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

Table 3

(a) The number of operations needed to factor 𝑛 with GNFS method. (b) Computing 𝜙(𝑛) without Factoring “𝑛.”

(a)

Digits	Number of operations	Time
100	9.6 × 108	16 minutes
200	3.3 × 1012	38 days
300	1.3 × 1015	41 years
400	1.7 × 1017	5313 years
500	1.1 × 1019	3.5 × 105 years
1024	1.3 × 1026	4.2 × 1012 years
2048	1.5 × 1035	4.9 × 1021 years

(b)

𝑞−𝑝	(𝑞+𝑝)^2=4𝑛+(𝑞−𝑝)^2	𝑞+𝑝=4𝑛+(𝑞−𝑝)^2
1	885	29.7489⋯
2	888	29.7993⋯
3	893	29.8831⋯
4	900	30