Reaching agreement on a secret value

A first device and a second device are disclosed for reaching agreement on a secret value. Herein, the second device comprises a receiver configured to receive information indicative of a reconciliation data h from the first device, a processor configured to compute a common secret s based on an integer value b, an equation, and system parameters. The processor is configured to compute b based on a key exchange protocol. The first device has a number a in approximate agreement with the number b. The first device comprises a processor configured to determine a common secret s based on an integer value a an equation, and system parameters, and determine a reconciliation data h. The first device further comprises a transmitter configured to transmit information indicative of the reconciliation data h to the second device.

CROSS-REFERENCE TO PRIOR APPLICATIONS

This application is the U.S. National Phase application under 35 U.S.C. § 371 of International Application No. PCT/EP2017/077843, filed on Oct. 31, 2017, which claims the benefit of European Patent Application No. 16197277.3, filed on Nov. 4, 2016. These applications are hereby incorporated by reference herein.

FIELD OF THE INVENTION

The invention relates to reaching agreement on a secret value. The invention in particular relates to two devices having already approximate agreement on a secret value, to reach exact agreement on the secret value.

BACKGROUND OF THE INVENTION

Many current applications make use of a key exchange protocol in which two parties A and B wish to generate a shared value. Such protocols may be related to the well-known Diffie-Hellman key-exchange protocol. In order to withstand cryptanalysis, the parties introduce some small errors in the computations in the protocol. As a result, parties A and B can obtain values, say vA, vBthat agree nearly, but not necessarily exactly. In order to arrive at exact agreement, one of the parties, say A, sends the other party, B, a bit value, say h, that is indicative of the secret value vAthat is has computed. Party A also computes a value sAfrom the value vA. Party B then computes a value sBfrom h and its own value vB. The design of the system may be such that the secret values sAand sBare equal if the values vAand vBwere sufficiently close to each other. An example of such a system is disclosed in J. Ding, X. Xie and X. Lin, “A simple provably secure key exchange scheme based on the learning with errors problem”, Cryptology ePrint Archive, Report 2012/688, 2012, http://eprint.iacr.org/2012/688.pdf (referred to hereinafter as “Ding”).

C. Peikert, “Lattice Cryptography for the Internet”, Proceedings of the 6th Workshop on Post-Quantum Cryptography, PQ Crypto 2014, Springer LNCS, Vol. 8772, 2014, pp. 197-219 (hereinafter referred to as “Peikert”), discloses a method in which the generated secret shared values saand sbare statistically unbiased, that is, are uniformly distributed. In Peikert's set-up, the secret value obtained by the two parties is one single bit.

Joppe Bos, Craig Costello, Léo Ducas, Ilya Mironov, Michael Naehrig, Valeria Nikolaenko, Ananth Raghunathan and Douglas Stebila, “Frodo: Take off the ring! Practical, Quantum-Secure Key Exchange from LWE”, IACR Cryptology ePrint Archive, Report 2016/659, https://eprint.iacr.org/2016/659 (hereinafter referred to as “Bos” or “Frodo”), discloses an extension of Peikert's method so that the parties agree on a secret value that is uniformly distributed over a set of integers. In cited prior art methods, one single reconciliation bit is sent. In both Peikert's method and Bos' method, if the parties need to agree on many bits, the method is applied in parallel on multiple instances of reaching agreement. In all of the above references, exact key agreement can be achieved if the initially obtained values vA, vBcomputed by the two parties do not differ too much.

SUMMARY OF THE INVENTION

It would be advantageous to have an improved way of reaching agreement between two devices on a secret value. To better address this concern, a first aspect of the invention provides a second device, for reaching agreement on a secret value with a first device, comprising:

a receiver configured to receive information indicative of a reconciliation data h from the first device, wherein 0≤h<2δ, wherein δ is an integer greater than 1; and

a processor configured to compute a common secret s based on an integer value b and an equation

wherein b satisfies 0≤b<q, B is a positive integer, and q is an integer multiple of 2B+δ+1, wherein q, B, δ, and c are system parameters.

