TUCoPS :: Crypto
:: dcsshaik.txtAstounding! The DeCSS Haiku! |

How to decrypt a DVD: in haiku form. (Thanks, Prof. D. S. T.) ------------------------ (I abandon my exclusive rights to make or perform copies of this work, U. S. Code Title Seventeen, section One Hundred and Six.) Muse! When we learned to count, little did we know all the things we could do some day by shuffling those numbers: Pythagoras said "All is number" long before he saw computers and their effects, or what they could do by computation, naive and mechanical fast arithmetic. It changed the world, it changed our consciousness and lives to have such fast math available to us and anyone who cared to learn programming. Now help me, Muse, for I wish to tell a piece of controversial math, for which the lawyers of DVD CCA don't forbear to sue: that they alone should know or have the right to teach these skills and these rules. (Do they understand the content, or is it just the effects they see?) And all mathematics is full of stories (just read Eric Temple Bell); and CSS is no exception to this rule. Sing, Muse, decryption once secret, as all knowledge, once unknown: how to decrypt DVDs. Arrays' elements start with zero and count up from there, don't forget! Integers are four bytes long, or thirty-two bits, which is the same thing. To decode these discs, you need a master key, as hardware vendors get. (This is a "player key" and some folks other than vendors know them now. If they didn't, there is also a way not to need one, to start off.) You'll read a "disk key" from the disc, and decrypt it with that player key. You'll read a "title key" for the video file that you want to play. With the disk key, you can decrypt the title key; that decrypts the show. Here's a description of how a player key will decrypt a disk key. You need two things here: An encrypted disk key, which is just six bytes long. (Only five of those are the _key itself_, because "zero" marks the end. So that's five real bytes, and eight times five is forty; in the ideal case, forty bits will yield just short of two trillion possible choices! Ian Goldberg once recovered a key that long in seven half-hours. But his office-mate David Wagner points out that it's _impossible_ to achieve what the DVD CCA seems to want to achieve, even by making the key some reasonable, "adequate" key-length: There's no way to write a "secure" software player which contains the key and runs on PCs, yet somehow prevents users from extracting it. If the player can decrypt, Wagner has noted, users can learn how.) This is a pointer, "KEY", to those bytes, and when we're done, they'll be clear-text. Oh, the other thing! Called "im", a pointer to six bytes: a player key. (Now those six bytes, the DVD CCA says under penalty of perjury, are its trade secret, and you are breaking the law if you tell someone that, for instance, the Xing player used the following: Eighty-one; and then one hundred three -- two times; then two hundred (less three); two hundred twenty four; and last and not least, the humble zero.) We will use these few internal variables: t1 through t6, unsigned integers. k, pointer to five unsigned bytes. i, integer. So here's how you do it: first, take the first byte of im -- that's byte zero; OR that byte with the number 0x100 (hexadecimal -- that's two hundred and fifty-six to you if you prefer decimal). Store the result in t1. Take byte one of im. Store it in t2. Take bytes two through five of im; store them in t3. Take its three low bits (you can get them by ANDing t3 with seven); store this in t4. Double t3, add eight, subtract t4; store the result in t3. Make t5 zero. Now we'll start a loop; set i equal to zero. i gets values from zero up to four; each time, do all of these steps: Use t2 for an index into Table Two: find a byte b1. Use t1 for an index into Table Three: find a byte b2. Take exclusive OR of b1 with b2 and store this in t4. Shift t1 right by a single bit (like halving); store this in t2. Take the low bit of t1 (so, AND it with one), shift it left eight bits, then take exclusive OR of that with t4; store this back in t1. Use t4 for an index into Table Four: find a byte and store it back in t4. Shift t3 right by three bits, take exclusive OR of this with t3, shift this right by one bit, and take exclusive OR of this with t3, shift this right by eight bits, and take exclusive OR of this with t3, shift this right by five bits, and (No