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Proving primality in essentially quartic random time
 Math. Comp
, 2003
"... Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1. ..."
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Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1.
Solving Discrete Logarithms from Partial Knowledge of the Key
"... Abstract. For elliptic curve based cryptosystems, the discrete logarithm problem must be hard to solve.Butevenwhenthisistruefrom a mathematical point of view, sidechannel attacks could be used to reveal information about the key if proper countermeasures are not used. In this paper, we study the di ..."
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Abstract. For elliptic curve based cryptosystems, the discrete logarithm problem must be hard to solve.Butevenwhenthisistruefrom a mathematical point of view, sidechannel attacks could be used to reveal information about the key if proper countermeasures are not used. In this paper, we study the difficulty of the discrete logarithm problem when partial information about the key is revealed by side channel attacks. We provide algorithms to solve the discrete logarithm problem for generic groups with partial knowledge of the key which are considerably better than using a squareroot attack on the whole key or doing an exhaustive search using the extra information, under two different scenarios. In the first scenario, we assume that a sequence of contiguous bits of the key is revealed. In the second scenario, we assume that partial information on the “Square and Multiply Chain ” is revealed.