Since the helper data is in the range of 0≤h<2δ, wherein δ is an integer greater than 1, the helper data h that the first device sends to the second device consists of multiple bits. In this system, exact key agreement can be achieved even while imposing a less strict condition on the approximate agreement between the values a and b. The device set forth allows to determine the common secret s using the helper data h that the second device receives from the first device, so that exact agreement is achieved.

Specifically, exact agreement is achieved when the first device uses a number a in approximate agreement with the number b, in the sense that a≡b+e (mod q), wherein e represents a difference between the numbers a and b, wherein the constraint

e≤q2B+1-q2B+δ+1
allows for a relatively large difference between a and b. This property allows the use of a more secure key exchange algorithm.

Alternatively, for a given approximate agreement condition, the system can be used to reach exact agreement on a secret value s that has at least one more bit than is the case in for example the prior art disclosed in Peikert or Bos.

In a particular example, the processor is configured to compute b based on a key exchange protocol. This key exchange protocol may be, for example, one of the key exchange protocols disclosed in Ding, Peikert, and Bos, or a variant thereof, which leads to an approximate agreement on a key. The device set forth allows to subsequently reach exact agreement in an efficient way, as outlined above.

In a particular example, q=2mand δ=m−B−1, wherein m is a positive integer. This configuration allows agreement to be reached on multiple bits while using relatively few reconciliation bits.

In a particular example, the processor is configured to compute the value b based on a value β and an equation b≡wβ (mod q), wherein wN≡1 (mod q), wherein N is an integer greater than 1 and is relatively prime to q. This allows to support a situation in which approximate agreement between the value α of the first device and the value β of the second device is present according to a condition α=β+Ne(mod q), wherein

According to an other aspect of the invention, a first device for reaching agreement on a secret value with a second device is disclosed, wherein the first comprises:

a processor configured to:

determine a common secret s based on an integer value a and an equation

wherein a satisfies 0≤a<q, B is a positive integer, q is an integer multiple of 2B+δ+1, wherein δ is an integer greater than 1, wherein q, B, δ, and c are system parameters, and

determine a reconciliation data h based on an equation

h=⌊((a+c)⁢⁢mod⁢⁢q)⁢⁢mod⁡(q2B)q2B+δ⌋;
and

a transmitter configured to transmit information indicative of the reconciliation data h to the second device.

Since the helper data h is in the range of 0≤h<2δ, wherein δ is an integer greater than 1, the helper data h that the first device sends to the second device consists of multiple bits. In this system, exact key agreement can be achieved even while imposing a less strict condition on the approximate agreement between the values a and b of the first device and the second device. The device set forth allows to generate and transmit the helper data h that the second device needs to determine the common secret s, so that exact agreement is achieved.

Specifically, exact agreement is achieved when the first device uses a number a in approximate agreement with the number b, in the sense that a≡b+e (mod q), wherein e represents a difference between the numbers a and b, wherein the constraint

e≤q2B+1-q2B+δ+1
allows for a relatively large difference between a and b. This property allows the use of a more secure key exchange algorithm.

Alternatively, for a given approximate agreement condition, the system can be used to reach exact agreement on a secret value s that has at least one more bit than is the case in for example the prior art disclosed in Peikert or Bos.

In a particular example, the processor is configured to compute a based on a key exchange protocol. This key exchange protocol may be, for example, one of the key exchange protocols disclosed in Ding, Peikert, and Bos, or a variant thereof, which leads to an approximate agreement on a key.

In a particular example, q=2m, wherein m is a positive integer, the common secret s corresponds to B most significant bits of a binary expansion of (a+c) mod 2m, and the reconciliation data h corresponds to next δ bits of the binary expansion. This is a particularly appealing representation of the data components that together form a. In an even more specific example, δ=m−B−1. This value allows to reconcile multiple bits at once, while using relatively few bits for the helper data h. For example, this value of δ allows to reconcile one more bit than with the method disclosed in Bos, under same approximate agreement conditions.