exclusive OR! Orange you glad I didn't say banana?) take the low byte (by AND with two hundred and fifty-five); now store this into t6. Phew! Shift t3 left eight bits, take OR with t6, and store this in t3. Use t6 for an index into Table Four: find a byte and store it in t6. Add t6, t5, t4; store the sum in t5. Take t5's low byte (AND t5 with two hundred fifty five) to put it in the ith byte of the vector called k. Now shift t5 right eight bits; store the result in t5 again. Now that's the last step in the loop. No sooner have we finished that loop than we'll start another; no rest for the wicked nor those innocents whom lawyers serve with paperwork. Reader! Think not that technical information ought not be called speech; think not diagrams, schematics, tables, numbers, formulae -- like the terrifying and uniquely moving, though cliche, Einstein equation "Energy is just the same as matter, but for a little factor, speed of light by speed of light, and we are ourselves frozen energy." Einstein's formula to convert from joules into kilogram-meters squared per second squared, for all its power, uses just five characters. But Einstein wrote to physicists: formal, concise, specific, detailed. And sometimes we write to machines to teach them how tasks are carried out: and sometimes we write to our friends to show a way tasks are carried out. We write precisely since such is our habit in talking to machines; we say exactly how to do a thing or how every detail works. The poet has choice of words and order, symbols, imagery, and use of metaphor. She can allude, suggest, permit ambiguities. She need not say just what she means, for readers can always interpret. Poets too, despite their famous "license" sometimes are constrained by rules: How often have we heard that some strange twist of plot or phrase was simply "Metri causa", for the meter's sake, solely done "to fit the meter"? Programmers' art as that of natural scientists is to be precise, complete in every detail of description, not leaving things to chance. Reader, see how yet technical communicants deserve free speech rights; see how numbers, rules, patterns, languages you don't yourself speak yet, still should in law be protected from suppression, called valuable speech! Ending my appeal on that note, I will describe the second loop. Store nine in i; i gets values from nine down to naught. Each time, do this: Use i+1 as an index into Table Zero: find a byte. Call that byte p1. Now use i for an index in Table Zero: find a byte and call that byte p0. Now use p1 as an index into k (a vector, remember?); thus find a byte b1. Use p1 as an index into KEY, as well: find a byte, use that byte as an index into Table One. Call the byte you find b2. Use p0 as an index into KEY to find a byte b3. Take exclusive OR of b1, b2, b3, and store that in KEY (not just anywhere, though!). In KEY at the byte that's indexed by p1. That's it for that loop and also for the task of disk key decryption. Title keys are next. It isn't hard to decrypt one. The rule's the same! Well, there's one slight change: where you use t6, it's Table "Five", not "Four". And this time im is the decrypted disk key, and KEY the title key. How would you like to hear how to decrypt _both_ the disk and title keys? All we'll need are the encrypted versions and a player key. No sweat! We'll call the title key TKEY, a pointer to six bytes, encrypted. We'll call the disk key DKEY, a pointer to six bytes, encrypted too. We will use a few internal variables, once again: so i, an integer, will serve again as loop index. im1 is six bytes, im2 is six bytes. Both vectors and the latter holds our player key. Now I told you once about a player key I think they gave to Xing -- I don't know this for sure; DVD CCA said so in court, though. Otherwise I'd say that this is just a "magic number" from on high, vouchsafed to mortals from the mouths of the Muses for our benefit. In reality I'm told it was discovered by M.o.R.E., some Europeans two-thirds of whom are today still anonymous. I don't want to make a long excursus right now on why that's not bad: reverse engineers in many fields are heroes of technology, for advancing the knowledge of their colleagues or of the public mind. Yet in software the recent trend has been to brand tinkerers as thieves! I urge you to read the Crypto-gram newletter on why that's not so. Bruce can make the point better there than I can here, sticking to