sen 2: 'which is the most' -> 'as it is the most widely'
sen 3: 'a parameter' -> 'the parameter'
Paragraph 3: mention here that p is 581 bits.
par 4: 'its twisted curve E' ' -> 'its twist E' ' (or Et if you choose to use this notation earlier).
par 4: Just refer to earlier, no need to redefine the pairing. Delete from 'A pairing' until the end of the paragraph. (This is the first mention of D-type. If you want to discuss it it needs to be defined in the earlier sections).
point on G_1: Problems:
-'G_1 defined over E' does not mean anything.
-You defined E above so you do not need to repeat it here (and it is inconsistent with your definition here unless b=1). I'll assume you want b=1.
-Above you say that G_1 is of order r. Is G_1 of order r or does r divide the order of G_1? The corrected version of the pairing intro I wrote above defines G_1 as the largest prime-order subgroup of E(F_p), I'll assume this is what you want here as well.
Comments from Chloe (Expert Reviewer)
sen 1: 'candidats' -> 'candidates'
sen 2: 'which is the most' -> 'as it is the most widely'
sen 3: 'a parameter' -> 'the parameter'
Paragraph 3: mention here that p is 581 bits.
par 4: 'its twisted curve E' ' -> 'its twist E' ' (or Et if you choose to use this notation earlier).
par 4: Just refer to earlier, no need to redefine the pairing. Delete from 'A pairing' until the end of the paragraph. (This is the first mention of D-type. If you want to discuss it it needs to be defined in the earlier sections).
point on G_1: Problems: -'G_1 defined over E' does not mean anything. -You defined E above so you do not need to repeat it here (and it is inconsistent with your definition here unless b=1). I'll assume you want b=1. -Above you say that G_1 is of order r. Is G_1 of order r or does r divide the order of G_1? The corrected version of the pairing intro I wrote above defines G_1 as the largest prime-order subgroup of E(F_p), I'll assume this is what you want here as well.