Editing Gottesman and Chuang Quantum Digital Signature (section)

==Protocol Description==

<u>'''Stage 1'''</u> Distribution
*'''Input''' L
*'''Output''' Seller: <math>\{k_0^i, k_1^i\}</math>, <math>1\le i\le M</math>, <math>|\{f(k_0^i)\rangle, |f(k_1^i)\rangle\}</math>
**'''Key Generation'''
#For m = 0,1
##For i=0,M
###Seller generates classical bits <math>k_m^i</math>
##Seller performs quantum one way map: <math>k_m^i\rightarrow |f(k_m^i)\rangle</math> 4 times for each element, so to have 4 copies denoted <math>|f(k_m^i)\rangle^j</math> with j from 1 to 4

**'''Key Distribution:''' (No Trusted Third Party Assumption)
#For m = 0,1
##For i=0,M
###Seller sends <math>|f(k_m^i)\rangle^j</math> to buyer for j=1,2 and verifier for j=3,4
###Buyer performs '''QSWAP TEST'''<math>(|f(k_m^i)\rangle^1, |f(k_m^i)\rangle^2)</math>
###If QSWAP TEST= '''False''', protocol aborted
###If QSWAP TEST= '''True''', Buyer sends any of <math>|f(k_m^i)\rangle^2, |f(k_m^i)\rangle^1</math> (supposedly equal) to verifier, we will denote it with b superscript
###Verifier performs similar steps and sends any of <math>|f(k_m^i)\rangle^3, |f(k_m^i)\rangle^4</math> (supposedly equal) to buyer, we will denote it with v superscript
###Both perform '''QSWAP TEST'''<math>(|f(k_m^i)\rangle^{b},|f(k_m^i)\rangle^{v})</math> on the copies they have 
###If QSWAP TEST= '''False''', protocol aborted
###If QSWAP TEST= '''True''', distribution successful

<u>'''Stage 2'''</u> Messaging
 
*'''Input''' Seller: Message b, Private Key for <math>k_b</math>
*'''Output''' Buyer: '''1-ACC''' (Message is valid and transferable), '''0-ACC''' (Message is valid but not transferable), '''REJ''' (Message is invlaid)
**'''Signing:''' 
# For i=1,M
#	Seller sends Buyer (b,<math>k_b^i</math>)
#	For l = 1,2,..,L
## Buyer performs <math>k_b^i\rightarrow f_{new}</math>
## Buyer performs '''QSWAP TEST'''<math>(f_{new},f_{k_b^i})</math>
## If QSWAP TEST= False, <math>s_B=s_B+1</math>
#If <math>s_B<c_1M</math>, result '''1-ACC'''
#If <math>c_1M<s_B<c_2M</math>, result '''0-ACC'''
#If <math>s_B>c_2M</math>, result '''REJ'''