haiku. So this number is, again, the player key: (trade secret haiku?) "Eighty-one; and then one hundred three -- two times; then two hundred (less three); two hundred twenty four; and last and not least, the humble zero." If you didn't know a valid player key, then you could find one out -- ask Frank Stevenson, or his fellow programmer wise Andreas Bogk. All we have to do is this: copy our DKEY into im1, use the rule above that decrypts a disk key (with im1 and its friend im2 as inputs) -- thus we decrypt the disk key im1. Use the rule above that decrypts a title key. TKEY and our new disk key im1 are inputs now -- we decrypt TKEY, and we're done. That was straightforward. Probably we didn't need to explain this part, but computers are very literal, so we might as well do so. This part is really exciting for movie fans: decrypt DVDs! Well, at least sectors of DVDs, but they are made up of sectors. Sectors (of two to the eleventh bytes) are the encryption units. Rejoice then, get some popcorn out, and butter if you aren't vegan. Margarine works well if you're vegan, or if you are watching your weight. I've heard you can put tarragon on your popcorn. I haven't tried it. Why did I tell you to rejoice? Because we are about to watch a movie, at least if we have a good MPEG-2 player close at hand. We need two things now, though, beyond our MPEG-2 players and popcorn: A vector "SEC" of two thousand forty-eight bytes, disk sector contents. (These start off in their encrypted form, but we will leave them decrypted.) And a vector KEY of six bytes, the decrypted title key we'll use. We will use these few internal variables: t1 through t6, unsigned integers. Remember those from before? END is a pointer to the end of the sector, which is SEC plus two thousand forty-eight. Take the first byte of KEY (that's byte zero), perform exclusive OR with byte eighty-four of SEC. Treating the result as an integer, take OR of that with two hundred fifty-six. Store the result in t1. Take the next byte (which is byte one) of KEY, perform exclusive OR with the next byte of SEC (byte eighty-five, right?); store the result in t2. Take bytes two through five of KEY and take exclusive OR of these with their counterparts abroad, bytes eighty-six through eighty-nine of our sector SEC. Store this in t3. (It will fit because it is four bytes, like t3.) I must quote myself because we're going to do some things once again: (Above, in the first part, we talked about t3:) "Take its three low bits (you can get them by ANDing t3 with seven); store this in t4. Double t3, add eight, subtract t4; store the result in t3." (Now increment SEC by one hundred twenty-eight!) "Make t5 zero." Now start a loop, and do these things as long as SEC doesn't equal END: Use t2 for an index into Table Two: find a byte b1. Use t1 for an index into Table Three: find a byte b2. Take exclusive OR of b1 with b2 and store this in t4. Shift t1 right by a single bit (like halving); store this in t2. Take the low bit of t1 (so, AND it with one), shift it left eight bits, then take exclusive OR of that with t4; store this back in t1. (The step that's coming up is _slightly_ different from the original.) Use t4 for an index into Table Five: find a byte and store it back in t4. Shift t3 right by three bits, take exclusive OR of this with t3, shift this right by one bit, and take exclusive OR of this with t3, shift this right by eight bits, and take exclusive OR of this with t3, shift this right by five bits, and (No exclusive OR! Orange you glad I didn't say banana?) take the low byte (by AND with two hundred and fifty-five); now store this into t6. Phew! Shift t3 left eight bits, take OR with t6, and store this in t3. Use t6 for an index into Table Four: find a byte and store it in t6. Add t6, t5, t4; store the sum in t5. (Again, here's a change from the original steps: please don't get confused.) In Table One use the byte SEC points to as an index, get a byte there, take exclusive OR of that byte with the low byte of t5; store the result where SEC points, and increment SEC by one. This is all that has changed in these steps. And we're almost done! Now shift t5 right eight bits; store the result in t5 again. That is the last step in the loop. Now I want a drink (mnemonics in crypto poems are great!); exercise from singing so long makes me thirst for a glass of soda, slice of pie. For this is the end of the decryption process; you can