System with and method of cryptographically protecting communications

A value transfer system comprising at least one Value Carrying Device (1) and at least one Value Accepting Device (2) being able to communicate with each other, the at least one Value Accepting Device (2) comprising a Value Accepting Device memory (53) for storing at least an aggregate value (28) of previous accepted values and being arranged to transfer a claiming message (13) representing at least a transaction value (20) to said at least one Value Carrying Device (1), the at least one Value Carrying Device (1) comprising a Value Carrying Device memory (52) for storing at least a balance value (7) and being arranged to transfer a proving message (14) to said at least one Value Accepting Device (2), characterised in that

• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) a cryptographically encoded aggregate value (38);
• the Value Accepting Device memory (53) also stores said cryptographically encoded aggregate value (38) which is derived by encoding said aggregate value (28) by using a “Peano” number scheme as follows:
• choosing a discrete maximum value for the encoding,
• selecting a cryptographic one-way function that maps starting numbers consisting of a predetermined number of bits to object numbers consisting of the same predetermined number of bits, a functional application to a number being defined as “successor operation” in the Peano number scheme;
• selecting a random number consisting of said predetermined number of bits as zero element in the Peano number scheme;
• determining a value encoded in a number by subtracting from said discrete maximum value the value of a Peano number determined by repeated functional applications of the one-way function starting with the zero element until a result of the functional application of the one-way function equals a code number to be decoded, wherein a code word is found not to be a valid encoding if none of the results of applying repetitively for a number of times equal to the chosen discrete maximum value the cryptographic one-way function starting with the selected zero element equals the code word;
• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) the following data:
• said discrete maximum value (44),
• a zero element cryptogram concealing said zero element in the Peano number scheme, and
• a correctness cryptogram to proof correctness of said discrete maximum value and said zero element;
• the at least one Value Carrying Device (1) is arranged to compute and include into the proving message (14) an encoded new aggregate value (43) computed on the basis of the cryptographically encoded aggregate value (38) and the transaction value (20);
• the at least one Value Carrying Device (1) is arranged to complete a value transfer only
• if it has proved the correctness of said discrete maximum value and said zero element by using the correctness cryptogram,
• if it determines that, based on said discrete maximum value and said zero element, the cryptographically encoded aggregate value (38) is validly encoded, and,
• after it has reduced the balance value (7) with the transaction value (20).

A value transfer system comprising at least one Value Carrying Device (1) and at least one Value Accepting Device (2) being able to communicate with each other, the at least one Value Accepting Device (2) comprising a Value Accepting Device memory (53) for storing at least an aggregate value (28) of previous accepted values and being arranged to transfer a claiming message (13) representing at least a transaction value (20) to said at least one Value Carrying Device (1), the at least one Value Carrying Device (1) comprising a Value Carrying Device memory (52) for storing at least a balance value (7) and being arranged to transfer a proving message (14) to said at least one Value Accepting Device (2), characterised in that

• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) a cryptographically encoded aggregate value (38);
• the Value Accepting Device memory (53) also stores said cryptographically encoded aggregate value (38) which is derived by encoding said aggregate value (28) by using a “Peano” number scheme as follows:
• choosing a discrete maximum value for the encoding;
• selecting a cryptographic one-way function that maps starting numbers consisting of a predetermined number of bits to object numbers consisting of the same predetermined number of bits, a functional application to a number being defined as “successor operation” in the Peano number scheme;
• selecting an encoding seed and appropriate cryptographic derivation algorithm to derive a randomised number consisting of said predetermined number of bits for use as zero element in the Peano number scheme;
• determining a value encoded in a number by subtracting from said discrete maximum value the value of a Peano number determined by repeated functional applications of the one-way function starting with the zero element until a result of the functional application of the one-way function equals a code number to be decoded, wherein a code word is found not to be a valid encoding if none of the results of applying repetitively for a number of times equal to the chosen discrete maximum value the cryptographic one-way function starting with the selected zero element equals the code word;
• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) the following data:
• said discrete maximum value,
• an encoding seed (39), and
• possible additional encoding data (41)
• the at least one Value Carrying Device (1) is arranged to compute and include into the proving message (14) an encoded new aggregate value (43) computed on the basis of the cryptographically encoded aggregate value (38) and the transaction value (20);
• the at least one Value Carrying Device (1) is arranged to perform the selected derivation algorithm in such fashion that the derived zero element is concealed and to complete a value transfer only
• if in using the derivation algorithm it computes the zero element from at least the encoding seed and determines said previous aggregate value as being validly encoded based on said maximum value and said derived zero element, and
• after it has reduced the balance value (7) with the transaction value (20).

