From 02f8c485c18cb2d3c2ac50ea70e99421996e56fc Mon Sep 17 00:00:00 2001 From: "David A. Harding" Date: Wed, 6 Sep 2023 10:03:37 +0200 Subject: [PATCH] Whitepaper: use MathML instead of screenshots MathML created using LaTeX from https://github.com/dhimmel/bitcoin-whitepaper/, a version of the paper formatted by Daniel Himmelstein --- images/mbc2_abin08.png | Bin 4291 -> 0 bytes images/mbc2_abin09.png | Bin 1320 -> 0 bytes images/mbc2_abin10.png | Bin 7276 -> 0 bytes images/mbc2_abin11.png | Bin 5473 -> 0 bytes whitepaper.adoc | 223 ++++++++++++++++++++++++++++++++++++++++- 5 files changed, 219 insertions(+), 4 deletions(-) delete mode 100755 images/mbc2_abin08.png delete mode 100755 images/mbc2_abin09.png delete mode 100755 images/mbc2_abin10.png delete mode 100755 images/mbc2_abin11.png diff --git a/images/mbc2_abin08.png b/images/mbc2_abin08.png deleted file mode 100755 index 9ea86e7d27b9c56b15f9e1990fe1dd9c292259c6..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 4291 zcmV;!5IpaRP)(_`g8%^e{{R4h=>PzAFaQARU;qF*m;eA5Z<1fd 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+ + + + q + z + + = + + { + + + + 1 + + + + 𝑖𝑓 + + p + + q + + + + + + + ( + q + + p + + ) + z + + + + + + 𝑖𝑓 + + p + > + q + + + + + } + + + +
+++++ Given our assumption that p > q, the probability drops exponentially as the number of blocks the attacker has to catch up with increases. With the odds against him, if he doesn't make a lucky lunge forward early on, his chances become vanishingly small as he falls further behind. @@ -123,15 +178,175 @@ The receiver generates a new key pair and gives the public key to the sender sho The recipient waits until the transaction has been added to a block and z blocks have been linked after it. He doesn't know the exact amount of progress the attacker has made, but assuming the honest blocks took the average expected time per block, the attacker's potential progress will be a Poisson distribution with expected value: -image::images/mbc2_abin09.png["eq2"] +++++ +
+ + + λ + = + z + + q + p + + + +
+++++ To get the probability the attacker could still catch up now, we multiply the Poisson density for each amount of progress he could have made by the probability he could catch up from that point: -image::images/mbc2_abin10.png["eq3"] +++++ +
+ + + + + + + k + = + 0 + + + + + + + + λ + k + + + e + + + λ + + + + + k + ! + + + + + { + + + + + ( + q + + p + + ) + + ( + z + + k + ) + + + + + + + 𝑖𝑓 + + k + + z + + + + + + 1 + + + + 𝑖𝑓 + + k + > + z + + + + + } + + + +
+++++ Rearranging to avoid summing the infinite tail of the distribution... -image::images/mbc2_abin11.png["eq4"] +++++ +
+ + + 1 + + + + + + k + = + 0 + + z + + + + + + λ + k + + + e + + + λ + + + + + k + ! + + + + ( + 1 + + ( + q + + p + + ) + + ( + z + + k + ) + + + ) + + + +
+++++ Converting to C code...