1 / 9

Governor’s School for the Sciences

Governor’s School for the Sciences. Mathematics. Day 6. POTD: Thomas Jefferson. 1743-1826 (America) Statesman Architect Naturalist Linguist Inventor Mathematician? (Probably not). Jefferson (?) Cipher.

luna
Download Presentation

Governor’s School for the Sciences

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Governor’s School for the Sciences Mathematics Day 6

  2. POTD: Thomas Jefferson • 1743-1826 (America) • Statesman • Architect • Naturalist • Linguist • Inventor • Mathematician? (Probably not)

  3. Jefferson (?) Cipher • Jefferson did invent a mechanical encoding device where the encoding of each letter in the message is based on a key word • Our variation uses the key word as a rotating set of shifts for the addition cipher • Changes letter distribution; decipherer needs to know the key word

  4. Example • Key word: MATH = 12,0,19,7 • Message: GSS ROCKSM: G S S R O C K S#: 6 18 18 17 14 2 10 18K:12 0 19 7 12 0 19 7E:18 18 11 24 0 2 19 25C: S S L Y A C T Z • Enciphered message: SSLYACTZ

  5. Hill Cipher • Work with two letters at a time, using a 2x2 matrix to do the encoding • Letters: a, b Encoded Letters: A,B • A = n1*a + n2*b, B = n3*a + n4*b • All arithmetic is done mod 26

  6. Example • GSS ROCKS = 6,18,18,17,14,2,10,18 • Encoding matrix = [1, 2; 3 5]; • Boardwork to get 16, 4, 0, 9, 18, 0, 20, 16 QEAJSAUQ • How about deciphering?

  7. Modular Matrix Math • If A is a 2x2 matrix [a,b;c,d] then det(A) = ad-bc A-1 = (1/det(A))*[d,-b;-c,a] • If A is a matrix with integer entries det(A)*A-1has integer entries • A-1 (mod 26) is computed as (det(A))-1*(det(A)*A-1) where (det(A))-1 is the multiplicative inverse of det(A)

  8. Examples • A = [1,2;3,5] det(A) = 5-6=-1=25 (det(A))-1 = 25 det(A)*A-1 = [5,-2;-3,1] A-1 = 25*[5,-2;-3,1] = [21,2;3,25] • A = [3,4;1,3] det(A) = 3*3-4*1 = 5 (det(A))-1 = 21 det(A)*A-1 = [3,-4;-1,3] A-1 = 21*[3,-4;-1,3] = [11,20;5,11]

  9. Quiz Bowl Time Team 3 vs. Dr. Collins team one p vs. Denominators of Doom

More Related