now go home. But wait! I hear a voice entreating me to stay: "O, the Tables tell!" Alas, I have not as yet declared to you the CSS Tables. This is a major issue, in that I don't know what these tables _mean_: our noble guide has told us in outline what they are for, or something of their structure. But to me, a humble poet of mathematics, they are opaque, they are certain combinations of ancient, noble Number. Their inner logic, aitia, telos, still unknown to me. Herein a clear free speech question: would courts see fit to muzzle me, then, from speaking numbers, technical data, which I did not make and more cannot memorize, cannot explain in detail, cannot understand? I have these numbers. They have meaning, this is clear: else why suppress them? I wish to speak or let the Muse announce through me Tables of Numbers. Professor Moglen! Help defend my right to share these numbers with you! You called the right to speak with PGP like that to use Navaho. Help me then in my haiku quest to share these bits, and not be censored. Preserve my right to speak here, in this extreme, these mystery Tables the products, pieces of technique, which but a very few will understand. Mary Jo White, the United States Attorney for S.D.N.Y., your logic erodes any meaningful power Internet speakers would retain against state censorship. Do you care? Have you any shame? Fight, brave amici, with effective, functional argumentation. I'll try to help by singing these octets, until court orders forbid. (Sad to say, I have removed hyphens in numbers: poetic license.) Table Zero is: Five, zero, one, two, three, four, oh, one, two, three, four. Table One is long: two to the eighth power bytes. Ready? Here they are: Fifty one; then one hundred fifteen; fifty nine; thirty eight; ninety nine; thirty five; one hundred seven; one hundred eighteen; sixty two; one hundred twenty six; fifty four; forty three; one hundred ten; then forty six; then one hundred two; one hundred and twenty three; then two hundred eleven; one hundred forty seven; two hundred nineteen; six; sixty seven; three; seventy five; then one hundred fifty; two hundred twenty two; one hundred fifty eight; two hundred fourteen; then eleven; and then seventy eight; fourteen; then seventy; then one hundred fifty five; eighty seven; twenty three; ninety five; then one hundred thirty; one hundred ninety nine; then one hundred thirty five; two hundred seven; eighteen; ninety; twenty six; eighty two; then one hundred forty three; two hundred two; one hundred thirty eight; then one hundred ninety four; thirty one; two hundred and seventeen; then one hundred fifty three; two hundred nine; zero; then seventy three; nine; sixty five; then one hundred forty four; then two hundred sixteen; one hundred fifty two; two hundred eight; one; and then seventy two; eight; sixty four; then one hundred forty five; sixty one; one hundred and twenty five; fifty three; thirty six; one hundred nine; forty five; one hundred one; then one hundred sixteen; sixty; one hundred twenty four; fifty two; thirty seven; one hundred eight; then forty four; then one hundred; one hundred seventeen; two hundred and twenty one; then one hundred and fifty seven; two hundred thirteen; four; seventy plus seven; thirteen; then sixty nine; one hundred and forty eight; then two hundred twenty; one hundred fifty six; then two hundred twelve; five; then seventy six; twelve; sixty eight; one hundred and forty nine; eighty nine; twenty five; then eighty one; one hundred and twenty eight; then two hundred one; then one hundred thirty seven; one hundred ninety three; sixteen; eighty eight; twenty four; eighty; then one hundred twenty nine; two hundred; then one hundred thirty six; then one hundred ninety two; seventeen; then two hundred fifteen; one hundred fifty one; then two hundred twenty three; two; seventy one; seven; seventy nine; one hundred forty six; two hundred eighteen; then one hundred fifty four; two hundred ten; then fifteen; seventy four; ten; sixty six; then one hundred fifty nine; eighty three; nineteen; ninety one; one hundred and thirty four; then one hundred ninety five; then one hundred thirty one; two hundred three; then twenty two; ninety four; then thirty; eighty six; one hundred thirty nine; two hundred six; then one hundred forty two; one hundred ninety eight; twenty seven; then one hundred seventy nine; two hundred and forty three; one hundred and eighty seven; one