ch10: Align target description with the actual implementation

See: 43f3ada27b/src/pow.cpp (L86-L88)

ch10.asciidoc

@@ -408,7 +408,7 @@ Jing's node then fills in the target, which defines the required Proof-of-Work t
 
 The final field is the nonce, which is initialized to zero.
 
-With all the other fields filled, the block header is now complete and the process of mining can begin. The goal is now to find a value for the nonce that results in a block header hash that is less than the target. The mining node will need to test billions or trillions of nonce values before a nonce is found that satisfies the requirement.
+With all the other fields filled, the block header is now complete and the process of mining can begin. The goal is now to find a value for the nonce that results in a block header hash that is equal to or less than the target. The mining node will need to test billions or trillions of nonce values before a nonce is found that satisfies the requirement.
 
 === Mining the Block
 
@@ -490,11 +490,11 @@ Each phrase produces a completely different hash result. They seem completely ra
 
 The number used as a variable in such a scenario is called a _nonce_. The nonce is used to vary the output of a cryptographic function, in this case to vary the SHA256 fingerprint of the phrase.
 
-To make a challenge out of this algorithm, let's set a target: find a phrase that produces a hexadecimal hash that starts with a zero. Fortunately, this isn't difficult! <<sha256_example_generator_output>> shows that the phrase "I am Satoshi Nakamoto13" produces the hash +0ebc56d59a34f5082aaef3d66b37a661696c2b618e62432727216ba9531041a5+, which fits our criteria. It took 13 attempts to find it. In terms of probabilities, if the output of the hash function is evenly distributed we would expect to find a result with a 0 as the hexadecimal prefix once every 16 hashes (one out of 16 hexadecimal digits 0 through F). In numerical terms, that means finding a hash value that is less than +0x1000000000000000000000000000000000000000000000000000000000000000+. We call this threshold the _target_ and the goal is to find a hash that is numerically less than the target. If we decrease the target, the task of finding a hash that is less than the target becomes more and more difficult.
+To make a challenge out of this algorithm, let's set a target: find a phrase that produces a hexadecimal hash that starts with a zero. Fortunately, this isn't difficult! <<sha256_example_generator_output>> shows that the phrase "I am Satoshi Nakamoto13" produces the hash +0ebc56d59a34f5082aaef3d66b37a661696c2b618e62432727216ba9531041a5+, which fits our criteria. It took 13 attempts to find it. In terms of probabilities, if the output of the hash function is evenly distributed we would expect to find a result with a 0 as the hexadecimal prefix once every 16 hashes (one out of 16 hexadecimal digits 0 through F). In numerical terms, that means finding a hash value that is less than +0x1000000000000000000000000000000000000000000000000000000000000000+. We call this threshold the _target_ and the goal is to find a hash that is numerically equal to or less than the target. If we decrease the target, the task of finding a hash that is less than the target becomes more and more difficult.
 
-To give a simple analogy, imagine a game where players throw a pair of dice repeatedly, trying to throw less than a specified target. In the first round, the target is 12. Unless you throw double-six, you win. In the next round the target is 11. Players must throw 10 or less to win, again an easy task. Let's say a few rounds later the target is down to 5. Now, more than half the dice throws will exceed the target and therefore be invalid. It takes exponentially more dice throws to win, the lower the target gets. Eventually, when the target is 2 (the minimum possible), only one throw out of every 36, or 2% of them, will produce a winning result.
+To give a simple analogy, imagine a game where players throw a pair of dice repeatedly, trying to throw equal to or less than a specified target. In the first round, the target is 11. Unless you throw double-six, you win. In the next round the target is 10. Players must throw 10 or less to win, again an easy task. Let's say a few rounds later the target is down to 5. Now, more than half the dice throws will exceed the target and therefore be invalid. It takes exponentially more dice throws to win, the lower the target gets. Eventually, when the target is 2 (the minimum possible), only one throw out of every 36, or 2% of them, will produce a winning result.
 