In a particular example, c=0. In that case, the common secret s equals a quotient of a and

(q2B),
rounded downwards to the closest integer.

In a particular example,

c=q2B+1.
In that case, the common secret s equals a quotient of a and

(q2B),
rounded to the closest integer, wherein rounding is performed upwards in case of a tie.

In a particular example,

c=q2B+1-1.
In that case, the common secret s equals a quotient of a and

(q2B),
rounded to the closest integer, wherein rounding is performed downwards in case of a tie.

In a particular example, the processor is configured to compute the value a based on a value α and an equation a≡wα (mod q), wherein wN≡1 (mod q), wherein N is an integer greater than 1, wherein N is relatively prime to q. This allows to support a situation in which approximate agreement between the value α of the first device and the value β of the second device is present according to a condition α≡β+Ne(mod q), wherein

According to another aspect of the invention, a system is presented that comprises the first device and the second device set forth hereinabove, wherein the number a is in approximate agreement with the number b, in the sense that a≡b+e (mod q), wherein e represents a difference between the numbers a and b, wherein

e≤q2B+1-q2B+δ+1.
This enables the two devices, who have approximate agreement about values a and b, to reach exact agreement on a common secret s, by transmitting the reconciliation data h from the first device to the second device.

According to another aspect of the invention, a method is to be performed by a second device for reaching agreement on a secret value with a first device, wherein the method comprising:

receiving information indicative of a reconciliation data h from the first device, wherein 0≤h<2δ, wherein δ is an integer greater than 1; and

computing a common secret s based on an integer value b and an equation

wherein b satisfies 0≤b<q, B is a positive integer, and q is an integer multiple of 2B+δ+1, wherein q, B, δ, and c are system parameters.

According to another aspect of the invention, a method is to be performed by a first device for reaching agreement on a secret value with a second device, wherein the method comprises:

determining a common secret s based on an integer value a and an equation

wherein a satisfies 0≤a<q, B is a positive integer, q is an integer multiple of 2B+δ+1, wherein δ is an integer greater than 1, wherein q, B, δ, and c are system parameters;

determining a reconciliation data h based on an equation

h=⌊((a+c)⁢mod⁢q)⁢mod⁡(q2B)q2B+δ⌋;
and

transmitting information indicative of the reconciliation data h to the second device.

It will be appreciated by those skilled in the art that two or more of the above-mentioned embodiments, implementations, and/or aspects of the invention may be combined in any way deemed useful. Modifications and variations of the methods, which correspond to the described modifications and variations of the devices, can be carried out by a person skilled in the art on the basis of the present description.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The following notation will be used in this disclosure: For any two integers x and v, with v≥2, thenSvdenotes the integer satisfying
0≤xv≤v−1 andxv≡xmodv.
Moreover, for any real number y, the notation └y┘ denotes the result of rounding y downwards to the closest integer, and the notation ┌y┐ denotes the result of rounding y upwards to the closest integer. For example:

In certain embodiments, two parties, A and B, are using a particular protocol in which party A computes a number a and party B computes a number b. The protocol should be such that, because of the way in which a and b have been computed, they approximately agree. This approximate agreement is expressed in terms of system constants q, B and δ, which are known to A and B, where q, δ and B are positive integers, and q is an integer multiple of 2B+δ+1, as follows: a and b both are integers in the interval [0, q) and satisfy
a≡b+e(modq)  (equation 1)
wherein

e≤q2B+1-q2B+δ+1.(equation⁢⁢2)
Using the present disclosure, the two parties can arrive at a common B-bits secret S by having one party, say party A, transmit δ bits of reconciliation data to party B. One more integer system parameter c; its relevance will be disclosed hereinafter. Integers h and v are defined by the following equation:

In particular,

s=⌊〈a+c〉q(q/2B)⌋.(equation⁢⁢4)
In the special case that q=2m, the secret value S corresponds to the B most significant bits of the binary expansion ofa+c2m, h corresponds to the next significant δ bits of the binary expansion ofa+c2m, and v corresponds to the m−B−δ least significant bits ofa+c2m.

