Patent Description:
In cryptography, digital signature schemes define a set of rules for how a receiving party to a message can verify that the message originated from the intended sender. Typically, such scheme provides a layer a validation for messages sent over a non-secure channel, such as the Internet.

Various types of digital signature schemes exist. One commonly used scheme uses the Rivest-Shamir-Adleman (RSA) algorithm. In this algorithm the private key in a public key/private key pair can be used, typically in a hash function, to "sign" a message. The receiving party can then use the public key and the same hash function to find a value. If the value matches the signature, then the message has not been tampered with. The security in such system is based on the problem of factoring large numbers.

A second commonly used scheme is the Elliptic Curve Digital Signature Algorithm (ECDSA), which is an expansion of the Digital Signature Algorithm (DSA) using Elliptic Curve cryptography. These algorithms use modular exponentiation and the discrete logarithm problem for security.

However, quantum computers are emerging as a potential computing platform. Quantum computers use "quantum bits" rather than binary digits utilized in traditional computers. Such quantum computers would theoretically be able to solve certain problems much more quickly than classical computers, including integer factorization, which is the strength behind the RSA algorithm, and discrete logarithms, which is the strength behind ECDSA.

In particular, Peter Shor formulated Shor's quantum algorithm in <NUM>. This algorithm is known to attack digital signatures based on integer factorization or discrete logarithms if a sufficiently powerful quantum computer can be built. Utilizing such algorithm, the risk of a quantum computer discovering the secret for a party in digital signature schemes is nonzero. Therefore, counter measures to Shor's algorithm are needed. In the prior art, the following documents propose different approaches:.

There is provided a method, a computing device, and a computer readable medium according to the claims.

The present disclosure provides a method for verification at a computing device of a signed message received from a first party over a public communications channel, the method comprising: extracting, by the computing device, a message digest "a" from the signed message, the message digest "a" belonging to a semigroup; obtaining, by the computing device, a public key [c,e] for the first party, elements of the public key including a checker "c" and an endpoint "e", checker "c" and endpoint "e" belonging to the semigroup and the endpoint comprising a multiplication of a private key "b" for the first party and the checker "c"; multiplying the message digest "a" and the endpoint "e" to create an endmatter "ae" = "abc"; extracting, by the computing device, a signature "d" from the signed message, the signature "d" belonging to the semigroup and being a multiplication of message digest "a" and private key "b"; multiplying the signature "d" and the checker "c" to create a signcheck "dc" = "abc"; and verifying that the endmatter "ae" matches the signcheck "dc", wherein the checker is a fixed value.

The present disclosure further provides a computing device configured for verification of a signed message received from a first party over a public communications channel, the computing device comprising: a processor; and a communications subsystem, wherein the computing device is configured to: extract a message digest "a" from the signed message, the message digest "a" belonging to a semigroup; obtain a public key [c,e] for the first party, elements of the public key including a checker "c" and an endpoint "e", checker "c" and endpoint "e" belonging to the semigroup and the endpoint comprising a multiplication of a private key "b" for the first party and the checker "c"; multiply the message digest "a" and the endpoint "e" to create an endmatter "ae" = "abc"; extract a signature "d" from the signed message, the signature "d" belonging to the semigroup and being a multiplication of message digest "a" and private key "b"; multiply the signature "d" and the checker "c" to create a signcheck "dc" = "abc"; and verify that the endmatter "ae" matches the signcheck "dc", wherein the checker is a fixed value.

The present disclosure further provides a computer readable medium for storing instruction code for verification of a signed message received from a first party over a public communications channel, the instruction code, when executed by a processor of a computing device, cause the computing device to: extract a message digest "a" from the signed message, the message digest "a" belonging to a semigroup; obtain a public key [c,e] for the first party, elements of the public key including a checker "c" and an endpoint "e", checker "c" and endpoint "e" belonging to the semigroup and the endpoint comprising a multiplication of a private key "b" for the first party and the checker "c"; multiply the message digest "a" and the endpoint "e" to create an endmatter "ae" = "abc"; extract a signature "d" from the signed message, the signature "d" belonging to the semigroup and being a multiplication of message digest "a" and private key "b"; multiply the signature "d" and the checker "c" to create a signcheck "dc" = "abc"; and verify that the endmatter "ae" matches the signcheck "dc", wherein the checker is a fixed value.

