Massey-Omura scheme

The Massey-Omura scheme is a cryptosystem that allows two parties to exchange messages in confidence without the existence of public keys or shared secret keys . It is based on the difficulty of the discrete logarithm .

The Massey-Omura scheme was developed in 1983 by cryptologists James Massey and Jim Omura .

requirements

A prerequisite for the Massey-Omura scheme is the common knowledge of all participants about a large prime number . ${\displaystyle p}$

In addition, each subscriber generated for the communication a key with which relatively prime to is, so we have: . ${\displaystyle T}$ ${\displaystyle e_{T}}$ ${\displaystyle e_{T}<p-1}$ ${\displaystyle p-1}$ ${\displaystyle \operatorname {ggT} (e_{T},p-1)=1}$

The number is determined for this (e.g. using the extended Euclidean algorithm ) . It is the multiplicative inverse of modulo . Thus: . ${\displaystyle d_{T}}$ ${\displaystyle e_{T}}$ ${\displaystyle p-1}$ ${\displaystyle e_{T}\cdot d_{T}=1{\bmod {(}}p-1)}$

Now applies to all messages : ${\displaystyle m<p}$

${\displaystyle (m^{e_{T}})^{d_{T}}=m^{e_{T}\cdot d_{T}}=m^{k\cdot (p-1)+1}=m^{k\cdot (p-1)}\cdot m=m{\bmod {p}}}$based on Fermat's Little Theorem , da ${\displaystyle m^{k\cdot (p-1)}\equiv 1{\bmod {p}}}$

procedure

As an example, subscriber A should transmit the confidential message to subscriber B. You have both , in addition, each only knows his own key and or and . ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle e_{A}}$ ${\displaystyle d_{A}}$ ${\displaystyle e_{B}}$ ${\displaystyle d_{B}}$

A now forms and sends the resulting number to B. ${\displaystyle m^{e_{A}}{\bmod {p}}}$

B raises the received message to the power and replies . ${\displaystyle e_{B}}$ ${\displaystyle (m^{e_{A}})^{e_{B}}{\bmod {p}}}$

A generated , which after the small Fermat's theorem corresponds and sends this back to B. Thus, A has the effect of exponentiation with the known only to him to "canceled." However, the message is still encrypted by the exponentiation . ${\displaystyle {\big (}(m^{e_{A}})^{e_{B}}{\big )}^{d_{A}}{\bmod {p}}}$ ${\displaystyle m^{e_{B}}{\bmod {p}}}$ ${\displaystyle e_{A}}$ ${\displaystyle m}$ ${\displaystyle e_{B}}$

B can now by exponentiation with the message win: . ${\displaystyle d_{B}}$ ${\displaystyle m}$ ${\displaystyle (m^{e_{B}})^{d_{B}}=m{\bmod {p}}}$

It is not possible to infer from all exchanged messages without knowing the key of the participants . ${\displaystyle m}$

Safety considerations

The Massey-Omura scheme is secure against passive eavesdropping on messages; H. Third parties cannot infer the original text from the messages exchanged. Furthermore, due to the assumed severity of the calculation of discrete logarithms , it is almost impossible, even with existing knowledge of the original text , to open up the key selected by a subscriber T and with the aid of a recorded message . ${\displaystyle m}$ ${\displaystyle e_{T}}$ ${\displaystyle d_{T}}$ ${\displaystyle m^{e_{T}}}$

However, the method is susceptible to a man-in-the-middle attack (Janus attack) by proceeding similarly to a man-in-the-middle attack on the Diffie-Hellman method .