By considering equation (1) modulo

q2B,
it follows that

b+c-h⁢q2B+δ≡v-e⁡(mod⁢q2B).(equation⁢⁢5)
As

0≤v≤q2B+δ-1
and as equation (2) is satisfied, it follows that

0≤v-e-q2B+δ+1+q2B+1≤q2B-1.(equation⁢⁢6)
By combining equation (5) and equation (6), it follows that

⁢(equation⁢⁢7)v-e-q2B+δ+1+q2B+1=〈b+c-h⁢q2B+δ-q2B+δ+1+q2B+1〉q/2B.
By combining equation (1) and equation (3), it follows that

s⁢q2B≡b+c-h⁢q2B+δ-(v-e)⁢(mod⁢q),(equation⁢⁢8)
and from equation (8) it follows that

By combining equation (9) and equation (7), and using the property S∈[0,2B), it follows that party B can compute s using the equation

s=〈⌊b+c-h⁢q2B+δ-q2B+δ+1+q2B+1q⁢/⁢2B⌋〉2B.(equation⁢⁢10)
By simplifying equation (10), it follows that party B can alternatively compute S using the equation

s=〈⌊b+cq⁢/⁢2B-h2δ-12δ+1+12⌋〉2B.(equation⁢⁢11)
Equations (10) and (11) show that S can be computed from b, h and the system parameters q, B and δ. So if party A sends information indicative of h to party B, then party B can retrieve S, which can be used as a common secret between party A and party B.
Since equation (3) implies

0≤h⁢q2B+δ<q2B,
it follows that 0≤h<2δ, so h can be represented by δ bits.

It is observed that if c=0, Equation (4) states that the secret S equals the quotient of a and (q/2B), rounded downwards to the closest integer. With the choice c=q/2B+1, the secret S equals the quotient of a and q/2B, rounded to the closest integer (modulo 2B) (and rounded upwards in case of a tie, that is, if a equals

k⁢q2B+q2B+1
for some integer k). With the choice c=q/(2B+1)−1, the secret S equals the quotient of a and q/2B, rounded to the closest integer modulo 2B, with rounding downwards in case of a tie. Other values of c may be used to obtain another result, as desired. For these special choices for c, the computation of S by party B can be simplified. Indeed, party B can obtain S using the equation

s=〈1+⌊bq⁢/⁢2B-h2δ-12δ+1⌋〉2B⁢⁢if⁢⁢c=q2B+1,(equation⁢⁢12)
and as

In case that q=2m, the common secret S may consist of the B most significant bits of a; the helper data h consists of the subsequent δ bits of a. Moreover, if a is uniformly distributed, then the common secret S given the helper data h is uniformly distributed as well. That is, an adversary cannot obtain information on the common secret S from the observation of the helper data h.

It is noted that the “approximate agreement” condition can be generalized. For example, it is possible to replace equation (1) by the condition:
a≡b+Ne(modq)  (equation 14)
for some integer N that is relatively prime to q, that is, the greatest common divisor of N and q is one. The condition on the absolute value of e as specified in equation (2) may be kept the same:

e≤q2B+1-q2B+δ+1.(equation⁢⁢2)
For example, if q=2mfor some integer m, then N can be any odd number. In such a case, the computation of the secret and the helper data can be performed using the following derivation. Let W be an integer such that wN≡1(mod q). Such an integer exists, because q and N are relatively prime. Let α:=waqand β:=wbq. Then α≡β+e(mod q). Thus, the parties can agree on a secret

s=⌊α+c(q⁢/⁢2B)⌋
as explained before, using α and β instead of a and b, respectively.

For δ=1 and q=2m, and obtaining the secret s as the integer closest to the quotient of a and 2m−B, one reconciliation bit h is sent, and the parties can agree on a B-bits secret s whenever, for example, |e|≤2m−B−2. If and δ=B−1, the parties can agree on a B-bits secret s whenever |e|≤2m−B−1−1. By increasing the number of reconciliation bits, the parties thus can agree on a secret value that is one bit longer.