-From the perspective of an observer who knows that the target of the dice game is 2, if someone has succeeded in casting a winning throw it can be assumed that they attempted, on average, 36 throws. In other words, one can estimate the amount of work it takes to succeed from the difficulty imposed by the target. When the algorithm is based on a deterministic function such as SHA256, the input itself constitutes _proof_ that a certain amount of _work_ was done to produce a result below the target. Hence, _Proof-of-Work_.
+From the perspective of an observer who knows that the target of the dice game is 2, if someone has succeeded in casting a winning throw it can be assumed that they attempted, on average, 36 throws. In other words, one can estimate the amount of work it takes to succeed from the difficulty imposed by the target. When the algorithm is based on a deterministic function such as SHA256, the input itself constitutes _proof_ that a certain amount of _work_ was done to produce a result equal to or below the target. Hence, _Proof-of-Work_.
 
 [TIP]
 ====
@@ -505,11 +505,11 @@ In <<sha256_example_generator_output>>, the winning "nonce" is 13 and this resul
 
 [TIP]
 ====
-The Proof-of-Work must produce a hash that is _less than_ the target. A higher target means it is less difficult to find a hash that is below the target. A lower target means it is more difficult to find a hash below the target. The target and difficulty are inversely related.
+The Proof-of-Work must produce a hash that is _equal to or less than_ the target. A higher target means it is less difficult to find a hash that is equal to or below the target. A lower target means it is more difficult to find a hash equal to or below the target. The target and difficulty are inversely related.
 ====
 
 
-((("targets")))Bitcoin's Proof-of-Work is very similar to the challenge shown in <<sha256_example_generator_output>>. The miner constructs a candidate block filled with transactions. Next, the miner calculates the hash of this block's header and sees if it is smaller than the current _target_. If the hash is not less than the target, the miner will modify the nonce (usually just incrementing it by one) and try again. At the current difficulty in the bitcoin network, miners have to try quadrillions of times before finding a nonce that results in a low enough block header hash.
+((("targets")))Bitcoin's Proof-of-Work is very similar to the challenge shown in <<sha256_example_generator_output>>. The miner constructs a candidate block filled with transactions. Next, the miner calculates the hash of this block's header and sees if it is equal to or smaller than the current _target_. If the hash is greater than the target, the miner will modify the nonce (usually just incrementing it by one) and try again. At the current difficulty in the bitcoin network, miners have to try quadrillions of times before finding a nonce that results in a low enough block header hash.
 
 A very simplified Proof-of-Work algorithm is implemented in Python in <<pow_example1>>.
 
@@ -590,7 +590,7 @@ Hashing Power: 127141 hashes per second
 
 As you can see, increasing the difficulty by 1 bit causes a doubling in the time it takes to find a solution. If you think of the entire 256-bit number space, each time you constrain one more bit to zero, you decrease the search space by half. In <<pow_example_outputs>>, it takes 84 million hash attempts to find a nonce that produces a hash with 26 leading bits as zero. Even at a speed of more than 120,000 hashes per second, it still requires 10 minutes on a laptop to find this solution.
 
-At the time of writing, the network is attempting to find a block whose header hash is less than:
+At the time of writing, the network is attempting to find a block whose header hash is equal to or less than:
 
 ----
 0000000000000000029AB9000000000000000000000000000000000000000000
@@ -734,7 +734,7 @@ In the next section, we'll look at the process each node uses to validate a bloc
 When a node receives a new block, it will validate the block by checking it against a long list of criteria that must all be met; otherwise, the block is rejected. These criteria can be seen in the Bitcoin Core client in the functions +CheckBlock+ and +CheckBlockHeader+ and include:
 
 * The block data structure is syntactically valid
-* The block header hash is less than the target (enforces the Proof-of-Work)
+* The block header hash is equal to or less than the target (enforces the Proof-of-Work)
 * The block timestamp is less than two hours in the future (allowing for time errors)
 * The block size is within acceptable limits
 * The first transaction (and only the first) is a coinbase transaction
@@ -860,7 +860,7 @@ In the last two years, the ASIC mining chips have become increasingly denser, ap
 [[extra_nonce]]
 ==== The Extra Nonce Solution
 
