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Quadratic Residues Essay

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An Identity based Encryption Scheme based on Quadratic Residues Introduction: In Public Key Crypto system, it is required to know the Public Key of Receiver to encrypt the messages. So that receiver can decrypt the message using his/her own private key. This system needs to maintain directories which holds Public Keys of each user. To eliminate the need of maintaining such directories, this paper suggests computing the public key using receiver’s identity such as Email address. However, implementation of such system which is secure and practical, is difficult. The system uses Quadratic Residues modulo a large composite integer. [2] What is Quadratic Residues? “If m and a are coprime integers, then a is called a quadratic residue modulo …show more content…

Hence, a is square modulo of M. Also, P and Q are congruent to 3 mod 4, (-1/P)=(-1/Q)= -1 Thus, a or -a will be Quadratic Residues modulo P and Q. Only authority can determine the square root of modulo M and provide it to Alice. Here, we call it r r=a^((M+5-(P+Q))/8) mod M Where, r^2=a mod M or r^2=-a mod M If Bob wants to send data to Alice, he will generate a Transport key and perform Symmetric Key Encryption using that key. Then, he sends it to Alice bit by bit. Let’s say, x is the single bit of transport key, coded as +1 or -1. Bob will choose a value t at a random modulo M, such that Jacobi Symbol (t/M)=x Then, he sends s=(t+a/t)mod M to Alice Alice recovers the bit x as follows, s+2r=t (1+r/t)* (1+r/t) mod M Also, it follows Jacobi Symbol ((s+2r)/M)=(t/M)=x Here, Alice knows the value of r, using that she will calculate value of x Bob need to double up the Key data as he is not aware whether Alice uses root of a or -a. He will select random t to send x bits, and send s=(t-a/t)mod M. There is no way to avoid sending this extra amount of data. [2] Practical Implementation Aspects: System is not very much expensive in terms of computation. If transport key is K bits long, then Alice need to compute K Jacobi Symbols and Bob will need to compute K divisions mod M and K Jacobi Symbols. Issue: Each bit of Transport key requires size up to M for bandwidth. Bob Does not know whether Alice uses square root of a or -a, hence, he must

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