Using the techniques described herein, it is possible to reach agreement about the secret without exchange of information about the m−B−δ≥1 least significant bits of a. By varying δ, it is possible to achieve a trade-off between bandwidth requirements for sending reconciliation data and the approximation requirements for successful exact agreement.

FIG. 1shows a bock diagram of an example of a second device150for reaching agreement on a secret value with a first device250. The second device150comprises an antenna100, a receiver101, a processor102, and a memory105. The antenna100is connected to the receiver101for receiving a signal. In operation, the memory105comprises a data block103and a computing block104. The antenna100may be used for sending and/or receiving signals wirelessly using an appropriate communications standard. In an alternative implementation, antenna100may be replaced by a wired (network) connection. Although in the present disclosure, only a receiver102is needed, practical implementations may also comprise a transmitter for transmitting signals, for example using the antenna100. The processor102controls operation of the device, including the receiver101and the memory. The data block103of the memory105may be used to store various data, including but not limited to system parameters (e.g. q, B, δ, and c), a secret s, received reconciliation data h, a value b, and other data such as contents to be encrypted or decrypted. The computing block104may comprise executable computer code that implements at least a method to reach agreement on a secret value with another device (for example first device250).

The receiver101is configured to receive information indicative of a reconciliation data h from the first device, wherein 0≤h<2δ, wherein δ is an integer greater than 1. The processor102is configured to compute a common secret s based on the integer value b and an equation

For example, s can be chosen to be the value for which the above equation holds and wherein 0≤s<2B.

The processor102may be configured to compute b before computing the common secret s. Such a computation may be based on a key exchange protocol. To that end, the second device150may exchange further information with the first device250or another device, such as a third party intermediary (not shown), via its receiver102or optional transmitter, in accordance with the key exchange protocol. The details of this key exchange protocol are beyond the scope of the present disclosure. It is a property of the second device that it can determine the common secret s using the reconciliation data h, if the first device250has computed the reconciliation data h as disclosed hereinafter with reference toFIG. 2, as long as the first device250uses a number a in approximate agreement with the number b, in the sense that a≡b+e (mod q), wherein e represents a difference between the numbers a and b, wherein

For specific values of c, the computation of the common secret s can be simplified (see also equations 12 and 13). The processor102of the second device can be configured to compute s by evaluating a formula

s=〈1+⌊b-h2B+δ-q2B+δ+1q2B⌋〉2B⁢⁢if⁢⁢c=q2B+1
Also, processor102of the second device can be configured to compute s by evaluating a formula

s=〈1+⌊b-h2B+δ-q2B+δ+1-1q2B⌋〉2B⁢⁢if⁢⁢c=q2B+1-1
which can be implemented alternatively as

In a particular example, q=2mand δ=m−B−1, wherein m is a positive integer. Herein, >B+3.

In another example, the processor102is configured to compute the value b based on a value β and an equation b≡wβ (mod q), wherein wN≡1 (mod q), wherein N is an integer greater than 1 and is relatively prime to q. This allows to support a larger difference between a and b, such as explained hereinabove with respect to equation 14.

FIG. 2shows a bock diagram illustrating an example of a first device250for reaching agreement on a secret value with a second device150. The first device250comprises an antenna200, a transmitter201, a processor202, and a memory205. The antenna200is connected to the transmitter201for receiving a signal. In operation, the memory205comprises a data block203and a computing block204. The antenna200may be used for sending and/or receiving signals wirelessly using an appropriate communications standard. In an alternative implementation, antenna200may be replaced by a wired (network) connection. Although for description of the present disclosure, only a transmitter202is needed, practical implementations may also comprise a receiver for receiving signals, for example using the antenna200. The processor202controls operation of the device, including the transmitter201and the memory205. The data block203of the memory205may be used to store various data, including but not limited to system parameters (e.g. q, B, δ, and c), a secret s, reconciliation data h, a value b, and other data such as contents to be encrypted or decrypted. The computing block204may comprise executable computer code that implements at least a method to reach agreement on a secret value with another device (for example second device150).

