(Wikipedia)  The four-square cipher is a manual symmetric encryption technique. It was invented by famous French cryptographer Felix Delastelle.

The technique encrypts pairs of letters (digraphs), and thus falls into a category of ciphers known as polygraphic substitution ciphers. This adds significant strength to the encryption when compared with monographic substitution ciphers which operate on single characters. The use of digraphs makes the four-square technique less susceptible to frequency analysis attacks, as the analysis must be done on 676 possible digraphs rather than just 26 for monographic substitution. The frequency analysis of digraphs is possible, but considerably more difficult – and it generally requires a much larger ciphertext in order to be useful.

So although we know K4 has been masked in some way that renders it impossible to use the English language to translate it, who can resist the lure of attempting a foursquare solution.  It is a matrix system like Sanborn likes.  It’s a different enciphering method than K1-3.  It distorts the plaintext letter frequency.  In a lot of ways, it seems like the perfect solution.

Unfortunately we know that it doesn’t take into account the masking technique which will interfere with any cryptoanalytic method.  It also has 97 letters and foursquare works on a series of digraphs.  Foursquare also allows the use of keywords in the Ciphertext quadrants.  Even if we were 100% sure a foursquare would take us directly to a solution, we couldn’t be sure of the keywords.

I tried it anyway.

I was hoping that it might force out some plaintext which would then help isolate some of the masking technique.  The best I got were some two letter words and it’s inevitable you’d get a few of those.

For my keywords I used Kryptos and Quagmire.

abcde            KRYPT
fghi/jk           OSABC
lmnop           DEFGH
qrstu             I/JLMNQ
vwxyz           UVWXZ

QUAGM       abcde
I/JREBC      fghi/jk
DFHKL         lmnop
NOPST          qrstu
VWXYZ       vwxyz

Each ciphertext is located in their quadrants and a line drawn horizontally and vertically.  The intersections that occur in the plaintext are naturally the plaintext letter.  By convention it would appear that plaintext 1 and ciphertext 1 are the top two sections and ciphertext 2 and plaintext 2 are the bottom.

Since there are 97 letters, I tried it in the original order and then shifted one letter to the right to cover the possibility that the masking technique involved removing some of the letters.

I didn’t find much, just to warn you.

Regular series with the extra R on the end:

IF-BF-QW-TW-ON-LZ-IT-LK-LR-II-NW-GR-BV-TS-RA-OO-RJ-QK-SZ
UT-AQ-RI-MO-ZZ-CX-KA-LE-LV-AS-HV-OR-TF-SL-RZ-EE-PW-OD-CY-LO
YX-KA-PF-AT-AB-EM-CG-MO-EH

Offset series missing the O at the beginning:

OI-BB-VH-AJ-EM-GT-RG-GQ-HO-YG-NP-WB-EQ-IB-OT-QD-RG-ZB-EA
GF-UT-GH-VE-ZW-SK-LT-BR-FL-XG-QQ-IO-YT-YA-UE-XU-SN-VB-OM-VE
TZ-LU-FL-OO-LX-AX-FX-FE-HG

So, not much.  It did become obvious that the end of the alphabet for ciphertext would remain the same no matter what the keyword was.  It’s likely that v, w, x, y, z would remain in the same position.  The largest distortion would be in the first 5-10 letters of the alphabet.

Otherwise we don’t have enough ciphertext to make any conclusions about the digraph frequency.

Maybe it would work after the masking was removed.

But it looked like a failure.

Kind of hard to move past that…