Cryptanalysis and Performance Evaluation of Enhanced Threshold Proxy Signature Scheme Based on RSA for Known Signers

Abstract

In these days there are plenty of signature schemes such as the <svg style="vertical-align:-2.3205pt;width:32.924999px;" id="M1" height="15.0875" version="1.1" viewBox="0 0 32.924999 15.0875" width="32.924999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path id="x1D461" d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" /></g><g transform="matrix(.017,-0,0,-.017,11.741,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,18.438,12.138)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,26.972,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> threshold proxy signature scheme (Kumar and Verma 2010). The network is a shared medium so that the weakness security attacks such as eavesdropping, replay attack, and modification attack. Thus, we have to establish a common key for encrypting/decrypting our communications over an insecure network. In this scheme, a <svg style="vertical-align:-2.3205pt;width:32.924999px;" id="M2" height="15.0875" version="1.1" viewBox="0 0 32.924999 15.0875" width="32.924999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x1D461"/></g><g transform="matrix(.017,-0,0,-.017,11.741,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,18.438,12.138)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,26.972,12.138)"><use xlink:href="#x29"/></g> </svg> threshold proxy signature scheme based on RSA, any <svg style="vertical-align:-0.1638pt;width:5.9250002px;" id="M3" height="10.3875" version="1.1" viewBox="0 0 5.9250002 10.3875" width="5.9250002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.125)"><use xlink:href="#x1D461"/></g> </svg> or more proxy signers can cooperatively generate a proxy signature while <svg style="vertical-align:-0.1638pt;width:31.612499px;" id="M4" height="11.125" version="1.1" viewBox="0 0 31.612499 11.125" width="31.612499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D461"/></g><g transform="matrix(.017,-0,0,-.017,9.633,10.862)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,23.385,10.862)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg> or fewer of them cannot do it. The threshold proxy signature scheme uses the RSA cryptosystem to generate the private and the public key of the signers (Rivest et al., 1978). Comparison is done on the basis of time complexity, space complexity, and communication overhead. We compare the performance of four schemes (Hwang et al. (2003), Kuo and Chen (2005), Yong-Jun et al. (2007), and Li et al. (2007), with the performance of a scheme that has been proposed earlier by the authors of this paper. In the proposed scheme, both the combiner and the secret share holder can verify the correctness of the information that they are receiving from each other. Therefore, the enhanced threshold proxy signature scheme is secure and efficient against notorious conspiracy attacks.