In a practical implementation, the processor202may be configured to determine a common secret s based on an integer value a and an equation

s=⌊(α+c)⁢⁢mod⁢⁢qq2B⌋
This may be alternatively written as:

Before or after determining the common secret s (or simultaneously), the processor202may determine a reconciliation data h based on an equation

h=⌊((a+c)⁢⁢mod⁢⁢q)⁢⁢mod⁢⁢(q2B)q2B+δ⌋
This may be alternatively written as:

The transmitter201may be configured to transmit, under control of the processor202, information indicative of the reconciliation data h to the second device. For example, the information indicative of the reconciliation data h can be a binary representation of the reconciliation data h or an encoded representation of the reconciliation data h.

In a particular example, the processor202is configured to compute a based on a key exchange protocol. To that end, the first device250may exchange further information with the second device150, or another device, such as a third party intermediary (not shown), in accordance with the key exchange protocol, using transmitter201or an optional receiver. The details of this key exchange protocol are beyond the scope of the present disclosure. It is a property of the first device250that it can provide the second device150with the additional reconciliation data h. The second device150can determine the common secret s using the reconciliation data h, by combining the reconciliation data h with the number b in a way as described herein with reference toFIG. 1, as long as the first device250uses a number a in approximate agreement with the number b that is used by the second device150, in the sense that a≡b+e (mod q), wherein e represents a difference between the numbers a and b, wherein

In a particular implementation example, q=2m, wherein m is a positive integer, the common secret s corresponds to B most significant bits of a binary expansion of (a+c) mod 2m, and the reconciliation data h corresponds to next most significant δ bits of the binary expansion of (a+c) mod 2m. For example, δ=m−B−1 may provide a relatively large number of bits that can be reconciled while allowing a relatively relaxed constraint regarding how approximate the agreement between a and b should be, and transmitting relatively few bits of reconciliation data δ. However, this value is only presented as an example.

The secret s can be derived from the value a in several different ways. For example, different behavior can be realized by varying the system parameter c. The same value of the system parameters (including c) should be used in both the first device and the second device for optimal performance. For example, c=0 can be chosen so that the common secret s equals a quotient of a and

(q2B),
rounded downwards to the closest integer. Alternatively,

c=q2B+1
may be chosen, so that the common secret s equals a quotient of a and

(q2B),
rounded to the closest integer, wherein rounding is performed upwards in case of a tie. Yet alternatively,

c=q2B+1-1
is chosen, so that the common secret s equals a quotient of a and

(q2B),
rounded to the closest integer, wherein rounding is performed downwards in case of a tie.

In a particular implementation example, the processor is configured to compute the value a based on a value α and an equation a≡wα (mod q), wherein wN≡1 (mod q), wherein N is an integer greater than 1, wherein N is relatively prime to q. This allows to support a larger difference between a and b, such as explained hereinabove with respect to equation 14.

The processors102and202can be any type of computer processor, capable of executing a program stored in memory and controlling peripherals such as a transmitter, receiver, memory, and the like. For example, the processor102or202can be a microcontroller or a microprocessor. Such a processor is an electronic device that is well known in the art. Also, the processor102,202may comprise a plurality of sub-processors that can cooperate to perform certain tasks in parallel. The memory105or205can be any type of memory that is capable of storing digital data, either in volatile or non-volatile manner. The memory105or205is computer readable, and can be used by the respective processor102,202to retrieve and/or store data. Such memory105,205is an electronic device. Well known examples include a Flash memory, a random access (RAM) memory, read-only memory (ROM) and a magnetic or optical drive. A combination of these types of memory may be used in each device.

In a particular example, one device contains all the components and functionality of both the first device and the second device. For example, the device can switch roles between the role of the first device and the second device.

The data transmission from the first device to the second device may be by means of direct communication. Alternatively, the transmission may be performed via a network, and the reconciliation data may pass several nodes on the network before reaching the second device. For example, the data transmission can use Wi-Fi, Bluetooth, 3G, 4G, LTE data network technology.

