EASTNORTHEAST - This is not exactly the hint Jim Sanborn (JS) gave for K4 on the 29th of January this year. He only gave NORTHEAST - which refers to the positions 26-34 of K4's plaintext. Beside BERLIN and CLOCK it is the third revealed plaintext word of K4. However, also this hint does not seem to help much. However, it just so happened, that a member in the yahoo kryptos group had a conversation with Jim Sanborn due to a submitted solution. Sandborn's answer to the question contained again the last clue which surprisingly was EASTNORTHEAST at position 22-34. Jim Sanborns compass rose at CIA There is disagreement if Jim revealed this on purpose or he did it accidentially, but the new extended clue seem to be serious and valid.Interestingly, EASTNORTHEAST is exactly the direction which is illustrated on the compass rose on one of the stones around kryptos, also created by Jim Sanborn. Actually, i dont really kn...
We start with a very brief primer to Binary Quadratic Forms (BQF). For more information ⦗1⦘ is a very good source to start with. A Binary Quadratic Form is a polynom of the form $Q(x,y) = ax^2+bxy+cy^2$ with integer coefficients. We focus on integer BQF, i.e., $x,y \in \mathbb{Z}$. Often one is only interested in the behaviour of the coefficients $a,b,c$ and in those cases one refers to a BQF simply by the triple $(a,b,c)$. Further, the integer $D$ defined as: $$D = b^2 - 4ac$$ is called the discriminant of $(a,b,c)$. A positive definite BQF $(a,b,c)$, i.e. $D < 0, a > 0$, is reduced if $-a < b < a < c$. Gauss observed, that all reduced BQF with the same discriminant form an abelian group and he gave a composition algorithm for those group elements. This group is called the class group and is usually denoted as $\mathcal{Cl}(D)$. The size of the group has a special name and is called the class number $h(D)$. Informally spoken, the class group $\mathcal{Cl}(D)$ ...