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Paulo Mateus   MMA\[Dash]MEIC
\
\>", "Text",
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Cell["Discrete log problem", "Section",
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 "Let p be a prime number \[Alpha] a primitive element (generator of ",
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 StyleBox["Objective: ",
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 "Find a, with 0 \[LessEqual] a \[LessEqual] p-2, such that ",
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 " mod p (sometimes we drop the \[OpenCurlyDoubleQuote]mod p\
\[CloseCurlyDoubleQuote] whenever it is obvious from context).\n\n",
 StyleBox["Exercise",
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 ": Show that the Discrete Log problem is NP (and also that it is co-NP).\n\
Show that \nA={(p,\[Alpha],\[Beta],k):",
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 " mod p < k} \[Element] NP\n\n\n\nB={(p,\[Alpha],\[Beta],k,a):",
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Thus, A is in NP since A is  a projection of B in the first 4 components.\n\n\
\n\nAs we shall see, there are several primes for which the problem is known \
to be polynomial-time. This idea is also the basis of ",
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 "Let p be a prime number for which the Discrete Log problem is hard (p=2q+1) \
and \[Alpha] a public known primitive element for ",
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\[RightArrow]",
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 StyleBox["Definition",
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 " A number n is said to be",
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Cell[TextData[{
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 " mod p (sometimes we drop the \[OpenCurlyDoubleQuote]mod p\
\[CloseCurlyDoubleQuote] whenever it is obvious from context).\n\nWe have \
seen that if p=2q+1 with q prime, then p is a safe prime against PH, note \
that p=3 mod 4 in this case.\n\nLet ",
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\[Beta]\[CloseCurlyQuote]=",
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all odd number with exception of q are invertible mod p-1\n\n\[Bullet] ",
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obvious...\n \n",
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 StyleBox[" Show what happen if, for a given x, one fix \[Gamma] or fix \
\[Delta] and try to finish the signature.\n\n1) Fix x and \[Gamma] and try to \
find \[Delta] such that (x,\[Gamma],\[Delta]) is a signature\n\n",
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rich\n \n ",
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means that gcd(p-1,\[Gamma])=d>1 (because we chose p-1=2q), so d=2 or d=q\n\n\
assume d =2 because for the other case is very improbable\n (that is, \
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This operations forms an abelian group (associativity is hard to check), \
moreover kP can be computed in polynomial time on |p|
\
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Cell[TextData[{
 StyleBox["Theorem",
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 "[Hasse] Let E be an elliptic curve defined over ",
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