Diffie-Hellman key exchange

The Diffie-Hellman key exchange is a cryptographic protocol for symmetric key exchange. There are various implementations of this protocol. The following interchange between Alice and Bob demonstrates the Elliptic Curve Diffie-Hellman key exchange.

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1) Alice and Bob publicly agree on an elliptic curve $E$ over a large finite field $F$ and a point $P$ on that curve.
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2) Alice and Bob each privately choose large random integers, denoted $a$ and $b$ .
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3) Using elliptic curve point-addition, Alice computes $a P$ on $E$ and sends it to Bob. Bob computes $b P$ on $E$ and sends it to Alice.
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4) Both Alice and Bob can now compute the point $a b P$ , Alice by multipliying the received value of $b P$ by her secret number $a$ , and Bob vice-versa.
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5) Alice and Bob agree that the $x$ coordinate of this point will be their shared secret value.

An evil interloper Eve observing the communications will be able to intercept only the objects $E$ , $P$ , $a P$ , and $b P$ . She can succeed in determining the final secret value by gaining knowledge of either of the values $a$ or $b$ . Thus, the security of the exchange depends on the hardness of that problem, known as the elliptic curve discrete logarithm problem. For large $a$ and $b$ , it is a computationally “difficult” problem.

As a side note, some care has to be taken to choose an appropriate curve $E$ . Singular curves and ones with “bad” numbers of points on it (over the given field) have simplified solutions to the discrete log problem.

单词	DiffieHellmanKeyExchange
释义	Diffie-Hellman key exchange The Diffie-Hellman key exchange is a cryptographic protocol for symmetric key exchange. There are various implementations of this protocol. The following interchange between Alice and Bob demonstrates the Elliptic Curve Diffie-Hellman key exchange. • 1) Alice and Bob publicly agree on an elliptic curve $E$ over a large finite field $F$ and a point $P$ on that curve. • 2) Alice and Bob each privately choose large random integers, denoted $a$ and $b$ . • 3) Using elliptic curve point-addition, Alice computes $a P$ on $E$ and sends it to Bob. Bob computes $b P$ on $E$ and sends it to Alice. • 4) Both Alice and Bob can now compute the point $a b P$ , Alice by multipliying the received value of $b P$ by her secret number $a$ , and Bob vice-versa. • 5) Alice and Bob agree that the $x$ coordinate of this point will be their shared secret value. An evil interloper Eve observing the communications will be able to intercept only the objects $E$ , $P$ , $a P$ , and $b P$ . She can succeed in determining the final secret value by gaining knowledge of either of the values $a$ or $b$ . Thus, the security of the exchange depends on the hardness of that problem, known as the elliptic curve discrete logarithm problem. For large $a$ and $b$ , it is a computationally “difficult” problem. As a side note, some care has to be taken to choose an appropriate curve $E$ . Singular curves and ones with “bad” numbers of points on it (over the given field) have simplified solutions to the discrete log problem.
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