1 | r""" |
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2 | Rivest, Shamir, Adleman public-key encryption scheme. |
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3 | |
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4 | The Rivest, Shamir, Adleman public-key encryption scheme. The Rivest-Shamir-Adleman (RSA) scheme has since that time reigned supreme as the most widely accepted and implemented general-purpose approach to public-key encryption. See also the `Wikipedia article <http://en.wikipedia.org/wiki/RSA>`_ on this scheme. |
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5 | |
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6 | REFERENCES: |
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7 | |
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8 | .. William Stallings. *Cryptography and Network Security, Priciples and Practices*, Fourth Edition. Prentice Hall, 16 November 2005. |
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9 | |
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10 | .. Anoop MS. *Public Key Cryptography, Applications Algorithms and Mathematical Explanations*. Tata Elxsi Ltd, India anoopms@tataelxsi.co.in. |
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11 | |
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12 | .. Minh Van Nguyen. *Number Theory and the RSA Public Key Cryptosystem*. nguyenminh2@gmail.com, July 2009. |
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13 | |
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14 | AUTHORS: |
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15 | |
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16 | -Ajeesh R (2011-07)- initial procedural version released as public domain software. |
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17 | |
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18 | """ |
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19 | |
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20 | ########################################################################### |
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21 | # Copyright (c) 2011 |
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22 | # AJEESH R <iamajeeshr@gmail.com> |
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23 | # |
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24 | # This program is free software; you can redistribute it and/or modify |
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25 | # it under the terms of the GNU General Public License as published by |
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26 | # the Free Software Foundation; either version 2 of the License, or |
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27 | # (at your option) any later version. |
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28 | # |
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29 | # This program is distributed in the hope that it will be useful, |
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30 | # but WITHOUT ANY WARRANTY; without even the implied warranty of |
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31 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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32 | # GNU General Public License for more details. |
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33 | # |
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34 | # http://www.gnu.org/licenses/ |
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35 | ########################################################################### |
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36 | |
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37 | from sage.crypto.cryptosystem import PublicKeyCryptosystem |
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38 | from sage.rings.arith import gcd |
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39 | from sage.rings.arith import xgcd |
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40 | from sage.rings.arith import is_prime |
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41 | from sage.rings.arith import power_mod |
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42 | from sage.rings.arith import euler_phi |
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43 | from sage.rings.integer import Integer |
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44 | from sage.rings.finite_rings.integer_mod import Mod as mod |
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45 | from sage.rings.integer_ring import ZZ |
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46 | |
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47 | class RSACryptosystem(PublicKeyCryptosystem): |
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48 | r""" |
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49 | The Rivest, Shamir, Adleman public-key encryption scheme. |
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50 | |
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51 | The RSA encryption and decryption algorithms as described in |
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52 | :func:`encrypt() <RSACryptosystem.encrypt>` and |
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53 | :func:`decrypt() <RSACryptosystem.decrypt>`, respectively, make use of the |
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54 | |
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55 | EXAMPLES: |
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56 | |
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57 | The following is an encryption/decryption example.:: |
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58 | |
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59 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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60 | sage: rc = RSACryptosystem(); rc |
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61 | The Rivest, Shamir, Adleman public-key encryption scheme. |
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62 | sage: p = 31; q = 61; |
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63 | sage: pubkey = rc.public_key(p, q); pubkey |
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64 | (4951760154835678088235319297L, 17) |
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65 | sage: prikey = rc.private_key(p, q); prikey |
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66 | (4951760154835678088235319297L, 4077920125612805357425763753) |
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67 | sage: P = 72697676798779827668 |
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68 | sage: C = rc.encrypt(P, pubkey); C |
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69 | 2467165704948727396791981601 |
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70 | sage: M = rc.decrypt(C, prikey); M |
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71 | 72697676798779827668 |
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72 | sage: M == P |
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73 | True |
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74 | |
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75 | Generate a pair of public/private keys. Use the public key to |
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76 | encrypt a plaintext. Then decrypt the resulting ciphertext using the |
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77 | private key. Finally, compare the decrypted message with the original |
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78 | plaintext.:: |
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79 | |
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80 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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81 | sage: rc = RSACryptosystem() |
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82 | sage: P = 72697676798779827668 |
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83 | sage: C = rc.encrypt(P, pubkey) |
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84 | sage: M = rc.decrypt(C, prikey) |
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85 | sage: M == P |
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86 | True |
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87 | """ |
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88 | |
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89 | def __init__(self): |
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90 | """ |
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91 | Construct the RSA public-key encryption scheme. |
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92 | |
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93 | OUTPUT: |
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94 | |
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95 | - A ``RSACryptosystem`` object representing the RSA public-key encryption scheme. |
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96 | |
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97 | See the class docstring of ``RSACryptosystem`` for detailed documentation. |
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98 | |
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99 | EXAMPLES:: |
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100 | |
