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The Secure Hill Cipher Jeff Overbey MA Dr. Jerzy Wojdyło April 29, 2003 Based on S. Saeednia. How to Make the Hill Cipher Secure. Cryptologia 24 (2000) 353–360. HILL

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The Scenario

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Alice

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The Scenario Bob

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The Scenario Oscar

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The Scenario

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Nosy NeighborSender Recipient Insecure Channel

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Hill Cipher Beforehand… Alice and Bob privately share matrix K, invertible over Z m Alice and Bob privately share matrix K, invertible over Z m To encrypt a matrix X over Z m … Compute Y = KX Compute Y = KX Send Y Send Y To decrypt Y… Compute X = K –1 Y Compute X = K –1 Y

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel “Hey Bob, wassup?”

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel Y = KX

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel HEYBOBWAS Known Plaintext

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel HEYBOBWAS Known Plaintext

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Why the Hill Cipher Isn’t Secure Nosy NeighborSender Recipient Insecure Channel HEY BOB, WASSUP? hahahaHAHAHAHA… KNOWN PLAINTEXT ATTACK

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Hill Cipher Beforehand… Alice and Bob privately share matrix K, invertible over Z m Alice and Bob privately share matrix K, invertible over Z m To encrypt a matrix X over Z m … Compute Y = KX Compute Y = KX Send Y Send Y To decrypt Y… Compute X = K –1 Y Compute X = K –1 Y Secure Hill Cipher Beforehand… Alice and Bob privately share matrix K, invertible over Z m Alice and Bob privately share matrix K, invertible over Z m To encrypt a matrix X over Z m … Choose a vector t over Z m Choose a vector t over Z m Form permutation matrix P t Form permutation matrix P t Compute K t = P t –1 KP t Compute K t = P t –1 KP t Compute Y = K t X and u = K t Compute Y = K t X and u = K t Send ( Y, u ) Send ( Y, u ) To decrypt (Y, u )… Compute t = K –1 u Compute t = K –1 u Compute P t Compute P t Compute K t –1 = P t –1 K –1 P t Compute K t –1 = P t –1 K –1 P t Compute X = K t –1 Y Compute X = K t –1 Y K t ) –1 = K –1 ) t Note that ( K t ) –1 = ( K –1 ) t

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Permutation Matrices I =

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Permutation Matrices I = P 23 =

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Permutation Matrices I = P 23 = P 12,23 = A permutation matrix is a matrix with exactly one 1 in each row and column and zeros elsewhere.

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Permutation Matrices

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Vector Representation P 23 = P 12,23 =

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Hill Cipher Beforehand… Alice and Bob privately share matrix K, invertible over Z m Alice and Bob privately share matrix K, invertible over Z m To encrypt a matrix X over Z m … Compute Y = KX Compute Y = KX Send Y Send Y To decrypt Y… Compute X = K –1 Y Compute X = K –1 Y Secure Hill Cipher Beforehand… Alice and Bob privately share matrix K, invertible over Z m Alice and Bob privately share matrix K, invertible over Z m To encrypt a matrix X over Z m … Choose a vector t over Z m Form permutation matrix P t Form permutation matrix P t Compute K t = P t –1 KP t Compute K t = P t –1 KP t Compute Y = K t X and u = K t Compute Y = K t X and u = K t Send ( Y, u ) Send ( Y, u ) To decrypt (Y, u )… Compute t = K –1 u Compute t = K –1 u Compute P t Compute P t Compute K t –1 = P t –1 K –1 P t Compute K t –1 = P t –1 K –1 P t Compute X = K t –1 Y Compute X = K t –1 Y

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The Easy Way to Encrypt Require t to be the vector representation of a permutation matrix Require t to be the vector representation of a permutation matrix N.B.: This is for example only—it is not practical “in the field.” N.B.: This is for example only—it is not practical “in the field.”

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An Example “Hey Bob, wassup?”

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An Example Choose a permutation vector: Form permutation matrix P t : 3 rows, since K is 3×3

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An Example Find P t –1 : Compute K t = P t –1 KP t :

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An Example Find P t –1 : Compute K t = P t –1 KP t : Permutation of K

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An Example Compute Y = K t X: Compute u = Kt:

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An Example Compute Y = K t X: Compute u = Kt:

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An Example Send (Y,u) to Bob

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The Scenario Nosy NeighborSender Recipient

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The Scenario Nosy NeighborSender Recipient

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The Scenario Nosy NeighborSender Recipient

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The Scenario Nosy NeighborSender Recipient

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An Example Receive (Y,u):

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An Example Compute K –1 : Compute t = K –1 u and derive P t :

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An Example Compute K t –1 = P t –1 K –1 P t : Compute X = K t –1 Y:

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An Example Compute K t –1 = P t –1 K –1 P t : Compute X = K t –1 Y: “Hey Bob, wassup?”

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An Example But that was just a regular Hill cipher with a fancy key… KNOWN PLAINTEXT ATTACK

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The Actual Really Secure Hill Cipher Jeff Overbey MA Dr. Jerzy Wojdyło April 29, 2003 Based on S. Saeednia. How to Make the Hill Cipher Secure. Cryptologia 24 (2000) 353–360. HILL

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The Actual Really Secure Hill Cipher or How to Secure the Secure Hill Cipher HILL

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Ensuring Security X should have two columns X should have two columns Why? Suppose K is n × n and X is n × s. Why? Suppose K is n × n and X is n × s. If s n… If s n… If 2 < s < n… If 2 < s < n… If s = 2… If s = 2… P t should be different for each encryption P t should be different for each encryption Theoretically, this can be ensured by choosing a different t for each encryption Theoretically, this can be ensured by choosing a different t for each encryption We did this by requiring t to be a vector representation of a permutation matrix, but this is not the best solution. We did this by requiring t to be a vector representation of a permutation matrix, but this is not the best solution.

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The Function Let n denote the set of n × n permutation matrices. : Z m n n Ideally, is onto is onto is 1-1 is 1-1

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The Function Let n denote the set of n × n permutation matrices. : Z m n n Is it possible? Z m n = m n Z m n = m n n = n ! n = n ! Z m n n Z m n n y = 26 n y = n ! An “ideal” function DNE.

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The Function Ideas? Awkward silence…

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Apologies for the horrible background image pun.

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