hundred sixty six; two hundred twenty seven; one hundred sixty three; two hundred and thirty five; two hundred and forty six; then one hundred ninety; two hundred fifty four; then one hundred eighty two; then one hundred and seventy one; two hundred thirty eight; then one hundred seventy four; two hundred thirty; two hundred fifty one; fifty five; then one hundred nineteen; sixty three; thirty four; then one hundred three; thirty nine; one hundred eleven; one hundred fourteen; fifty eight; then one hundred twenty two; fifty; forty seven; one hundred six; forty two; ninety eight; one hundred twenty seven; one hundred eighty five; two hundred forty nine; one hundred seventy seven; one hundred sixty; two hundred thirty three; one hundred and sixty nine; then two hundred twenty five; then two hundred forty; one hundred eighty four; two hundred forty eight; one hundred seventy six; one hundred and sixty one; two hundred and thirty two; then one hundred sixty eight; two hundred twenty four; two hundred forty one; ninety three; twenty nine; eighty five; one hundred thirty two; then two hundred five; then one hundred forty one; then one hundred and ninety seven; twenty; then ninety two; twenty eight; then eighty four; then one hundred thirty three; two hundred four; one hundred forty; one hundred ninety six; twenty one; one hundred eighty nine; two hundred fifty three; one hundred and eighty one; one hundred and sixty four; then two hundred and thirty seven; then one hundred and seventy three; two hundred twenty nine; two hundred forty four; one hundred eighty eight; two hundred fifty two; one hundred eighty; one hundred sixty five; two hundred thirty six; one hundred seventy two; two hundred and twenty eight; then two hundred forty five; fifty seven; one hundred twenty one; forty nine; thirty two; one hundred five; forty one; then ninety seven; one hundred twelve; fifty six; one hundred twenty; then forty eight; thirty three; one hundred four; forty; ninety six; then one hundred thirteen; one hundred eighty three; then two hundred and forty seven; one hundred ninety one; one hundred and sixty two; then two hundred thirty one; one hundred sixty seven; two hundred thirty nine; two hundred and forty two; one hundred and eighty six; then two hundred fifty; one hundred seventy eight; one hundred seventy five; two hundred and thirty four; one hundred and seventy; then two hundred twenty six; two hundred fifty five. That's the whole Table. Just when you thought it was safe, here is Table Two, which has the same length: Zero; one; two; three; four; five; six; seven; nine; eight; eleven; ten; then thirteen; twelve; fifteen; fourteen; eighteen; nineteen; then sixteen; seventeen; twenty two; twenty three; twenty; twenty one; then twenty seven; then twenty six; twenty five; twenty four; thirty one; thirty; twenty nine; twenty eight; thirty six; thirty seven; thirty eight; thirty nine; then thirty two; thirty three; thirty four; thirty five; forty five; forty four; forty seven; forty six; forty one; then forty; forty three; forty two; fifty four; fifty five; fifty two; fifty three; fifty; fifty one; forty eight; forty nine; sixty three; sixty two; sixty one; sixty; fifty nine; fifty eight; fifty plus seven; fifty six; seventy three; then seventy two; seventy five; seventy four; seventy seven; seventy six; seventy nine; seventy eight; sixty four; sixty five; sixty six; sixty plus seven; sixty eight; sixty nine; and then seventy; seventy one; ninety one; ninety; eighty nine; eighty eight; ninety five; ninety four; ninety three; ninety two; eighty two; then eighty three; eighty; eighty one; eighty six; then eighty seven; then eighty four; eighty five; one hundred nine; then one hundred eight; then one hundred eleven; then one hundred ten; then one hundred five; then one hundred four; one hundred and seven; one hundred six; one hundred; one hundred one; one hundred two; one hundred three; then ninety six; ninety seven; ninety eight; ninety nine; one hundred and twenty seven; one hundred twenty six; then one hundred twenty five; one hundred twenty four; one hundred twenty three; one hundred twenty two; one hundred and twenty one; one hundred and twenty; one hundred eighteen; one hundred nineteen; one hundred sixteen; then one hundred and seventeen; then one hundred fourteen; one hundred fifteen; one hundred twelve; one hundred and thirteen; one hundred forty six; one hundred and forty seven; one