**Chapter 7**
1. Encipher the following message using .
TWO PLUS TWO IS GREATER THAN FIVE, FOR SUFFICIENTLY LARGE TWO.
2. Encipher the following message using .
MY GIRLFRIEND’S GOT A GUN - I’LL SPEND THE REST OF MY LIFE PLAYING HIDE AND SEEK^{1}
3. Verify the keyspace for 2x2 matrix encryption. You may do this by either writing a computer program to check the determinants of all possible 2x2 matrices to see which are invertible or by applying the formula provided in Overbey, Jeffrey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, *Cryptologia*, Vol. 29, No. 1, January 2005, pp. 59-72 Available online at http://jeff.actilon.com/keyspace-final.pdf,.
4. Can every possible digraphic substitution be realized by matrix encryption? Hint: compare keyspaces.
5. Verify the keyspace for 3x3 matrix encryption. You are strongly encouraged to solve this by applying the formula provided in the reference given in exercise 3, although you may write a program and patiently wait…
6. The unicity point for a random N-gram substitution cipher is .90 log(26^{N})!^{2} How do you think this compares to that for encryption with an NxN matrix? Explain your reasoning. In chapter 10, I’ll reveal a formula for computing unicity points and an exercise will ask you to go back and calculate it for 2x2, 3x3, and 4x4 matrix encryption, but for now I want you to simply think about it. Perhaps you’ll figure out the key component in the formula before it is unveiled!
7. Can an enciphering matrix have the form ? Justify your answer!
8. H. Gary Knight provided the following 2x2 matrix encryption ciphertext as his very first problem for readers of his column in *Cryptologia*.^{3}
QHDIW QQQEI WFRLI YLUIO WQUVC NQDHV SNTQV YRLEP RVMND ERMOA
GTNFQ QGWBS TJXCR IWQUH PBQME XMTXH WFXJS ACOZA SPKGS PAOYV NSJQK JXHZU PACAA I.
Prior to enciphering, Knight converted the letters to numbers using the assignments A=1, B=2, C=3,… Z=26. He gave two probable words, SUBMARINE and OBSERVING, as hints. Feel free to apply Levine’s attack or brute force a solution with a computer.
9. Levine’s attack depends on parity (even or odd). Is it significant (i.e., cryptanalytically useful) that all vowels are even, if we start our numbering with A=0? We have A=0, E=4, I=8, O=14, U=20, Y=24.
10. Using the assignments A=1, B=2, C=3,…, Y=25, Z=0, and a 3x3 matrix for encryption, Levine obtained the following ciphertext message:
MIU GNJ WWU YHZ DNS WVK RFV LLK AMP IGS MIU
WKN OEM IEK ORW WAE KZB APL KYP MEU ZMO QIX
FHS SJI DDJ KFY BWW HQP KLI NKG TMJ ROB TZE
One typo has been corrected for reproduction here.
In the May 1961 paper^{4} where Levine detailed his crib attack, he provided the probable plaintext THREE CONGRUENCES and showed how it may be used to recover the message. Go ahead and do this without referring to the original paper. Note: in the 2x2 example there were 4 possible forms, but for the 3x3 case, there will be 8.
11. Decipher the following message enciphered using a 2x2 matrix.
WVUQU HSCZR LXLVH IIGZR FVVRE TFAQK KVOYM DFAIT UHESO JOKFN FPOJS LXGWI FFUKU QCEHA JOSFG PDDXL NMYUO MDYMT RDBHN MWIRD NJSMT VXMEC EPKZO FWORX KLYDE PEIUR DYWML NKROF XGCEK TNZXG
MAGKQ NUWBU UHSBY UMDWO OJCGM SANSL ERTVD VSBRD OKFNF POJEI URXJX KWMJZ UGEPR DTYET DVNNL VBZVD WICOT YKQYV KICEO UFPCE SBQAU UWVGH JFIUM AGKQN XGWIF NOHJD AKAWH FMTGV OF
12. Decipher the following message enciphered using a 3x3 matrix.
WMUHC EIHVA SFKJE QSPMU VKXVY UVILX AVNTF CPRMK TVTNP KMMUR ONHNG JRVCJ TNPXJ VKSSP AYWTM TXYDH YKMEB ODHQD GWKOE GENQX PMYDP TNCPO XWGDL FCTMN MSWMG VWFPE VQQSF YOPLF CJVOG PSFRS
BBGAV SWDAG EAKNA YKENH OTJEK TZYUV QPTTN PYBCI HZJIR KBDMZ ZNWRP UZSPJ OIGYZ OFVBS VOMFU NXDGE CFZJR TUTYN YSEAL NQJJC OMRKH PRMAY KVEJU EAHFB FQDRU QMMET MVXIT GZAYW UXADN VNTNP
UBDEX ZUNQV IKLFC EBXQL JVMKU IZUDR NQPXF GUHAA VBXYX CLKJG LZKLS NCRRM XVNFT WBBTG JRPVA IMIYM EQUSM ZNBEP RTYAK YILXV BWETL NVNPC RJXUQ YZCJE SBOEV YZMPU XJ
13. Decipher the following message enciphered using a 4x4 matrix.
OFQXB SGQRQ PAMHI DTYML RIYWF EEKQL ALYJB WMEWI NKZWA QEHWQ XEFCX VNFES ECKKW IHQBW XMJRU ULNPZ LAQFG WIUHZ RLEUB IKRGJ HFWBR YQSJN PQYUE LNMRY FPOMC QZOMH IDYGH IDJPJ QJZBC ICMGB
MHTWR BETEK VUSCN ZOZHY NCYZE SUPAD XISGJ FRRGK CPJRU ULSFE ZHMST TRCTZ FDKKT YFAEL RWJQB GLXJR UUAJK WGEVE ANDDO KKWSV DBVKL YQEVE LPYRJ LGOOW PAOWQ CXQWA YPPFP STFJM CPLWF IXEOX
JDCUG EVEHH YIRAW REGTU WYAMQ IFXCH WKVXX TEJGK RWJQD KJAIT NAWTE ZXGQM BOPHQ HWCGM DGNTD PEQHT WPYQB DRPTV EIOFG DBSWY XLZJJ VSJPT XBMCX VHXSU SCYEH NKOWH QKQFJ E
14. How many plaintext/ciphertext pairs are needed to uniquely determine a 3x3 matrix? If your answer is denoted by n, will any n pairs suffice? If not, what condition must hold to make them sufficient?
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