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101 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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102 | sage: rc = RSACryptosystem() |
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103 | sage: rc == loads(dumps(rc)) |
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104 | True |
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105 | """ |
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106 | # no internal data for now; nothing to initialize |
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107 | pass |
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108 | |
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109 | def __eq__(self, other): |
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110 | """ |
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111 | Compare this ``RSACryptosystem`` object with ``other``. |
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112 | |
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113 | INPUT: |
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114 | |
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115 | - ``other`` -- a ``RSACryptosystem`` object. |
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116 | |
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117 | OUTPUT: |
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118 | |
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119 | - ``True`` if both ``self`` and ``other`` are ``RSACryptosystem`` |
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120 | objects. ``False`` otherwise. |
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121 | |
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122 | Two objects are ``RSACryptosystem`` objects if their string |
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123 | representations are the same. |
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124 | |
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125 | EXAMPLES:: |
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126 | |
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127 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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128 | sage: rc1 = RSACryptosystem() |
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129 | sage: rc2 = RSACryptosystem() |
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130 | sage: rc1 == rc2 |
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131 | True |
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132 | """ |
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133 | if self.__repr__() == other.__repr__(): |
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134 | return True |
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135 | else: |
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136 | return False |
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137 | |
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138 | def __repr__(self): |
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139 | """ |
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140 | A string representation of this RSA public-key encryption scheme. |
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141 | |
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142 | OUTPUT: |
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143 | |
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144 | - A string representation of this RSA public-key encryption scheme. |
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145 | |
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146 | EXAMPLES:: |
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147 | |
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148 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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149 | sage: RSACryptosystem() |
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150 | The Rivest, Shamir, Adleman public-key encryption scheme. |
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151 | """ |
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152 | return "The Rivest, Shamir, Adleman public-key encryption scheme." |
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153 | |
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154 | def decrypt(self, C, K): |
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155 | r""" |
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156 | Apply the RSA public-key encryption scheme to decrypt the ciphertext ``C`` |
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157 | using the private key ``K``. |
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158 | |
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159 | INPUT: |
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160 | |
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161 | - ``C`` -- a ciphertext resulting from encrypting a plaintext using |
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162 | the RSA public-key encryption algorithm. |
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163 | |
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164 | - ``K`` -- a private key `(n, d)` where `n` is the product of two Mersenne |
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165 | primes `p` and `q`, `d` is computed using the extended euclidean algorithm. |
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166 | |
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167 | OUTPUT: |
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168 | |
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169 | - The plaintext resulting from decrypting the ciphertext ``C`` using |
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170 | the RSA public-key decryption algorithm. |
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171 | |
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172 | ALGORITHM: |
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173 | |
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174 | The RSA public-key decryption algorithm is described as follows: |
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175 | |
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176 | #. Let `C` be the ciphertext `C = (M^e) mod n`. |
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177 | #. Let `(n, d)` be the private key whose corresponding |
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178 | public key is `(n, e)`. |
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179 | #. The plaintext is `M = (C^d) mod n`. |
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180 | |
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181 | EXAMPLES: |
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182 | |
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183 | The following is a decryption example. Here we decrypt a string of numbers.:: |
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184 | |
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185 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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186 | sage: rc = RSACryptosystem() |
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187 | sage: p = 31; q = 61; |
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188 | sage: C = 2467165704948727396791981601 |
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189 | sage: K = rc.private_key(p, q); K |
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190 | (4951760154835678088235319297L, 4077920125612805357425763753) |
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191 | sage: P = rc.decrypt(C, K); P |
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192 | 72697676798779827668 |
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193 | |
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194 | TESTS: |
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195 | |
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196 | The private key `K = (n, d)` must be such that `p` and `q` are |
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197 | distinct prime numbers.:: |
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198 | |
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199 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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200 | sage: rc = RSACryptosystem() |
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201 | sage: C = 2467165704948727396791981601 |
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202 | sage: K = rc.private_key(31, 31); |
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203 | Traceback (most recent call last): |
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204 | ... |
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205 | ValueError: p and q must be distinct primes. |
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206 | sage: K = rc.private_key(31, 61) |
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207 | sage: P = rc.decrypt(C, K); P |
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208 | 72697676798779827668 |
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209 | """ |
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210 | # private key |
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211 | (n, d) = K |