hundred forty four; then one hundred forty five; one hundred fifty; one hundred fifty one; one hundred forty eight; one hundred forty nine; one hundred fifty five; one hundred fifty four; one hundred and fifty three; one hundred and fifty two; then one hundred fifty nine; one hundred fifty eight; one hundred and fifty seven; one hundred fifty six; one hundred and twenty eight; then one hundred twenty nine; one hundred thirty; then one hundred thirty one; one hundred thirty two; one hundred thirty three; one hundred thirty four; one hundred and thirty five; one hundred and thirty seven; one hundred thirty six; one hundred thirty nine; one hundred thirty eight; one hundred forty one; one hundred forty; one hundred forty three; one hundred forty two; one hundred eighty two; one hundred eighty three; one hundred and eighty; one hundred eighty one; one hundred and seventy eight; one hundred seventy nine; one hundred seventy six; one hundred and seventy seven; then one hundred ninety one; one hundred ninety; one hundred eighty nine; one hundred eighty eight; one hundred eighty seven; one hundred eighty six; one hundred and eighty five; then one hundred eighty four; then one hundred sixty four; one hundred sixty five; one hundred sixty six; one hundred sixty seven; one hundred sixty; one hundred sixty one; one hundred and sixty two; then one hundred sixty three; then one hundred seventy three; one hundred and seventy two; one hundred seventy five; one hundred seventy four; one hundred sixty nine; one hundred sixty eight; one hundred and seventy one; one hundred seventy; then two hundred nineteen; two hundred eighteen; two hundred seventeen; then two hundred sixteen; two hundred twenty three; then two hundred twenty two; two hundred twenty one; two hundred twenty; two hundred ten; then two hundred eleven; two hundred eight; two hundred nine; two hundred and fourteen; two hundred fifteen; two hundred twelve; two hundred thirteen; two hundred one; then two hundred; two hundred three; two hundred two; then two hundred five; then two hundred four; two hundred and seven; two hundred six; one hundred and ninety two; one hundred and ninety three; then one hundred ninety four; one hundred ninety five; one hundred ninety six; one hundred ninety seven; one hundred ninety eight; one hundred and ninety nine; then two hundred fifty five; then two hundred fifty four; two hundred fifty three; two hundred fifty two; two hundred fifty one; two hundred and fifty; two hundred forty nine; two hundred and forty eight; then two hundred forty six; then two hundred and forty seven; two hundred forty four; two hundred and forty five; then two hundred forty two; two hundred forty three; two hundred forty; two hundred forty one; two hundred thirty seven; two hundred thirty six; two hundred and thirty nine; two hundred and thirty eight; then two hundred thirty three; two hundred thirty two; two hundred thirty five; two hundred thirty four; two hundred twenty eight; two hundred twenty nine; two hundred and thirty; two hundred thirty one; two hundred and twenty four; then two hundred twenty five; then two hundred twenty six; two hundred twenty seven. That's the end of the Table Two listing. Table Three repeats itself sixty-four times with this eight-byte sequence: Zero, thirty six, seventy three, one hundred nine, one hundred and forty six, then one hundred eighty two, and then two hundred nineteen, and last of all, two to the eighth, less one (or two hundred fifty five). Dr. Touretzky has a more concise account of what Table Four is for, and where it comes from; but for now, I think I will just list it: Zero; one hundred twenty eight; sixty four; one hundred ninety two; thirty two; then one hundred sixty; ninety six; two hundred twenty four; sixteen; then one hundred forty four; eighty; two hundred eight; then forty eight; then one hundred seventy six; one hundred twelve; then two hundred forty; eight; one hundred thirty six; then seventy two; two hundred; forty; one hundred sixty eight; then one hundred four; then two hundred thirty two; twenty four; one hundred and fifty two; eighty eight; two hundred and sixteen; fifty six; then one hundred eighty four; one hundred twenty; two hundred forty eight; four; one hundred thirty two; sixty eight; one hundred ninety six; thirty six; one hundred and sixty four; then one hundred; two hundred twenty eight; twenty; then one hundred forty eight; eighty four; two