FIG. 3illustrates a method to be performed by a second device for reaching agreement on a secret value with a first device.FIG. 4illustrates a method to be performed by the first device for reaching agreement on a secret value with the second device.FIG. 5illustrates how the first device501and the second device502can cooperate to reach agreement. The steps illustrated inFIG. 5that correspond to the steps illustrated inFIG. 3andFIG. 4have been indicated using the same reference numerals.

Referring toFIG. 3andFIG. 5, the second device starts the method at step301. The start may be triggered by an appropriate internal or external signal, or an input provided by a user, for example. For example, the method starts when a first device tries to set up communication with the second device. In step302, the system parameters parameters q, B, δ, and c are determined. For example, these system parameters are retrieved from the memory103. Optionally, as indicated by arrow503, this step can involve negotiating between the first and second device about the system parameters to be used; for example, messages can be exchanged about a set of parameters that is supported by both devices. B is a positive integer, δ is an integer greater than 1, and q is an integer multiple of 2B+δ+1. In step303, the number b is determined. For example, this number is computed from data that is made available to the second device. Alternatively, the number b is received from an external source, for example a trusted party, preferably in an encrypted form. The number b could be obtained as part of a lattice-based key exchange protocol. As indicated by arrow504, the value b is in approximate agreement with a corresponding value a of the first device501. In step304, the second device receives the information indicative of the reconciliation data h, as indicated by arrow505. The information may be transmitted to the second device in an encrypted form, and be decrypted by the second device, for example. The reconciliation data is in the range of 0≤h<2δ. In step305, the second device computes s based on an equation

s=〈⌊b+c-h⁢q2B+δ-q2B+δ+1+q2B+1q2B⌋〉2B
Other representations of s are also possible.

In step306, optionally a key is determined based on the common secret s.

Then, the method is ended in step307. Optionally, the second device can now start using the common secret s and/or the key based on the common secret s. The possible uses can by any one or more of many, including cryptographic processing of data, such as content, e.g. encryption, decryption, digital signature creation and verification. For example, the common secret s can be used for secure exchange of messages between the first device and the second device, as indicated by arrow506. Also, the common secret s and/or the key derived therefrom can be stored in the memory of the second device for later use.

Referring toFIG. 4andFIG. 5, the first device starts the method at step401. The start may be triggered by an appropriate internal or external signal, or an input provided by a user, for example. For example, the method starts when a second device tries to set up communication with the first device. In step402, the system parameters q, B, δ, and c are determined. For example, these system parameters are retrieved from memory. Optionally, as indicated by arrow503, this step can involve negotiating between the first and second device about the system parameters to be used; for example, messages can be exchanged to determine a set of parameters that is supported by both devices. B is a positive integer, δ is an integer greater than 1, and q is an integer multiple of 2B+δ+1. In step403, the first device determines a number a. This determination may be based on a key exchange protocol, for example. For example, this number a is computed from data that is made available to the first device. Alternatively, the number a is received from an external source, for example a trusted party, preferably in an encrypted form. The number a could be obtained as part of a lattice-based key exchange protocol. As indicated by arrow504, the value a is in approximate agreement with a corresponding value b of the second device502. In step404, the first device determines a reconciliation data h. This reconciliation data may be based on the equation

The reconciliation data can be in the range of 0≤h<2δ. In step405, the first device transmits information indicative of the reconciliation data h, as indicated by arrow505, to the second device. The information may be encrypted by the first device to transmit the information to the second device in an encrypted form, for example. In step406, the first device determines the common secret s. This step may be performed before the other steps. In an alternative implementation, the common secret s may be determined before determining the number a, wherein the first device may derive the number a from the common secret s. The common secret s may be computed based on an equation

s=⌊(a+c)⁢⁢mod⁢⁢qq2B⌋
In other notation,

s=⌊〈a+c〉qq2B⌋
Other representations of s are also possible. In step407, optionally a key is determined based on the common secret s. Alternatively, the common secret s may be based on a key determined beforehand. Then, the method is ended in step408. Optionally, the first device can use the common secret s and/or the key. The possible uses can by any one or more of many, including cryptographic processing of data, such as content, e.g. encryption, decryption, digital signature creation and verification. For example, the common secret s or the key can be used for secure exchange of messages between the first device and the second device, as indicated by arrow506. Also, the common secret s and/or the key can be stored in the memory of the second device for later use.

