

A, B, S :  principal 
Ka, Kb :  key 
PK, SK :  principal > key (keypair) 
1.  A  >  S  :  B, {Ka}PK(S) 
2.  S  >  B  :  A 
3.  B  >  S  :  A, {Kb}PK(S) 
4.  S  >  A  :  B, {Kb}Ka 
1.  I(A)  >  S  :  B, {Ki}PK(S) 
2.  S  >  B  :  A 
3.  B  >  S  :  A, {Kb}PK(S) 
4.  S  >  I(A)  :  B, {Kb}Ki 
1.  A  >  S  :  B, {Ka}PK(S) 
2.  S  >  I(B)  :  A 
3.  I(B)  >  S  :  A, {Ki}PK(S) 
4.  S  >  I(A)  :  B, {Ki}Ka 
5.  A  >  B  :  {X}Kb 
i.1.  I(A)  >  S  :  B, {Ki}PK(S) 
i.2.  S  >  B  :  A 
i.3.  B  >  S  :  A, {Kb}PK(S) 
i.4.  S  >  I(A)  :  B, {Kb}Ki 
ii.1.  A  >  S  :  B, {Ka}PK(S) 
ii.2.  S  >  I(B)  :  A 
ii.3.  I(B)  >  S  :  A, {Kb}PK(S) 
ii.4.  S  >  I(A)  :  B, {Kb}Ka 
Hence, the protocol reads:
(x+y)+y = x (1) x+(x+y) = y (1')
1.  A  >  S  :  B, {Ka}PK(S) 
2.  S  >  B  :  A 
3.  B  >  S  :  A, {Kb}PK(S) 
4.  S  >  A  :  B, Kb + Ka 
{x * {y}PK(U)}PK(U) = {x*y}PK(U) (2)Moreover, we assume a partial division operator (associated to *).
x^3 mod n
(with x < n),
i.3.  B  >  I(S)  :  A, {Kb}PK(S) 
ii.1.  I  >  S  :  D, {Ki * {Kb}PK(S)}PK(S) ( = {Ki*Kb}PK(S) by (2) ) 
ii.2.  S  >  I(D)  :  I 
ii.3.  I(D)  >  S  :  I, {Kd}PK(S) 
ii.4.  S  >  I  :  D, Kd + (Ki * Kb) 

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