In accordance with the present disclosure, semigroups, which are a category of mathematical objects in algebra, may be used as a basis for multiplicative signature schemes. In one embodiment of the present disclosure, one example semigroup, namely a plactic monoid, may be used for a multiplicative signature scheme.

Multiplicative signatures, hashed multiplicative signatures, and Plactic monoids and their use in multiplicative signatures, are described below.

Semigroups are a category of mathematical objects in algebra. Each semigroup S has a set of elements, and a binary operation defined on the set. The binary operation must be associative. This means that: <MAT>.

In equation <NUM> above, a, b and c are in the semigroup S. Equation <NUM> indicates that when computing the product abc of three elements a, b and c, it does not matter if one multiplies a and b first, getting some value d=ab, and then multiplying d by c to get abc=dc, or if one first multiplies b and c to get a value e=bc and then multiplying a and e to get abc=ae.

Any set equipped with an associative binary operation is a semigroup.

Two examples of semigroup includes positive integers {<NUM> ,<NUM>,<NUM>,. } under addition, and positive integers {<NUM> ,<NUM>,<NUM>,. } under multiplication. These two examples share the same set, but have a different binary operation. As will be appreciated by those skilled in the art, there are many other semigroups besides the two defined above.

When discussing a general semigroup S, it is often assumed that the operation is written as multiplication. Furthermore, when a and b are variables represented with values in S, the product is written as ab, omitting any multiplication sign. However, in particular specific semigroups, such as positive integers under addition, a symbol "+" for a binary operation is used and the operation may be written as a+b instead of ab.

In some cases, semigroups S are commutative, which means that ab=ba for all a,b in S. In the example using the positive integer semigroups defined above, both examples are commutative.

Other semigroups are non-commutative. For example, matrices under multiplication would be non-commutative. In that case, in the product abc, the positions of a, b, and c matter for such product. Thus, abc may be different than bac and cab.

Associativity means that in the product abc, the order in which the two multiplications are carried out does not matter. Thus, either ab or be could be computed first, but the final result is the same.

In accordance with the present disclosure, semigroups have a multiplication operator.

Semigroups are however not required to have a division operator. In some cases, a division operator may be formed, and is written as "/". A division operator is a binary operator having left and right input. If / is a binary operator on semigroup S, / may be defined as a strong divider if: <MAT>.

Where equation <NUM> above is valid for all a,b in S.

The operator / may be defined as a partial strong divider if equation <NUM> above only holds for a subset of a,b values within S.

In semigroup nomenclature, the operation is generally written as ab/b instead of (ab)/b, which means that multiplications are done before divisions.

Further, a weak divider may also be defined for a semigroup. In particular, sometimes a semigroup has multiplication in which ab=db for many different values of d. In this case, there cannot be a strong divider. A "/" is a weak divider if: <MAT>.

In equation <NUM>, the weak divider is defined for all a, b and S.

A partial weak divider utilizes equation <NUM>, but is only valid for a subset of values a, b within S.

In equations <NUM> and <NUM> above, the divider/ is also called a right divider. Similarly, a binary operation "\" is called a left divider. The operator \ is a strong left divider if b\ba=a. Further, the binary operator \ is a weak left divider if b(b\ba) = ba.

In various semigroups, a divider operation may be known. For example, for positive integers under multiplication, it is the usual Euclidean division algorithm. For positive integers under addition, the division may become subtraction. Dividers are known for some matrix subgroups, where Bareiss elimination can be used.

Further, many of the constructions of a semigroup use a concept known as a semiring. A semiring R has two binary operations, namely addition and multiplication, each forming a semigroup on R. Addition is also commutative. Multiplication is distributed over addition, meaning that a(b+c)=ab+ac and (a+b)c=ac+bc for all a,b,c, in R. A basic example of a semiring includes positive integers under the usual addition and multiplication operations.

