Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In 2001, Cocks presented an elegent Identity Based Encryption (IBE) system based on standard quadratic residuocity problem modulo an RSA composite N. This is the first IBE system that does not use pairings on elliptic curves. Cocks’ IBE, hover, requires 2l elements of ℤ/Nℤ and 2l additional bits for an l-bit plaintext. At FOCS’07, Boneh-Gentry-Hamburg (BGH) presented a space-efficient IBE system without pairings thus solving a long standing open problem. The ciphertext length was reduced to just a single element of ℤ/Nℤ plus l + 1 additional bits. Hover, the encryption time of the concrete instantiation of their IBE is not ideal. The encryptor must solve l + 1 equations of the form $RX^2+SY^2=1\bmod N$ given R,S ∈ ℤ/Nℤ; while the decryptor needs a solution of l of these equations. Solving such equations seems to be the main bottleneck.   first show that the encryptor can find a random solution to an equation of the above type using only one inversion in ℤ/Nℤ.  then present a variant of the concrete instantiation of “BasicIBE” of BGH where (1) the private key consists of a single element of ℤ/Nℤ instead of l elements; (2) the encryptor needs to solve only $2\lceil \sqrt{\ell}\rceil$ equations of the form $Rx^2+Sy^2=1\bmod N$; (3) the decryptor can decrypt without solving any such equations and (4) the ciphertext size increases from a single element of ℤ/Nℤ to $2\lceil \sqrt{\ell}\rceil$ elements of ℤ/Nℤ.
['Mahabir Prasad Jhanwar', 'Rana Barua']
I am writing a research paper on a idea A Variant of Boneh-Gentry-Hamburg's Pairing-Free Identity Based Encryption Scheme
[{"from": "human", "value": "I am writing a research paper on a idea A Variant of Boneh-Gentry-Hamburg's Pairing-Free Identity Based Encryption Scheme"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In 2001, Cocks presented an elegent Identity Based Encryption (IBE) system based on standard quadratic residuocity problem modulo an RSA composite N. This is the first IBE system that does not use pairings on elliptic curves. Cocks\u2019 IBE, hover, requires 2l elements of \u2124/N\u2124 and 2l additional bits for an l-bit plaintext. At FOCS\u201907, Boneh-Gentry-Hamburg (BGH) presented a space-efficient IBE system without pairings thus solving a long standing open problem. The ciphertext length was reduced to just a single element of \u2124/N\u2124 plus l\u2009+\u20091 additional bits. Hover, the encryption time of the concrete instantiation of their IBE is not ideal. The encryptor must solve l\u2009+\u20091 equations of the form $RX^2+SY^2=1\\bmod N$ given R,S\u2009\u2208\u2009\u2124/N\u2124; while the decryptor needs a solution of l of these equations. Solving such equations seems to be the main bottleneck.   first show that the encryptor can find a random solution to an equation of the above type using only one inversion in \u2124/N\u2124.  then present a variant of the concrete instantiation of \u201cBasicIBE\u201d of BGH where (1) the private key consists of a single element of \u2124/N\u2124 instead of l elements; (2) the encryptor needs to solve only $2\\lceil \\sqrt{\\ell}\\rceil$ equations of the form $Rx^2+Sy^2=1\\bmod N$; (3) the decryptor can decrypt without solving any such equations and (4) the ciphertext size increases from a single element of \u2124/N\u2124 to $2\\lceil \\sqrt{\\ell}\\rceil$ elements of \u2124/N\u2124."}]