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212 | |
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213 | # perform the decryption |
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214 | return power_mod(C, d, n) |
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215 | |
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216 | def encrypt(self, P, K): |
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217 | r""" |
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218 | Apply the RSA public-key encryption scheme to encrypt the plaintext ``P`` using |
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219 | the public key ``K``. |
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220 | |
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221 | INPUT: |
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222 | |
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223 | - ``P`` -- a non-empty string of plaintext. The string ``""`` is |
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224 | an empty string, whereas ``" "`` is a string consisting of one |
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225 | white space character. The plaintext can be a string of numbers or |
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226 | a string of ASCII characters. |
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227 | |
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228 | - ``K`` -- a public key, which is the product of two Mersenne primes and e, 0 < e < euler_phi(n) |
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229 | such that also satisfy the requirement `\gcd(e, euler_phi(n))=1`. |
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230 | |
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231 | OUTPUT: |
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232 | |
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233 | - The ciphertext resulting from encrypting ``P`` using the public |
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234 | key ``K``. |
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235 | |
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236 | ALGORITHM: |
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237 | |
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238 | The RSA public-key encryption algorithm is described as follows: |
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239 | |
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240 | #. Let `(n, e)` be a public key, where `n = pq` is the product of two |
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241 | distinct Mersenne primes `p` and `q` and `e` is a random positive |
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242 | integer that is co-prime to euler_phi(n). |
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243 | #. Let `P = a string` be the message (plaintext). |
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244 | #. The ciphertext is `C = (P^e) mod n`. |
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245 | |
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246 | |
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247 | EXAMPLES: |
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248 | |
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249 | The following is an encryption example. Here, we encrypt a string of number.:: |
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250 | |
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251 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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252 | sage: rc = RSACryptosystem() |
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253 | sage: p = 101; q = 103; |
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254 | sage: P = 4 |
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255 | sage: C = rc.encrypt(P, K); C |
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256 | 4932417489295796679844298689 |
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257 | |
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258 | Now encrypt another string of numbers. The result is random; no seed is |
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259 | provided to the encryption function so the function generates its |
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260 | own random seed.:: |
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261 | |
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262 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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263 | sage: rc = RSACryptosystem() |
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264 | sage: p = 31; q = 61; |
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265 | sage: P = 72697676798779827668 |
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266 | sage: K = rc.public_key(p, q); |
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267 | sage: C = rc.encrypt(P, K); |
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268 | sage: C |
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269 | 2467165704948727396791981601 |
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270 | |
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271 | TESTS: |
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272 | |
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273 | The plaintext cannot be an empty string. :: |
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274 | |
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275 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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276 | sage: rc = RSACryptosystem() |
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277 | sage: rc.encrypt("", 3) |
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278 | Traceback (most recent call last): |
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279 | ... |
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280 | ValueError: The plaintext cannot be an empty string. |
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281 | """ |
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282 | # public key |
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283 | (n, e) = K |
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284 | |
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285 | # sanity check |
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286 | if P == "": |
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287 | raise ValueError("The plaintext cannot be an empty string.") |
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288 | |
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289 | else: |
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290 | return power_mod(P, e, n) |
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291 | |
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292 | def private_key(self, p, q): |
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293 | r""" |
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294 | Return the RSA private key corresponding to the |
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295 | distinct Mersenne primes ``p`` and ``q``. |
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296 | |
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297 | INPUT: |
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298 | |
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299 | - ``p`` -- a prime number. |
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300 | |
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301 | - ``q`` -- a prime number. |
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302 | |
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303 | OUTPUT: |
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304 | |
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305 | - The RSA private key `(n, d)` where |
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306 | `de is congruent to (1 mod euler_phi(n))`. |
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307 | |
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308 | Both ``p`` and ``q`` must be distinct primes. Let `p` be a |
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309 | positive prime. Then `p` is a Mersenne prime if `p` is equal to `(2^p)-1`. |
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310 | |
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311 | EXAMPLES: |
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312 | |
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313 | Obtain two distinct primes and compute the RSA |
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314 | private key corresponding to two Mersenne primes generated from the distinct primes.:: |
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315 | |
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316 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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317 | sage: rc = RSACryptosystem() |
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318 | sage: rc.private_key(19, 23) |
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319 | (4398037598209L, 1173935455331) |
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320 | |
