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b i 0 b map x i x a i a i 0 a b a length b length a map x i x b i b i 0 b map x i x a i a i 0 a b var result for var i 0 i a length i for var j 0 j b length j result i j result i j result i j 0 result i j a i b j return result a b throw new error not implemented var str new proxy makecontext array a b a 0 b 0 a b array from array b 0 map x a 0 join has target key return key match a z0 9a z_ key str get target prop receiver if typeof prop string return prop else return target prop function reduce numerator denominator var gcd function gcd a b return b gcd b a b a gcd gcd numerator denominator return numerator gcd denominator gcd var numdenom n 0 9 d 0 9 var rat new proxy makecontext array a b reduce a 0 b 1 a 1 b 0 a 1 b 1 a b reduce a 0 b 1 a 1 b 0 a 1 b 1 a b reduce a 0 b 0 a 1 b 1 a b reduce a 0 b 1 a 1 b 0 has target key return key match numdenom get target prop receiver if typeof prop string var m prop match numdenom return reduce bigint m 1 bigint m 2 else return target prop var sets makecontext set disjoint union a b new set a map v 0 v b map v 1 v product a b new set a flatmap v b map w v w var inj makecontext set union a b new set a b difference a b new set a filter v b has v intersection a b new set a filter v b has v var bind makecontext object array function a b a flatmap b leave a comment sea robins posted in uncategorized by mike stay on 2020 may 18 they have six legs they bark leave a comment watch a salamander grow from a single cell posted in uncategorized by mike stay on 2020 may 15 leave a comment kawah ijen by night posted in uncategorized by mike stay on 2020 april 30 leave a comment mrs wingate posted in uncategorized by mike stay on 2020 april 24 when i was a senior in high school i had calculus right after lunch the room was also on the west side of the building so the combination of dense math a full belly and the warm sun often conspired to make me fall asleep one day woke up and didn t understand a term my teacher mrs wingate was using it was a word i knew normal but in a way i d never heard it i raised my hand to ask what it meant and she replied mike go back to sleep the reason she was so forgiving was apparently because i was the only one in the class who came in after school to ask more questions some she knew how to answer like how to calculate the surface volume of a hypersphere others she didn t like whether you can raise an imaginary number to an imaginary power but she dug through her files and found me an article on polar coordinates and euler s formula she had a small library of mathematical books i borrowed a book of mathematical fiction it was volume 4 in some series bound in navy blue with an infinity symbol on the spine it had one story about a device to twist things into a fourth dimension for among other things robbing bank vaults there was another about a topologist who dissatisfied with the möbius strip s single side discovered a complicated topological surface with no sides but whenever he tried to make one out of paper it would disappear she also had a book about flexagons when she retired she gave my family the book on flexagons because every one of my five younger siblings also borrowed the book and no other students ever had she taught us so well that taking the ap calculus exam was a downright enjoyable experience i can t thank her enough for leading me to one of the principal joys of my life leave a comment anomalous magnetic moment still anomalous posted in uncategorized by mike stay on 2020 april 5 supercomputer simulations show effect of hadron light by light scattering is far too small to account for the 3 7σ deviation from standard model predictions there s still hope for new physics at accessible energies leave a comment how we recovered over 300k of bitcoin posted in uncategorized by mike stay on 2020 april 3 story time around twenty years ago i was working as a reverse engineer and cryptanalyst for accessdata while getting my physics degree at byu it was the late 90s and early 2000s the us government had been gradually easing off export restrictions on software containing cryptography but most software still contained pretty worthless password protection we d buy desktop office software i d reverse engineer it to figure out what algorithm it was using for access control and then break the crypto it was a never ending stream of interesting but not impossible math puzzles i wrote about forty password crackers during my time there we would sell them to home users system administrators and local and federal law enforcement agencies i got to go down to the federal law enforcement training center in glynco a few times to teach the secret service fbi and atf about cryptography and the use of our products two projects really stand out in my memory the first was microsoft word 97 before word 97 the files were encrypted by xoring the bytes with a repeating 16 byte string derived from the password the most common bytes in a word file were usually either 0x00 0xff or 0x20 space so we d just look at the most common character in each column and try the 3 16 variations on those recovering the key was usually instantaneous but to help people feel like they d gotten their money s worth we d put on a little animated show like a hollywood hacking scene with lots of random characters that gradually revealed the right password microsoft 97 changed that it might have been possible to find out the encryption format through msdn but we were small and couldn t afford a subscription it also wasn t clear that we d be allowed to write the cracker if we did get the info from them to figure out how it worked i used softice to stop the software immediately after typing in the password then walked slowly up the stack until i could find the algorithm this was before ida pro so i printed out a few dozen pages of assembly code taped it to the wall and drew all over it with a red crayon i was very pleased when i finally worked it all out at the time microsoft was only allowed to export 40 bit cryptography so they did as much as they were legally permitted they d repeatedly md5 hash the password with salt randomly chosen bytes stored in the file to get the 40 bit key then add salt to that and repeatedly hash it again it took roughly half a second to test a password on the computers of the time we had to resort to a dictionary attack because breaking it outright was pretty much impossible we did eventually write a cracker for larger companies and agencies that brute forced the 40 bit keyspace using those fancy mmx instructions on the pentium i knew of one place that ran the software for nine months before finally getting in the other really fun one one was zip archives the developer of pkzip phil katz made the decision unusual at the time to document his file format and include it with his software making it a favorite of developers roger schlafly designed the stream cipher used for encrypting the archives the zip standard quickly became by far the most popular compression format on windows and many other formats like java s jar format and openoffice s document formats are really zip files with a particular directory structure inside infozip is an open source version of the software its implementation was used as the basis for nearly all other branded zip software like winzip at the time i was trying to crack it winzip had 95 of the market eli biham and paul kocher had published a known plaintext attack on the