Typical Diffie-Hellman groups, such as elliptic curves and modular multiplication groups, are cyclic groups, which are known to be isomorphic to modular addition groups. As used herein, isomorphic means that they share the same underlying group structure, even though they have different representations. Modular integer addition groups would be insecure if used as Diffie-Hellman groups since the division problem is easy. Fortunately, finding isomorphism is difficult (except by Shor's quantum computer algorithm), even though it is known to exist.

Accordingly, it can be said that elliptic curve groups and modular multiplication groups are "structurally insecure" as Diffie-Hellman groups, because they share the same structure as insecure Diffie-Hellman groups (modular addition groups). It should be noted by those skilled in the art that just saying a scheme is structurally insecure does not mean that it is insecure. However, such scheme may be suspected to be insecure because the only thing between such scheme and attack is isomorphism, which is known to exist. Such existential threat is commonplace in cryptography, but nonetheless it may be beneficial to find a cryptographic scheme for which there is no known existential threat. Such a scheme would be deemed to be structurally secure.

One specific example of a structurally secure scheme is known as the Vernam cipher, also known as a one-time pad. It has been proven that the confidentiality of such cipher is unconditionally secure. Such a scheme is therefore structurally secure. However, the one-time pad has other security issues (as it does not provide message integrity or authentication) and has practicality issues (that the one-time pad must be equal in size to the message, among other factors).

However, a one-time pad is not a signature scheme, but it is structurally secure in the sense above.

In accordance with the present disclosure, structurally secure signature schemes are sought.

Similarly, a semigroup may be structurally insecure if it has the same structure as a known weak semigroup. Therefore, in accordance with one embodiment of the present disclosure, to avoid structurally insecure semigroups two options are provided. A first is referred to as trial elimination, and the second is referred to as restriction.

With regard to trial elimination, the structure of a particular semigroup, up to isomorphism, may be examined. In many cases, it will be obvious that a weak semigroup of a structure exists. In that case, the semigroup may be eliminated, and examination may move to another semigroup. Because the semigroups are plentiful and even semigroups with varying structures are plentiful, this process can be continued.

Next, trial elimination may examine whether the structure of the semigroup includes a weak semigroup. As used herein, "weak" means that the use of the semigroup in a cryptographic system would permit a computationally feasible attack on the cryptographic system, and thus use of such system would make the cryptographic system computationally vulnerable.

If yes, the semigroup may be eliminated.

If no weak semigroups are known, then the semigroup may be kept as a possibility for a signature scheme.

A second strategy is referred to herein as restriction. In restriction, a property of the semigroup structure may be considered.

Specifically, semigroups may have various properties such as, commutativity, where st=ts for all s and t within the semigroup. Another property may be regularity in which, for all s within the semigroup there exists a t with sts=s. Other properties are provided for in Table <NUM> below.

The list of properties in Table <NUM> above is not exhaustive. Other properties may also be considered.

Based on the properties in Table <NUM> above, a determination may be made to consider whether the property is favorable to security or not. If the property is not favorable, all semigroups with that property are avoided for the selection of the semigroup for the signature scheme. A property is not favorable to security if such property renders the keys computationally vulnerable to being discovered.

Alternatively, if the property is favorable for security then the semigroup may be kept and may again be further analyzed with other properties. Alternatively, if the properties that are being examined have all been examined, then such semigroup may be selected as a possibility for a signature scheme.

In a further embodiment, semigroups can be constructed using building blocks such as other semigroups, or other types of algebraic objects, such as semirings, and even arbitrary functions. These constructed semigroups can have the same or better security than the individual building blocks.

Thereafter, each semigroup construction can be used to build a signature scheme. The semigroup construction can use the same or diverse types of building blocks to form such semigroup.