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321 | Choose two distinct primes, compute the RSA |
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322 | private key corresponding to two Mersenne primes generated from the distinct primes , and test that the |
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323 | resulting private key `(n, d)` satisfies |
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324 | `\mod(de, euler_phi(n)) = 1`.:: |
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325 | |
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326 | sage: p = 31 |
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327 | sage: q = 61 |
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328 | sage: p = (2 ^ p)-1; q = (2 ^ q)-1 |
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329 | sage: is_prime(p); is_prime(q) |
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330 | True |
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331 | True |
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332 | |
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333 | TESTS: |
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334 | |
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335 | Both of the input ``p`` and ``q`` must be distinct primes. :: |
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336 | |
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337 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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338 | sage: rc = RSACryptosystem() |
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339 | sage: rc.private_key(78307, 78307) |
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340 | Traceback (most recent call last): |
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341 | ... |
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342 | ValueError: p and q must be distinct primes. |
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343 | """ |
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344 | if p == q: |
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345 | raise ValueError("p and q must be distinct primes.") |
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346 | if is_prime(p) and is_prime(q): |
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347 | p = (2**p)-1 |
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348 | q = (2**q)-1 |
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349 | |
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350 | n = p * q |
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351 | |
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352 | e = 3 |
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353 | while 1: |
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354 | if gcd(euler_phi(n), e) == 1: |
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355 | break |
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356 | else: |
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357 | e = e + 2 |
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358 | |
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359 | # Generate a number d such that de = 1 (mod euler_phi(n)) |
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360 | bezout = xgcd(e, euler_phi(n)) |
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361 | d = Integer(mod(bezout[1], euler_phi(n))) |
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362 | while mod(d * e, euler_phi(n)) != 1: |
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363 | d = Integer(mod(bezout[1], euler_phi(n))) |
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364 | else: |
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365 | raise ValueError("p and q must be distinct primes.") |
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366 | return (n, d) |
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367 | |
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368 | def public_key(self, p, q): |
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369 | r""" |
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370 | Return RSA public key corresponding to the |
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371 | distinct Mersenne primes ``p`` and ``q``. |
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372 | |
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373 | INPUT: |
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374 | |
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375 | - ``p`` -- a prime number. |
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376 | |
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377 | - ``q`` -- a prime number. |
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378 | |
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379 | OUTPUT: |
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380 | |
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381 | - The RSA public key `n = pq` and `e`. |
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382 | |
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383 | Both ``p`` and ``q`` must be distinct primes. Let `p` be a |
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384 | positive prime. Then `p` is a Mersenne prime if `p` is equal to `(2^p)-1`. |
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385 | |
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386 | EXAMPLES: |
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387 | |
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388 | Obtain two distinct Mersenne primes and compute the RSA |
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389 | public key corresponding to those two Mersenne primes:: |
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390 | |
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391 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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392 | sage: rc = RSACryptosystem() |
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393 | sage: rc.public_key(31, 61) |
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394 | (4951760154835678088235319297L, 17) |
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395 | |
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396 | Choose two distinct primes, compute the RSA |
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397 | public key corresponding to two Mersenne primes generated from the distinct primes.:: |
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398 | |
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399 | sage: p = 31 |
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400 | sage: q = 61 |
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401 | sage: p = (2 ^ p)-1; q = (2 ^ q)-1 |
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402 | sage: is_prime(p); is_prime(q) |
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403 | True |
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404 | True |
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405 | |
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406 | TESTS: |
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407 | |
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408 | The input ``p`` and ``q`` must be distinct primes. :: |
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409 | |
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410 | sage: from sage.crypto.public_key.rsa_cryptosystem import RSACryptosystem |
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411 | sage: rc = RSACryptosystem() |
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412 | sage: rc.public_key(3, 3) |
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413 | Traceback (most recent call last): |
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414 | ... |
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415 | ValueError: p and q must be distinct primes. |
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416 | """ |
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417 | if p == q: |
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418 | raise ValueError("p and q must be distinct primes.") |
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419 | if is_prime(p) and is_prime(q): |
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420 | p = (2**p)-1 |
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421 | q = (2**q)-1 |
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422 | |
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423 | n = p * q |
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424 | # Generate a number e so that gcd(euler_phi(n), e) = 1 |
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425 | e = 3 |
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426 | while 1: |
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427 | if gcd(euler_phi(n), e) == 1: |
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428 | break |
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429 | else: |
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430 | e = e + 2 |
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431 | else: |
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432 | raise ValueError("p and q must be distinct primes.") |
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433 | return (n, e) |
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