-((("nonce values")))Since 2012, bitcoin mining has evolved to resolve a fundamental limitation in the structure of the block header. In the early days of bitcoin, a miner could find a block by iterating through the nonce until the resulting hash was below the target. As difficulty increased, miners often cycled through all 4 billion values of the nonce without finding a block. However, this was easily resolved by updating the block timestamp to account for the elapsed time. Because the timestamp is part of the header, the change would allow miners to iterate through the values of the nonce again with different results.  Once mining hardware exceeded 4 GH/sec, however, this approach became increasingly difficult because the nonce values were exhausted in less than a second. As ASIC mining equipment started pushing and then exceeding the TH/sec hash rate, the mining software needed more space for nonce values in order to find valid blocks. The timestamp could be stretched a bit, but moving it too far into the future would cause the block to become invalid. A new source of "change" was needed in the block header. The solution was to use the coinbase transaction as a source of extra nonce values. Because the coinbase script can store between 2 and 100 bytes of data, miners started using that space as extra nonce space, allowing them to explore a much larger range of block header values to find valid blocks. The coinbase transaction is included in the merkle tree, which means that any change in the coinbase script causes the merkle root to change. Eight bytes of extra nonce, plus the 4 bytes of "standard" nonce allow miners to explore a total 2^96^ (8 followed by 28 zeros) possibilities _per second_ without having to modify the timestamp. If, in the future, miners could run through all these possibilities, they could then modify the timestamp. There is also more space in the coinbase script for future expansion of the extra nonce space.
+((("nonce values")))Since 2012, bitcoin mining has evolved to resolve a fundamental limitation in the structure of the block header. In the early days of bitcoin, a miner could find a block by iterating through the nonce until the resulting hash was equal to or below the target. As difficulty increased, miners often cycled through all 4 billion values of the nonce without finding a block. However, this was easily resolved by updating the block timestamp to account for the elapsed time. Because the timestamp is part of the header, the change would allow miners to iterate through the values of the nonce again with different results.  Once mining hardware exceeded 4 GH/sec, however, this approach became increasingly difficult because the nonce values were exhausted in less than a second. As ASIC mining equipment started pushing and then exceeding the TH/sec hash rate, the mining software needed more space for nonce values in order to find valid blocks. The timestamp could be stretched a bit, but moving it too far into the future would cause the block to become invalid. A new source of "change" was needed in the block header. The solution was to use the coinbase transaction as a source of extra nonce values. Because the coinbase script can store between 2 and 100 bytes of data, miners started using that space as extra nonce space, allowing them to explore a much larger range of block header values to find valid blocks. The coinbase transaction is included in the merkle tree, which means that any change in the coinbase script causes the merkle root to change. Eight bytes of extra nonce, plus the 4 bytes of "standard" nonce allow miners to explore a total 2^96^ (8 followed by 28 zeros) possibilities _per second_ without having to modify the timestamp. If, in the future, miners could run through all these possibilities, they could then modify the timestamp. There is also more space in the coinbase script for future expansion of the extra nonce space.
 
 [[mining_pools]]
 ==== Mining Pools
@@ -885,11 +885,11 @@ Successful blocks pay the reward to a pool bitcoin address, rather than individu
 
 ((("mining pools", "operation of")))Miners participating in a pool split the work of searching for a solution to a candidate block, earning "shares" for their mining contribution. The mining pool sets a higher target (lower difficulty) for earning a share, typically more than 1,000 times easier than the bitcoin network's target. When someone in the pool successfully mines a block, the reward is earned by the pool and then shared with all miners in proportion to the number of shares they contributed to the effort.
 