FIG. 6shows an example illustrating conceptually a possible relationship between a, s, h, m, B, and q for the specific case of q=2mand c=0. In the drawing, the binary representation of a is illustrated, from the most significant bit (on the left-hand side) to the most significant bit (on the right-hand side). At numeral601, it is illustrated that the common secret s is represented by the B most significant bits of a. At numeral602, it is illustrated that the reconciliation data h is represented by the (B+1)th to (B+δ)th most significant bits of a. At numeral603, it is illustrated that the remaining (B+δ+1)th to q-th bits, i.e. the m−B−δ least significant bits of a, are not represented in either the common secret s nor the reconciliation data h. This feature may allow data savings regarding the number of bits of the reconciliation data h and/or increased tolerance regarding the approximate agreement between a and b.

In this disclosure, a reconciliation method is presented that can send more than one reconciliation bit. The techniques disclosed herein may be used, for example, to cause parties to agree on a particular number of bits, while imposing less stringent conditions on “how approximate” the approximate agreement should be. Allowing for less stringent conditions on the approximate agreement can improve the security of the system. Alternatively, with about the same approximation conditions (i.e., with similar security guarantees), an instance of the method allows the two parties to agree on a secret value that is one bit longer. Hereinafter, some of the advantages of the method and its impact are disclosed by means of numerical examples.

Bos discloses a quantum-secure key exchange method. One party sends to another party a small seed and an n×nmatrix with elements from Zq. In response, anm×n matrix and a binaryn×mmatrix with reconciliation bits are sent. The parties both construct ann×mmatrix; from each entry of said matrix, B common bits are extracted. The total number of extracted bits (labeled “length” in the tables below) thus equalsn·m·B, while the total number of transmitted bits equals
n(n+m)┌log2(q)┐+m·n.
Table 1 is a condensed version of the proposed instantiations in Table 2 of Bos.

TABLE 2Improvement that can be achieved by the reconciliation schemedisclosed hereinSchemenqBnmLengthBandwidthRatioChallenge352211266725.84 KB0.76Classical59221237714712.48 KB0.88Recommended75221557828021.22 KB0.94Paranoid86421557828024.37 KB0.94

It will be appreciated that the invention also applies to computer programs, particularly computer programs on or in a carrier, adapted to put the invention into practice. The program may be in the form of a source code, an object code, a code intermediate source and object code such as in a partially compiled form, or in any other form suitable for use in the implementation of the method according to the invention. It will also be appreciated that such a program may have many different architectural designs. For example, a program code implementing the functionality of the method or system according to the invention may be sub-divided into one or more sub-routines. Many different ways of distributing the functionality among these sub-routines will be apparent to the skilled person. The sub-routines may be stored together in one executable file to form a self-contained program. Such an executable file may comprise computer-executable instructions, for example, processor instructions and/or interpreter instructions (e.g. Java interpreter instructions). Alternatively, one or more or all of the sub-routines may be stored in at least one external library file and linked with a main program either statically or dynamically, e.g. at run-time. The main program contains at least one call to at least one of the sub-routines. The sub-routines may also comprise calls to each other. An embodiment relating to a computer program product comprises computer-executable instructions corresponding to each processing step of at least one of the methods set forth herein. These instructions may be sub-divided into sub-routines and/or stored in one or more files that may be linked statically or dynamically. Another embodiment relating to a computer program product comprises computer-executable instructions corresponding to each means of at least one of the systems and/or products set forth herein. These instructions may be sub-divided into sub-routines and/or stored in one or more files that may be linked statically or dynamically.