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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Coppersmith's Attack ： ウィキペディア英語版 Coppersmith's Attack Coppersmith's attack describes a class of attacks on the public-key cryptosystem RSA based on Coppersmith's theorem (see below). The public key in the RSA system is a tuple of integers $(N,e)$, where ''N'' is the product of two primes ''p'' and ''q''. The secret key is given by an integer ''d'' satisfying $ed\backslash equiv\; 1\; \backslash pmod$; equivalently, the secret key may be given by $d\_p\backslash equiv\; d\; \backslash pmod$ and $d\_q\backslash equiv\; d\; \backslash pmod$ if the Chinese remainder theorem is used to improve the speed of decryption, see CRT-RSA. Encryption of a message ''M'' produces the ciphertext $C\backslash equiv\; M^e\backslash pmod$ which can be decrypted using $d$ by computing $C^d\backslash equiv\; M\; \backslash pmod$. Coppersmith's theorem has many applications in attacking RSA specifically if the public exponent ''e'' is small or if partial knowledge of the secret key is available. ==Low Public Exponent Attack== In order to reduce encryption or signature-verification time, it is useful to use a small public exponent ($e$). In practice, common choices for $e$ are 3, 17 and 65537 $(2^+1)$.〔(Imad Khaled Salah,Abdullah Darwish and Saleh Oqeili. Mathematical Attacks on RSA Cryptosystem )〕 These values for ''e'' are Fermat primes, sometimes referred to as $F\_0,\; F\_2$ and $F\_4$ respectively $(F\_x=2^+1)$. They are chosen because they make the modular exponentiation operation faster. Also, having chosen such $e$, it is simpler to test whether $\backslash gcd(e,\; p-1)=1$ and $\backslash gcd(e,\; q-1)=1$ while generating and testing the primes in step 1 of the key generation. Values of $p$ or $q$ that fail this test can be rejected there and then. (Even better: if ''e'' is prime and greater than 2 then the test $p\backslash ,\backslash bmod\backslash ,\; e\; \backslash ne1$ can replace the more expensive test $\backslash gcd(p-1,e)=\; 1$.) If the public exponent is small and the plaintext $m$ is very short, then the RSA function may be easy to invert which makes certain attacks possible. Padding schemes ensure that messages have full lengths but additionally choosing public exponent $e\; =\; 2^\; +\; 1$ is recommended. When this value is used, signature-verification requires 17 multiplications, as opposed to about 25 when a random $e$ of similar size is used. Unlike low private exponent (see Wiener's Attack), attacks that apply when a small $e$ is used are far from a total break which would recover the secret key ''d''. The most powerful attacks on low public exponent RSA are based on the following theorem which is due to Don Coppersmith. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Coppersmith's Attack」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.