cipher but the known plaintext was compressed plaintext to get the huffman codes at the start of a deflated file you basically need the whole file the attack was practically useless to law enforcement agencies the zip cipher has 96 bits of internal state split into three 32 bit chunks called key0 key1 and key2 key0 and key2 are the internal state of two copies of crc32 a linear feedback shift register basically to update the state with a new byte of data you shift all the bytes down by one byte dropping the low byte then xor in a constant from a lookup table indexed by the byte of data xored with the low byte key1 is the internal state of a truncated linear congruential generator tlcg to update its internal state you add the data byte multiply by a constant i ll call c and add 1 the cipher works like this you feed in a data byte to the first crc32 then take the low byte of that and feed it into the tlcg then take the high byte of that and feed it into the second crc32 then take the state and roughly square it then output the second byte of the result as the stream byte to initialize the 96 bit state you start from a known state and encrypt the password then encrypt ten bytes of salt pkzip got its salt bytes by allocating memory without initializing it so it would contain bits of stuff from other programs you d been running or images or documents whatever that worked fine on windows but on many unix systems memory gets initialized automatically when it s allocated infozip chose the salt bytes by using c s rand function it would initialize the state of the random number generator by xoring the timestamp with the process id then generate ten bytes per file had they left it that way knowing the timestamp and process id would be enough information to recover the header bytes which in turn would let you mount biham and kocher s attack it seems the infozip authors knew about that attack because they went one step further and encrypted the headers once using the password that way an attacker had only doubly encrypted plaintext and their attack wouldn t work i noticed that because the password was the same in both passes the first stream byte was the same each pass just like toggling a light switch twice leaves it where it started xoring a byte twice with the same stream byte leaves it unaffected this let me mount a very powerful ciphertext only attack given five encrypted files in an archive i could derive the internal state of c s rand function without having to look at the timestamp or knowing the process id that let me generate the original unencrypted headers because only a few bits of each part of the cipher affected the next part i could also guess only a few bits of the state and check whether decrypting the next bytes twice gave me the answer i was expecting i had to guess fewer and fewer bits of key material as i went on each extra file also let me exclude more potential key material at the time it took a few hours on a single desktop machine i published a paper on it and got to present it in japan at fast software encryption 2001 i left accessdata in 2002 i worked for a neural networking startup for a year spent three years on a master s degree in computer science at the university of auckland with cris calude started a phd with the mathematical physicist john baez at uc riverside worked on google s applied security team for six years and finished the phd and after a few more years ended up as cto of a startup about six months ago i received out of the blue a message on linkedin from a russian guy he had read that paper i d written 19 years ago and wanted to know if the attack could work on a file with only two files a quick analysis said not without an enormous amount of processing power and a lot of money because i only had two files to work with a lot more false positives would advance at each stage there would end up being 2 73 possible keys to test nearly 10 sextillion i estimated it would take a large gpu farm a year to break with a cost on the order of 100k he astounded me by saying he could spend that much to recover the key back in january of 2016 he had bought around 10k or 15k of bitcoin and put the keys in an encrypted zip file now they were worth upwards of 300k and he couldn t remember the password luckily he still had the original laptop and knew exactly when the encryption took place because infozip seeds its entropy using the timestamp that promised to reduce the work enormously only 10 quintillion and made it quite feasible a matter of a couple of months on a medium gpu farm we made a contract and i got to work i spent a while reconstructing the attack from the overview in the paper to my chagrin there were certain tricky details i had skimmed over in my paper but i worked them out again then i discovered i d made a terrible mistake while estimating the work in my original attack i would guess the high bytes of key1 c key1 c² key1 c³ and key1 c by the time i guessed that fourth byte i knew the complete state of the rest of the cipher i just had to convert those four bytes into the original key1 and i was done i would run through the 32 bit state space for key1 and check each one to see if it gave the proper high bytes here though i had quintillions of keys to check if i had to do 2 32 tests on each one it would take a few hundred thousand years i vaguely remembered that there had been some work done on cryptanalyzing tlcgs using lattice reduction so i dug out the original paper it was perfect i just needed to define a lattice with basis vectors given by 2 32 and the powers of the constant from the tlcg then do lattice reduction given the reduced basis i could recover the original state from the high bytes just by doing a matrix multiplication at least that was the idea i would need five bytes to constrain it to a single outcome and at that point in the attack i only had four the process described in the paper rarely gave the right answer however i knew the answer had to be close to the right one so i ran through all the possible key1 values and checked the difference from the answer it gave me and the true one the difference was always one of 36 vectors with this optimization i could run through just 36 possibilities instead of four billion we were still in business the next thing we ran into was the problem of transferring data between the machines with the gpus on them my business partner nash foster was working on the gpu implementation advising me on how fast different operations are and writing a lot of the support structure for the application into which i put my crypto cracking code how would we get these petabytes of candidate keys to the gpus that would test them they d be almost idle twiddling their thumbs waiting for stuff to work on it occurred to me that each stage of my attack was guessing a lot of bits and then keeping only one candidate out of roughly 65k if i had some way of using that information to derive bits rather than just guess and check i d save a lot of work and more importantly a lot of network traffic the problem with that idea was that the math was too complicated it involved mixing the math of finite fields with the math of integer rings and those don t play nicely together i thought about other cryptanalytic attacks i knew and one that seemed promising was a meet in the middle attack a meet in the middle attack applies to a block cipher when it uses one part of the key material to do the first half of the encryption and the other par...