vQ = Q + 2Q
+    vQ = Q + 2(52, 7)
+    vQ = (52, 7) + (25, 17)
+    vQ = (11, 20)
+\end{verbatim}
+
+Putting them together:
+
+\begin{verbatim}
+    (x, y) = uG + vQ
+    (x, y) = (62, 4) + (11, 20)
+    (x, y) = (62, 63)
+\end{verbatim}
+
+Clearly step 5 is the bulk of the work. For the final step,
+
+6. Verify that r = x mod n
+
+\begin{verbatim}
+    r = x mod n
+    62 = 62 mod 79
+    62 = 62
+\end{verbatim}
+
+Our signature is valid!
+
+\section{Conclusion}
+
+For those of you who saw all the equations and skipped to the bottom,
+what have we just learned?
+
+We have developed some intuition about the deep mathematical
+relationship that exists between public and private keys. We have seen
+how even in the simplest examples the math behind signatures and
+verification quickly gets complicated, and we can appreciate the
+enormous complexity which must be involved when the parameters
+involved are 256-bit numbers. We have seen how the clever application
+of the simplest mathematical procedures can create the one-way ``trap
+door'' functions necessary to preserve the information asymmetry which
+defines ownership of a bitcoin. And we have newfound confidence in the
+robustness of the system, provided that we carefully safeguard the
+knowledge of our private keys.
+
+In other words, this is why it is commonly said that Bitcoin is
+``backed by math.''
+
+If you hung in through the complicated bits, we hope it gave you the
+confidence to take the next step and try out the math on your own (a
+modular arithmetic calculator makes the finite field math much
+easier). We found that going through the steps of signing and
+verifying data by hand provides a deeper understanding of the
+cryptography that enables Bitcoin's unique form of ownership.
+
+***
+
+Special thanks to Steven Phelps for help with this article
+3 months ago
+
+\begin{thebibliography}{99}
+\bibitem[1]{1} 
+\verb|http://blog.chain.com/post/95218566791/the-math-behind-bitcoin|
+\bibitem[2]{2} 
+\verb|arstechnica.com/gaming/2010/12/ps3-hacked-through-poor-implementation|
+\verb|-of-cryptography|
+\end{thebibliography}
+\end{document}
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diff --git a/src/input/bitcoin5.eps b/src/input/bitcoin5.eps
new file mode 100644
index 0000000..e8023c5
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+showpage
+%%Trailer
+end
+%%EOF
-- 
1.7.5.4

