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Class-5-26-10

# Class-5-26-10 - MODERN CRYPTOGRAPHY next BASIC FACTS FOR...

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MODERN CRYPTOGRAPHY next

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BASIC FACTS FOR RSA EULER FERMAT next
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THE CONSTRUCTION OF BIG PRIME NUMBERS next
Quadratic Residues next Tables of squares has no solutions has no solutions Mod 5 1 2 3 4 1 4 4 1 x 2 x 0 1 2 3 4 4 6 5 1 2 2 1 4 Mod 7 x 2 x 0 1 2 3 4 6 5 7 11 9 8 0 1 4 9 5 3 3 5 1 4 9 Mod 11 x 2 x QR [5] = { 1 , 4 }

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How many Quadratic Residues ? next A useful result
THE LEGENDRE AND JACOBI SYMBOLS Definition we set The previous theorem gives (false) Not easy to compute from this definition when n is large and its prime factors are not known However see next page next

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Don’t forget that for a =0 J(a,n) =0 THE LAW OF QUADRATIC RECIPROCITY has a solution has a solution For instance when (p-1)(q-1)/4 is even A fast way of computing the Jacobi symbol !!! next
The Strassen Solovay Primality Test 1) Pick at random in the interval {1,2,3,. ..,n-1} i 2) For each a = a check the two equalities 3) if any of these tests fails then n is not a prime The bigger is k the higher the probability of reaching a correct conclusion i

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Class-5-26-10 - MODERN CRYPTOGRAPHY next BASIC FACTS FOR...

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