A Value Accepting Device (2) arranged to communicate with at least one Value Carrying Device (1) and comprising a Value Accepting Device memory (53) for storing at least an aggregate value (28) of previous accepted values and being arranged to transfer a claiming message (13) representing at least a transaction value (20) to said at least one Value Carrying Device (1), the Value Accepting Device (2) being arranged to receive a proving message (14) from said at least one Value Carrying Device (1), characterised in that

• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) a cryptographically encoded aggregate value (38);
• the Value Accepting Device memory (53) also stores said cryptographically encoded aggregate value (38) which is derived by encoding said aggregate value (28) by using a “Peano” number scheme as follows:
• choosing a discrete maximum value for the encoding;
• selecting a cryptographic one-way function that maps starting numbers consisting of a predetermined number of bits to object numbers consisting of the same predetermined number of bits, a functional application to a number being defined as “successor operation” in the Peano number scheme;
• selecting a random number consisting of said predetermined number of bits as zero element in the Peano number scheme;
• determining a value encoded in a number by subtracting from said discrete maximum value the value of a Peano number determined by repeated functional applications of the one-way function starting with the zero element until a result of the functional application of the one-way function equals a code number to be decoded, wherein a code word is found not to be a valid encoding if none of the results of applying repetitively for a number of times equal to the chosen discrete maximum value the cryptographic one-way function starting with the selected zero element equals the code word;
• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) the following data:
• said discrete maximum value (44),
• a zero element cryptogram concealing said zero element in the Peano number scheme, and
• a correctness cryptogram to proof correctness of said discrete maximum value and said zero element.

A Value Accepting Device (2) arranged to communicate with at least one Value Carrying Device (1) and comprising a Value Accepting Device memory (53) for storing at least an aggregate value (28) of previous accepted values and being arranged to provide a claiming message (13) representing at least a transaction value (20) to said at least one Value Carrying Device (1), the Value Accepting Device (2) being arranged to receive a proving message (14) from said at least one Value Carrying Device (1), characterised in that

• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) a cryptographically encoded aggregate value (38);
• the Value Accepting Device memory (53) also stores said cryptographically encoded aggregate value (38) which is derived by encoding said aggregate value (28) by using a “Peano” number scheme as follows:
• choosing a discrete maximum value for the encoding;
• selecting a cryptographic one-way function that maps starting numbers consisting of a predetermined number of bits to object numbers consisting of the same predetermined number of bits, a functional application to a number being defined as “successor operation” in the Peano number scheme;
• selecting an encoding seed and appropriate cryptographic derivation algorithm to derive a randomised number consisting of said predetermined number of bits for use as zero element in the Peano number scheme;
• determining a value encoded in a number by subtracting from said discrete maximum value the value of a Peano number determined by repeated functional applications of the one-way function starting with the zero element until a result of the functional application of the one-way function equals a code number to be decoded, wherein a code word is found not to be a valid encoding if none of the results of applying repetitively for a number of times equal to the chosen discrete maximum value the cryptographic one-way function starting with the selected zero element equals the code word;
• the at least one Value Accepting Device (2) is arranged to further include into the claiming message (13) the following data:
• said discrete maximum value,
• an encoding seed (39), and
• possible additional encoding data (41).