hundred twelve; fifty two; then one hundred eighty; one hundred sixteen; two hundred forty four; twelve; one hundred forty; then seventy six; two hundred four; forty four; one hundred seventy two; one hundred eight; then two hundred thirty six; twenty eight; then one hundred fifty six; ninety two; two hundred and twenty; sixty; one hundred eighty eight; then one hundred twenty four; two hundred fifty two; two; one hundred thirty; sixty six; then one hundred ninety four; thirty four; one hundred and sixty two; ninety eight; two hundred and twenty six; eighteen; then one hundred forty six; eighty two; then two hundred ten; fifty; then one hundred seventy eight; one hundred and fourteen; two hundred forty two; ten; one hundred thirty eight; and then seventy four; two hundred two; forty two; one hundred seventy; one hundred six; two hundred thirty four; twenty six; one hundred and fifty four; ninety; then two hundred eighteen; then fifty eight; then one hundred eighty six; then one hundred twenty two; two hundred fifty; six; one hundred thirty four; seventy; then one hundred ninety eight; thirty eight; one hundred and sixty six; then one hundred two; then two hundred thirty; twenty two; one hundred fifty; eighty six; then two hundred fourteen; fifty four; one hundred eighty two; one hundred and eighteen; two hundred forty six; fourteen; then one hundred forty two; seventy eight; two hundred six; forty six; one hundred seventy four; one hundred ten; then two hundred thirty eight; thirty; one hundred fifty eight; ninety four; two hundred twenty two; sixty two; then one hundred ninety; one hundred twenty six; then two hundred fifty four; one; one hundred twenty nine; sixty five; then one hundred ninety three; thirty three; one hundred and sixty one; ninety seven; two hundred twenty five; seventeen; one hundred forty five; eighty one; then two hundred nine; forty nine; one hundred seventy seven; one hundred thirteen; two hundred forty one; nine; one hundred thirty seven; then seventy three; two hundred one; forty one; one hundred sixty nine; one hundred five; two hundred thirty three; twenty five; one hundred and fifty three; eighty nine; two hundred seventeen; fifty seven; one hundred eighty five; then one hundred twenty one; two hundred forty nine; five; one hundred thirty three; sixty nine; one hundred and ninety seven; thirty seven; one hundred sixty five; one hundred one; two hundred twenty nine; twenty one; one hundred and forty nine; eighty five; two hundred thirteen; then fifty three; then one hundred eighty one; one hundred seventeen; two hundred forty five; thirteen; one hundred forty one; seventy plus seven; two hundred five; forty five; one hundred seventy three; one hundred nine; then two hundred and thirty seven; twenty nine; then one hundred and fifty seven; ninety three; two hundred twenty one; then sixty one; then one hundred eighty nine; one hundred twenty five; two hundred fifty three; three; one hundred and thirty one; sixty seven; one hundred ninety five; thirty five; one hundred sixty three; ninety nine; two hundred and twenty seven; then nineteen; one hundred forty seven; eighty three; then two hundred and eleven; fifty one; one hundred seventy nine; one hundred and fifteen; two hundred forty three; eleven; one hundred thirty nine; seventy five; two hundred three; forty three; one hundred seventy one; one hundred seven; two hundred thirty five; twenty seven; one hundred fifty five; ninety one; two hundred and nineteen; fifty nine; one hundred eighty seven; one hundred twenty three; two hundred and fifty one; seven; then one hundred thirty five; seventy one; one hundred ninety nine; thirty nine; one hundred and sixty seven; one hundred three; two hundred and thirty one; twenty three; one hundred and fifty one; eighty seven; two hundred fifteen; fifty five; then one hundred eighty three; then one hundred nineteen; two hundred and forty seven; fifteen; one hundred forty three; and then seventy nine; two hundred seven; forty plus seven; one hundred seventy five; one hundred eleven; then two hundred thirty nine; thirty one; then one hundred fifty nine; ninety five; two hundred and twenty three; sixty three; one hundred ninety one; one hundred twenty seven; two hundred fifty five. And that's the end of the fourth Table. (You'll get Table Five if you flip each bit in the Table Four, supra.) Have mercy on me, Lord, and lesser judges, and on Jon Johansen.