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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asked Jul 18 at 15:15

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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

asked Jul 18 at 15:15

user863967

62

add a comment | 

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

asked Jul 18 at 15:15

user863967

62

• 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

asked Jul 18 at 15:15

user863967

62

share|cite|improve this question

share|cite|improve this question

asked Jul 18 at 15:15

user863967

62

asked Jul 18 at 15:15

user863967

62

asked Jul 18 at 15:15

user863967

62

62

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