﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.00,0:00:02.00,Default,,0000,0000,0000,,The answer is the first one. Dialogue: 0,0:00:02.00,0:00:05.00,Default,,0000,0000,0000,,Bob should compute yA to the xB power modulo q. Dialogue: 0,0:00:05.00,0:00:07.00,Default,,0000,0000,0000,,The second one would compute the same thing. Dialogue: 0,0:00:07.00,0:00:09.00,Default,,0000,0000,0000,,This is in fact exactly what Alice computed, Dialogue: 0,0:00:09.00,0:00:12.00,Default,,0000,0000,0000,,but Bob can't do this because he doesn't know xA. Dialogue: 0,0:00:12.00,0:00:14.00,Default,,0000,0000,0000,,The third one wouldn't compute the same key, Dialogue: 0,0:00:14.00,0:00:19.00,Default,,0000,0000,0000,,so the correctness property is that Alice and Bob obtain the same key, Dialogue: 0,0:00:19.00,0:00:21.00,Default,,0000,0000,0000,,and we can show this by just plugging in the values. Dialogue: 0,0:00:21.00,0:00:25.00,Default,,0000,0000,0000,,The key Alice computed was yB to the xA. Dialogue: 0,0:00:25.00,0:00:28.00,Default,,0000,0000,0000,,The value of yB is g to the xB, so that's equivalent Dialogue: 0,0:00:28.00,0:00:31.00,Default,,0000,0000,0000,,to g to the xB xA mod q. Dialogue: 0,0:00:31.00,0:00:34.00,Default,,0000,0000,0000,,The key that Bob would compute--and we'll write that as key BA Dialogue: 0,0:00:34.00,0:00:38.00,Default,,0000,0000,0000,,since we haven't yet shown that they're equivalent using this equation. Dialogue: 0,0:00:38.00,0:00:44.00,Default,,0000,0000,0000,,Well, yA is g to the xA, so this is g to the xA xB mod q. Dialogue: 0,0:00:44.00,0:00:47.00,Default,,0000,0000,0000,,And we already showed that powers of powers are commutative, Dialogue: 0,0:00:47.00,9:59:59.99,Default,,0000,0000,0000,,so these two are equivalent.