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The Secure Hill Cipher Jeff Overbey MA464-01 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 1000 0100 0010 0001 I =

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Permutation Matrices 1000 0100 0010 0001 I =10000010 0100 0001 P 23 =

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Permutation Matrices 1000 0100 0010 0001 I =10000010 0100 0001 P 23 =01000010 1000 0001 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 1000 0010 0100 0001 P 23 =01000010 1000 0001 P 12,23 =13 2 4 23 1 4

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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 MA464-01 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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20-751 ECOMMERCE TECHNOLOGY FALL 2003 COPYRIGHT © 2003 MICHAEL I. SHAMOS Cryptography.

20-751 ECOMMERCE TECHNOLOGY FALL 2003 COPYRIGHT © 2003 MICHAEL I. SHAMOS Cryptography.

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