-Pools are open to any miner, big or small, professional or amateur. A pool will therefore have some participants with a single small mining machine, and others with a garage full of high-end mining hardware. Some will be mining with a few tens of a kilowatt of electricity, others will be running a data center consuming a megawatt of power. How does a mining pool measure the individual contributions, so as to fairly distribute the rewards, without the possibility of cheating? The answer is to use bitcoin's Proof-of-Work algorithm to measure each pool miner's contribution, but set at a lower difficulty so that even the smallest pool miners win a share frequently enough to make it worthwhile to contribute to the pool. By setting a lower difficulty for earning shares, the pool measures the amount of work done by each miner. Each time a pool miner finds a block header hash that is less than the pool target, she proves she has done the hashing work to find that result. More importantly, the work to find shares contributes, in a statistically measurable way, to the overall effort to find a hash lower than the bitcoin network's target. Thousands of miners trying to find low-value hashes will eventually find one low enough to satisfy the bitcoin network target.
+Pools are open to any miner, big or small, professional or amateur. A pool will therefore have some participants with a single small mining machine, and others with a garage full of high-end mining hardware. Some will be mining with a few tens of a kilowatt of electricity, others will be running a data center consuming a megawatt of power. How does a mining pool measure the individual contributions, so as to fairly distribute the rewards, without the possibility of cheating? The answer is to use bitcoin's Proof-of-Work algorithm to measure each pool miner's contribution, but set at a lower difficulty so that even the smallest pool miners win a share frequently enough to make it worthwhile to contribute to the pool. By setting a lower difficulty for earning shares, the pool measures the amount of work done by each miner. Each time a pool miner finds a block header hash that is equal to or less than the pool target, she proves she has done the hashing work to find that result. More importantly, the work to find shares contributes, in a statistically measurable way, to the overall effort to find a hash lower than the bitcoin network's target. Thousands of miners trying to find low-value hashes will eventually find one low enough to satisfy the bitcoin network target.
 
-Let's return to the analogy of a dice game. If the dice players are throwing dice with a goal of throwing less than four (the overall network difficulty), a pool would set an easier target, counting how many times the pool players managed to throw less than eight. When pool players throw less than eight (the pool share target), they earn shares, but they don't win the game because they don't achieve the game target (less than four). The pool players will achieve the easier pool target much more often, earning them shares very regularly, even when they don't achieve the harder target of winning the game. Every now and then, one of the pool players will throw a combined dice throw of less than four and the pool wins. Then, the earnings can be distributed to the pool players based on the shares they earned. Even though the target of eight-or-less wasn't winning, it was a fair way to measure dice throws for the players, and it occasionally produces a less-than-four throw.
+Let's return to the analogy of a dice game. If the dice players are throwing dice with a goal of throwing equal to or less than four (the overall network difficulty), a pool would set an easier target, counting how many times the pool players managed to throw equal to or less than eight. When pool players throw equal to or less than eight (the pool share target), they earn shares, but they don't win the game because they don't achieve the game target (equal to or less than four). The pool players will achieve the easier pool target much more often, earning them shares very regularly, even when they don't achieve the harder target of winning the game. Every now and then, one of the pool players will throw a combined dice throw of equal to or less than four and the pool wins. Then, the earnings can be distributed to the pool players based on the shares they earned. Even though the target of eight-or-less wasn't winning, it was a fair way to measure dice throws for the players, and it occasionally produces a four-or-less throw.
 
-Similarly, a mining pool will set a (higher and easier) pool target that will ensure that an individual pool miner can find block header hashes that are less than the pool target often, earning shares. Every now and then, one of these attempts will produce a block header hash that is less than the bitcoin network target, making it a valid block and the whole pool wins.
+Similarly, a mining pool will set a (higher and easier) pool target that will ensure that an individual pool miner can find block header hashes that are equal to or less than the pool target often, earning shares. Every now and then, one of these attempts will produce a block header hash that is equal to or less than the bitcoin network target, making it a valid block and the whole pool wins.
 
 ===== Managed pools
 

code/proof-of-work-example.py

@@ -21,8 +21,8 @@ def proof_of_work(header, difficulty_bits):
     for nonce in xrange(max_nonce):
         hash_result = hashlib.sha256((str(header) + str(nonce)).encode()).hexdigest()
 
-        # check if this is a valid result, below the target
-        if long(hash_result, 16) < target:
+        # check if this is a valid result, equal to or below the target
+        if long(hash_result, 16) <= target:
             print("Success with nonce %d" % nonce)
             print("Hash is %s" % hash_result)
             return (hash_result, nonce)