As used herein, a semiring is a pair of semigroups sharing the sets, with one operation written additively and the other multiplicatively. Further, in a semiring, distributive laws hold. In particular, a(b+c)=ab+ac and (a+b)c=ac+bc. Unless noted otherwise, addition in a semiring is assumed to be commutative.

To avoid confusion, in the embodiments below, constructions from building blocks are distinguished by labelling the building blocks with the adjective "base". For example, if a given semiring is taken as a building block, such as a semiring R of positive integers, then we construct a semiring S of <NUM> x <NUM> square matrices whose entries belong to R. Since both R and S are semirings, to avoid confusion we say that R is the base semiring. In this case, each semiring element (of S) is a matrix whose entries belong to the base semiring R.

Table <NUM> below provides a partial list of example constructions that can be used to build a semigroup. In many cases, such semigroup may be built from other building blocks, such as other semigroups or sometimes through a semiring.

Utilizing the embodiments of Table <NUM> above, in a direct product of the base semigroups, the resulting semigroup is at least as secure as the strongest base semigroup. This is the strongest link construction.

In other constructions, each construction may boost the security compared to the base objects. Thus, the aim is for security amplification.

In one example, consider a semigroup based on resultants of bivariate polynomials. The semigroup is first described mathematically. The details of using such semigroup in a cryptographic system are then described.

Let Z be the ring of integers. Let Z[x,y] be the set of bivariate polynomials with integer coefficients. Normally, Z[x,y] is treated like a ring R, under polynomial and addition, but here we give Z[x,y] a different semiring structure B. Addition in B, written as +B, is multiplication in R. Multiplication in B, written as *, uses the resultant operation, so (f*g)(x,y) = Rest(f(x,t),g(t,y)).

Now, B is a semiring with non-commutative multiplication. This follows from the well-known theory of resultants. For example, Res(f,gh)=Res(f,g)Res(f,h), proves the distributive law.

In particular, a semigroup S can be formed, with multiplication written *, by using <NUM> by <NUM> square matrices with entries in B, and where S multiplication is B-matrix multiplication (using operations in B).

An example of multiplication in S. Let <MAT> <MAT>.

Then: <MAT>
which equals the matrix: <MAT>.

So, now there are eight resultants to compute. One way to compute a resultant is to compute the determinant of the Sylvester matrix.

In this disclosure, the horizontal, ascending version of the Sylvester matrix is defined. The Sylvester matrix is a square matrix with sides length equal to the sum of the degrees in the active variables, in this case variable t. The t coefficients of each input polynomial are arranged horizontally, in ascending order, from the lowest degree term to the highest. Zeros fill the remaining entries of the row. Each polynomial is used in a number of rows matching the degree of the other polynomial. Each use of the polynomial is shifted once to the right, until it reaches the right side of the matrix.

Other arrangements, such as vertical or descending, for the Sylvester matrix are also possible (and sometimes used in textbooks), but they at most change the sign.

The determinant in this case is 72xy<NUM>+<NUM>. Computing all <NUM> determinants similarly, one gets that: <MAT>.

Finally, one can expand each entry, which are given above as standard polynomial products, into sums, as follows: <MAT>.

For better security, a starting polynomial (e.g. private key b) should be chosen with a higher degree and larger coefficient. Also, matrices with more rows and columns may be used. It should then be much more difficult to determine b from a*b and a.

The typical known algorithms for matrix division, such as Bareiss reduction, work over matrices with entries in a commutative ring. But here the matrix entries are not commutative, and not even a ring. For example, subtraction is not possible. Perhaps B can be extended to a ring, by introducing formal differences (in a manner similar to how negative integers can be introduced as formal difference of positive integers). But then division and the non-commutativity need to be dealt with. These difficulties may represent a significant hurdle to cryptanalysis.

Those skilled in the art may notice that that a*b was initially obtained in a form whose entries were products of resultants over the entries a and b. If the entries of a*b can be factored, then the factors can try to be matched to the entries of b, and then division in the semiring B be performed, to extract the entries of b.

