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Paulo Mateus   MMA\[Dash]MEIC
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 " ,where p is a positive polynomial, is said to be\n\[Bullet] strongly \
collision-free: if it is (hard) impossible to find in polynomial-time x,x\
\[CloseCurlyQuote] \[Element] ",
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\[Element] ",
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 ": Show that the strongly collision-free implies weakly \n\nAssume h is not \
weakly, then for a given x it is possible to find x\[CloseCurlyQuote] in PT, \
so randomly choose an x, and the provide the pair (x,x\[CloseCurlyQuote]) \
that collides, clearly then h is not stronfly collision-free.\n\n",
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 " Assume P=NP, and and let ",
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collision of x\n\nInput x\noutput x\[CloseCurlyQuote]\nx\[CloseCurlyQuote]=\
\[CurlyEpsilon] //empty string;\nwhile(|x|!=|x\[CloseCurlyQuote]|){ \n   \
if(",
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//divide and conquer or bisection or whatever...\n   else \
x\[CloseCurlyQuote]=x\[CloseCurlyQuote]1 \n}\n\nClearly this program runs in \
polynomial time in n, and for sure the solution is a collision\n\nDone!\n\n\n\
\nmd5 and sha1 are not strongly collision resistant but they are still weakly \
collision-free!\n\nSHA2 is still strongly collision resistant.\n\n",
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that allows for if-then-else commands.\n\nh(X)=h(Y) it is publicly known for \
md5 and sha1\n\nm1 ZZZZZZZZZif(\[DownArrow]X==Y)then{aspirin}else{morphine}\n\
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ZZZZZZZZZif(\[DownArrow]Y==Y)then{aspirin}else{morphine}\n\n(m1, ",
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Cell["Oblivious transfer and Bit commitment", "Section",
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Cell[TextData[{
 "Oblivious transfer (OT) is a fundamental  concept upon which all privacy \
can be set! Namely verifiable secret sharing, an even ZK can be constructed \
directly from OT. It is quite unituitive that OT is enough to construct all \
privacy protocols.",
 StyleBox["\n\nDefinition ",
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 "An all-or-nothing OT protocol is a two party protocol between the sender S \
and the receive R where S wants to send a message to R such that:\n\n1) R \
receives the message with probability 1/2, but R knows that S tried to send \
some message\n2)",
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Bob B. \nA wants to commit to some bit b (yes or no) and so she sends a \
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some function op(\[Gamma])=b in such a way that:\n\n1) ",
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in the quantum case is open)",
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Elections (checking the mode of private preferences) are all particular cases \
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say \[Eta], then f can be implemented with a Boolean circuit.\n\nAll Boolean \
circuits can be constructed using NAnd (NAnd is  universal gate) and Fan-Out \
(copying a bit), since only NAnd requires two inputs, only there one input \
can belong to one agent and the other to another agent.\n\n",
 StyleBox["Garbling",
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