Private Key Cryptography Nitin

8/2/2019 Private Key Cryptography Nitin

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Private Key Cryptography

Traditional private/secret/single keycryptography uses one key.

Key is shared by both sender and receiver.

The risk in this system is that if either partyloses the key or it is stolen, the system isbroken.


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CryptographyPublic Key Algorithm

Cryptographybased on the Knapsack Problem


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Was broken by Shamir in 1984.

Shamir showed how to use integerprogramming to solve the particular classof Subset Sum problems in polynomialtime.


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Private Key Cryptography

Uses two keys a public key and aprivate key

Asymmetric since parties are not equal

Public-key/two-key/asymmetric cryptographyinvolves the use of two keys: a public-key, which may be known by anybody, and

can be used to encrypt messages, and verifysignatures

a private-key, known only to the recipient, used to


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Public Key Cryptography

Public key cryptosystem was introduced in1976 by Diffie and Hellman.In PKCdifferent keys are used for encryption and

decryption.


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MerkleHellman knapsackcryptosystem

The Merkle

Hellman knapsackcryptosystem was one of the earliest publickey cryptosystems invented by RalphMerkle and Martin Hellman in 1978.

The Merkle-Hellman system is based on the subsetsum problem (a special case of the knapsackproblem).

The problem is as follows: given a set ofnumbers A and a number b, find a subset of A,which sums to b.

In general, this problem is known to be NP-


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Easy knapsack problem

An easy knapsack problem is one in whichset A={a1,a2,.. an} is a super increasingsequence.

A super increasing sequence is one inwhich the sum next to term of sequence isgreater than the sum of all preceding terms:

Example: A={1,2,4,8, 2n-1} is super

increasing sequence.


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Polynomial time algorithm for easyKnapsack problem


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Merkle-Hellman Additive Knapsackcryptosystem


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Choose S={s1,s2,..,sm} as asuperincreasing sequence, S is a simpleknapsack.

Choose a multiplier w and a modulus n.

The multiplier should have no commonfactors with the modulus.

Then replace every integer si in the simple

knapsack with the termhi = w*si mod n

Then, H=[h1,h2,.hm] is a hard knapsack


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Merkle hellman knapsackencryption algorithm

The merkle hellman algo begins with a binarymessage i.e binary sequence P=[p1,p2,.pk]

Divide the message into blocks of m bits

The encipherment of message P is asequence of targets,where eaach target is the

sum of some of the terms of the hardknapsack H.

Each term of the ciphertext is Pi * H, the target

derived using block Pi as the selection vector.


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Knapsack Decryption algorithm

Compute W as multiplicative inverse of wmod n such that Ww=1( mod n)

The connection b/w easy and hardknapsack is H= w* S mod n

The ciphertext message produced by theencryption algorithm is C=H*P= w*S*P modn


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Knapsack Decryption algorithm

To decipher,multiply C by by W,since

W*C=W*H*P=W*w*S*P=S*P mod n

The plain text Pi could be found by using

polynomial time algorithm for easyknapsack.


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Example

Alice private key:

S={1,2,4,9},w=15 and n=17

now hi =w*si mod n1*15=15 mod 17=15

2*15=30mod 17=13

4*15=60mod 17=99*15=135mod 17=16

The hard knapsack H={15,13,9,16}


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Encryption of message

We use knapsack S and Hw=15,n=17 and m=4 The message

P=0100101110100101Is encoded with the knapsack H

P=0100 1011 1010 0101[0,1,0,0]*[15,13,9,16]=13

[1,0,1,1]*[15,13,9,16]=40[1,0,1,0]*[15,13,9,16]=24[0,1,0,1]*[15,13,9,16]=29The message is ecrypted as the integers 13,40,24,29,

using H


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Now multiplicative inverse W = 8

( 15 mod 17 is 8,since 15*8 mod 17=120 mod17=(17*17)+1 mod 17 =1)

To decipher, we multiply these messages by

8 mod 17 bcoz 8 is 15 mod 17.13*8=104 mod 17 = 2 = [0 1 0 0]

40*8=320 mod 17 = 2 = [1 0 1 1]

24*8=192 mod 17 = 5 = [1 0 1 0]

29*8=232 mod 17 = 11 =[0 1 0 1]

The recovered msg is 0100101110100101


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Knapsack problem


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Private Key Cryptography Nitin