This attack strategy requires polynomial factorization. Polynomial factorization, for large integer coefficients and high degree polynomials can be difficult for conventional (non-quantum) computers.

A quantum computer may make polynomial factorization easier. However, to address the quantum computer risk, another measure can be used. Ensure the input matrices a and b have entries which are products too. The product a*b matrix entries can still be factored, but now there may be many more factors, and there may not be any easy way to match factors of the a*b entries to those of b.

Table <NUM> above listed various well-known construction of semi-groups that may be used to build up semigroups (from more base semigroups) with better security.

Two users that may communicate are referred to herein as Alice and Charlie. In multiplicative signature schemes, generally Alice may send Charlie a signed message, where Charlie will have access to Alice's public key to verify the signed document. If Alice and Charlie to use such semigroups for multiplicative signature schemes, they need to be able to send and represent semigroup elements to each other. So, that means that they must have some means of converting a semigroup element into a sequence of bytes. Such byte representations are commonly used in cryptography. They are used in Rivest-Shamir-Adleman (RSA), Elliptic Curve Cryptography (ECC) and many other systems.

A system for the semigroup based on resultants, which was described mathematically above, is described below.

In some cases, a new byte-encoding scheme may be used for such semigroup. Alternatively, rather the devising an entirely new byte-encoding scheme, in one embodiment some existing byte-encoding scheme that can do two things: encode integers, and encode sequences of other objects, may be used. Abstract Syntax Notation <NUM> (ASN. <NUM>) can do this (or more precisely ASN. <NUM> Basic Encoding Rules can do this). Another encoding systems is Javascript Object Notations (JSON).

In one embodiment, a matrix may be represented as a sequence of its rows. Further, a row may be represented as a sequence of entries. A bivariate polynomial in variables x and y may be represented as a sequence of y coefficients, in order of increasing degree, starting from degree zero, with each coefficient being a univariate polynomial in variable x. (But represent a zero polynomial as an empty sequence. ) A univariate polynomial in x may be represented as a sequence of coefficients in ascending degree starting from degree zero, with each coefficient being an integer (and represent a zero polynomial by an empty sequence).

Suppose that a sequence of objects a, b, c is represented as [a,b,c], where each object a,b,c being replaced by its representation. Suppose integers are represented in the usual decimal form.

Consider the example a*b from Equation <NUM> above. Its representation is: [[[[-<NUM>],[<NUM>,<NUM>,<NUM>],[<NUM>,<NUM>,<NUM>,-<NUM>]],[[-<NUM>,<NUM>,<NUM>],[],[-<NUM>,<NUM>,<NUM>]]],[[[-<NUM>],[-<NUM>],[<NUM>,-<NUM>],[<NUM>,-<NUM>]],[[<NUM>,<NUM>],[],[<NUM>,<NUM>]]]].

From these nested sequences and integers, a byte encoding is relatively easy. The most naïve is to just use ASCII text.

As noted above, Alice would likely use larger parameters than the shown in the example, in order to achieve better security. So, Alice would use larger integers, higher-degree polynomials, and matrices with more entries. But they could still use the encoding scheme described above, even they use much larger parameters. Larger parameters do mean that Alice must exchange a greater number of bytes with Charlie.

In addition to the embodiments above using resultants, which describes a new semigroup, and the brief sketch, there are also the semigroups listed in the separate technical and research reports. Again, these various semigroups can be combined into larger semigroups. In some cases, the semigroups involve semirings.

The list below shows, by name, various such semigroups:.

Consider the semiring of ideals of a standard polynomial ring. The theory of Groebner bases provides a unique representation of each ideal, in terms of its basis.

Then addition and multiplication are straightforward: to add just take the union of the bases, and to multiply the ideals just multiply the bases. Then re-normalize the basis using Buchberger's algorithm. This give a semiring, so an additive semigroup and multiplicative semigroup. The additive semigroup is idempotent, so the wedge problem is easy. The multiplicative semigroup has a known efficient division algorithm, the idea quotient algorithm, which is not quite as efficient as the multiplication algorithm. Therefore, this semigroup is probably not suitable for direct use in multiplicative signature schemes.

