Can EX=A where $EinBbbZ^6times6$, $XinBbbZ^6times34$, $AinBbbZ^6times34$ be solved in E and X when A is given?

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  • I am trying to solve a cryptography challenge (https://www.mysterytwisterc3.org/en/challenges/level-ii/hilly-part-2) where I must calculate:

  • $E in BbbZ^6times6$

  • 34 (column) vectors $X_i in BbbZ^6$

  • verifying $EX_i = A_i$ where $A_i in BbbZ^6$ are given ($i = 1,2,ldots, 34)$

The encryption scheme is such that:



  • $det(E) neq 0$

  • E = s + K where $K in BbbZ^6times6$ is the key of the encryption and s = trace(K) i.e. $E_i,j = s + K_i,j$

  • s $neq 0$

  • $det(K) neq 0$ and $gcd(det(K), 26) neq 1$

(In addition, 8 of the 36 elements of K are given.)



I do not know which method could possibly state if a solution exists or not, and if yes, allow to calculate E and the Xi.



I have looked at the documents about systems of bilinear equations, but all refer to a field.

I also looked at the lattice theory, as E being invertible its columns can be considered as the basis of a rank 6, dimension 6 lattice.

The problem would then become: given 34 points in $ BbbZ^6$, can one state if they all belong to the same rank 6 lattice of $BbbZ^6$ or not ?



The challenge is such that I have about 2 million sets of 34 Ai vectors to test, so a reasonably fast algorithm would be welcome.




systems-of-equations cryptography integer-lattices linear-diophantine-equations




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    • I am trying to solve a cryptography challenge (https://www.mysterytwisterc3.org/en/challenges/level-ii/hilly-part-2) where I must calculate:

    • $E in BbbZ^6times6$

    • 34 (column) vectors $X_i in BbbZ^6$

    • verifying $EX_i = A_i$ where $A_i in BbbZ^6$ are given ($i = 1,2,ldots, 34)$

    The encryption scheme is such that:



    • $det(E) neq 0$

    • E = s + K where $K in BbbZ^6times6$ is the key of the encryption and s = trace(K) i.e. $E_i,j = s + K_i,j$

    • s $neq 0$

    • $det(K) neq 0$ and $gcd(det(K), 26) neq 1$

    (In addition, 8 of the 36 elements of K are given.)



    I do not know which method could possibly state if a solution exists or not, and if yes, allow to calculate E and the Xi.



    I have looked at the documents about systems of bilinear equations, but all refer to a field.

    I also looked at the lattice theory, as E being invertible its columns can be considered as the basis of a rank 6, dimension 6 lattice.

    The problem would then become: given 34 points in $ BbbZ^6$, can one state if they all belong to the same rank 6 lattice of $BbbZ^6$ or not ?



    The challenge is such that I have about 2 million sets of 34 Ai vectors to test, so a reasonably fast algorithm would be welcome.




    systems-of-equations cryptography integer-lattices linear-diophantine-equations




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      • I am trying to solve a cryptography challenge (https://www.mysterytwisterc3.org/en/challenges/level-ii/hilly-part-2) where I must calculate:

      • $E in BbbZ^6times6$

      • 34 (column) vectors $X_i in BbbZ^6$

      • verifying $EX_i = A_i$ where $A_i in BbbZ^6$ are given ($i = 1,2,ldots, 34)$

      The encryption scheme is such that:



      • $det(E) neq 0$

      • E = s + K where $K in BbbZ^6times6$ is the key of the encryption and s = trace(K) i.e. $E_i,j = s + K_i,j$

      • s $neq 0$

      • $det(K) neq 0$ and $gcd(det(K), 26) neq 1$

      (In addition, 8 of the 36 elements of K are given.)



      I do not know which method could possibly state if a solution exists or not, and if yes, allow to calculate E and the Xi.



      I have looked at the documents about systems of bilinear equations, but all refer to a field.

      I also looked at the lattice theory, as E being invertible its columns can be considered as the basis of a rank 6, dimension 6 lattice.

      The problem would then become: given 34 points in $ BbbZ^6$, can one state if they all belong to the same rank 6 lattice of $BbbZ^6$ or not ?



      The challenge is such that I have about 2 million sets of 34 Ai vectors to test, so a reasonably fast algorithm would be welcome.




      systems-of-equations cryptography integer-lattices linear-diophantine-equations




      share|cite|improve this question











      • I am trying to solve a cryptography challenge (https://www.mysterytwisterc3.org/en/challenges/level-ii/hilly-part-2) where I must calculate:

      • $E in BbbZ^6times6$

      • 34 (column) vectors $X_i in BbbZ^6$

      • verifying $EX_i = A_i$ where $A_i in BbbZ^6$ are given ($i = 1,2,ldots, 34)$

      The encryption scheme is such that:



      • $det(E) neq 0$

      • E = s + K where $K in BbbZ^6times6$ is the key of the encryption and s = trace(K) i.e. $E_i,j = s + K_i,j$

      • s $neq 0$

      • $det(K) neq 0$ and $gcd(det(K), 26) neq 1$

      (In addition, 8 of the 36 elements of K are given.)



      I do not know which method could possibly state if a solution exists or not, and if yes, allow to calculate E and the Xi.



      I have looked at the documents about systems of bilinear equations, but all refer to a field.

      I also looked at the lattice theory, as E being invertible its columns can be considered as the basis of a rank 6, dimension 6 lattice.

      The problem would then become: given 34 points in $ BbbZ^6$, can one state if they all belong to the same rank 6 lattice of $BbbZ^6$ or not ?



      The challenge is such that I have about 2 million sets of 34 Ai vectors to test, so a reasonably fast algorithm would be welcome.





      systems-of-equations cryptography integer-lattices linear-diophantine-equations





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      asked Jul 18 at 15:15









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