However, the semiring maybe useful as an intermediate step of a more complicated construction. For example, it can be used as a base semiring in forming matrices, or in forming a semigroup algebra. The fact that strong subtraction in not possible in the semiring of ideals might make a known algorithm such as the Bareiss algorithm for matrix division infeasible.

Any semigroup may be converted into a multiplicative signature scheme. Examples of such semigroups are provided above.

Based on the above, if a secure, post quantum resistant signature scheme is possible, it can be created utilizing the methods and systems in accordance with the present disclosure, along with some subgroup.

Reference is now made to <FIG>. In the embodiment of <FIG>, the following terminology is used:.

In accordance with Table <NUM>, the variables a, b, c, d and e are elements of a semigroup. As indicated above, the main requirement of a semigroup and its elements is that any two elements can be multiplied, for some form of multiplication that obeys the associative law a(bc) = (ab)c.

The elements a, b, c, d and e are not necessarily numbers. The multiplication is therefore not necessarily the traditional multiplication of numbers. For example, one semigroup which could be used is the plactic monoid, as described below.

A public key is a pair [c,e] of the elements. A public key is also called a signature verification key. Element c is the checker and in accordance with the embodiments here in, is fixed system wide. Element e is the endpoint. Generally, element e is specific to a single signer.

For simplicity in the present disclosure, signers are named using their public keys. In reality, a public key infrastructure (PKI) would be used to establish each signer's public key [c,e], binding the cryptographic value [c,e] to a more legible name of the signer.

The signer's public key is generally embedded into a certificate which certifies that the [c,e] belongs to the signer. A typical PKI distributes some certificates manually as root certificates, and then transfers trust to other certificates using digital signatures.

A signed matter is a pair [a,d] of elements. Element a is the matter and element d is the signature. The matter is usually derived as a digest of a meaningful message. A matter is sometimes common to many signers, for example when short messages need to be signed.

With regards to terminology, it is often said that d is a signature on matter a, or that d is a signature over a.

Reference is now made to <FIG> in which a user, referred to herein as "Alice" wishes to send a signed message to a second user "Charlie". In this regard, at block <NUM>, Alice may generate a public key [c,e] using her private key, referred to as element b. in particular, a private key b for public key [c,e] is an element b such that: <MAT>.

A public key [c,e] is viable if there exists at least one private key b for [c,e].

At block <NUM>, Alice can choose private key b before choosing a public key[c,e] by computing the endpoint e from equation <NUM> above. This results in a viable public key.

From block <NUM>, the process proceeds to block <NUM> in which Alice may generate a signature d. In particular, Alice can sign matter a using private key b in accordance with the equation <NUM>.

Alice may then provide the signed matter [a,d] to Charlie as shown with message <NUM>. In some embodiments, the public key [c,e] is also provided in message <NUM>. However, in other cases, the public key may be published through other mechanisms.

Charlie receives message <NUM> and may obtain the signed matter [a,d] from the message. Charlie may further obtain the public key for Alice [c,e] either from a message <NUM> or through other mechanisms.

Further, as the checker is common system wide in the present embodiments, in some cases message <NUM> may only contain the endpoint. In other cases, the endpoint may be published through other mechanisms, without the checker, as the checker would be known to both Alice and Charlie. Other options are possible.

At block <NUM> Charlie may compute the endmatter ae by multiplying elements a and e.

At block <NUM>, Charlie may compute the signcheck dc by multiplying elements d and c.

In particular, it is sometimes useful to discuss both sides of equation one separately because they can be different in invalid signatures, because they require separate computations, and because they can help identify when an existing signature scheme is similar to a multiplicative signature scheme.

Therefore, at block <NUM>, a check can be made to determine whether the endmatter and the signcheck are the same. Specifically, the signed matter [a,d] is verifiable for [c,e] because multiplication is associative in accordance with equation <NUM> below.

Therefore, based on the results of block <NUM>, Charlie can verify whether Alice signed matter a.

As will be appreciated by those in the art, the signer should keep the private key b private so that no one else can generate signatures under [c,e].

In cryptography, a hash function is an algorithm that maps an arbitrary length input to a fixed sized output with a one way function. In particular, a hash function is deterministic, meaning that it gives the same results for the same message. Further, as a hash function is one-way, it is infeasible to obtain the message given the hash value.

In some embodiments, hash functions may be a keyed hash function. In this case, the hash function uses both a cryptographic key k and a cryptographic hash function.

A hashed multiplicative signature scheme modifies the multiplicative signature scheme of <FIG> as outlined below. Table <NUM> provides for a summary of this scheme.

Based on Table <NUM>, hashed multiplicative signatures signers and verifiers compute the matter from the hash as: <MAT>.

In equation <NUM> above, m is the message and f(m) is the hash function that is applied to the message.

Therefore, reference is now made to <FIG>. In the embodiment of <FIG>, Alice generates a hash of the message at block <NUM> to create the message digest a utilizing equation <NUM> above.

At block <NUM>, Alice may generate a public key [c,e] using her private key, referred to as element b. In particular, a private key b for public key [c,e] is an element b such that equation <NUM> above is satisfied.

At block <NUM>, utilizing equation <NUM> above and equation <NUM>, the raw signature d can be computed by Alice.

Alice may then provide the signed message in the form [m,d,f] in message <NUM> to Charlie. In some embodiments, message <NUM> may further include Alice's public key [c,e]. However, as c is known system wide, it may in some cases not form part of message <NUM>. Further, the endpoint for Alice may be published in other ways besides in message <NUM> and therefore, in some cases the public key is not provided in message <NUM>.

Further, in some embodiments, the hash function f may take the form of equation <NUM>.

In equation <NUM> above, h is a keyed hash function having a key k. In this case, the hash function h may be fixed across the whole system and in this case, the key k is sufficient to specify function f. This allows f to have a short specification so that the signature [d,f] is not too long. For example, the signature could be [d,k].

In this case, message <NUM> could transmit the signed message as [m,d,k] rather than [m,d,f].

Further, in some embodiments, the key k can be fixed for the whole system. In this case, it may be unnecessary for a signer to transmit the key k to the verifier. In this case, the signed message [m,d,f] reduces to [m,d] in message <NUM>.

When the key k is not fixed, it can be chosen in various ways. Sometimes Alice may choose the key k randomly from a key space. Sometimes Alice may choose key k as a deterministic, pseudorandom function of the message. For example, equation <NUM> may be used: <MAT>.

In equation <NUM>, the key k is therefore derived based on the private key of Alice.

Further, in some cases, the multiplicative signature scheme from the embodiment of <FIG> could be considered a special case of the hashed multiplication signature scheme of <FIG> when the hash function is unity.

Further, in some embodiments, the hash function used with <FIG> is not necessarily a standard hash function, since it needs to map messages into semigroup elements using the multiplicative signature scheme of the embodiment of <FIG>. Rather, some form of embedded hashing may be used.

The use of embedding is common in other types of signatures as well. For example, in RSA signatures, such embedded hashing is often called full domain hashing.

As described below, in the case of plactic signatures, a message may be mapped into a semistandard tableau. A traditional hash can be represented as a byte string. Standard techniques allow this byte string to be made as long as necessary. A simple way to turn a byte string into a semistandard tableau is to convert it into a string of characters or integers or any sortable entries, and then apply the Robinson-Schensted algorithm, which converts the string into a tableau.

A benefit of hashing is that a long message m can have a short hash, which usually means that the signature d=ab is short. Further, hashing algorithms are typically faster than secure semigroup multiplication.

Referring again to <FIG>, Charlie receives message <NUM> and at block <NUM> may generate the matter a=f(m).

At block <NUM> Charlie may compute the endmatter ae by multiplying elements a and e. In particular, in this case the endmatter is f(m)e.

At block <NUM>, a check can be made to determine whether the endmatter and the signcheck are the same. Specifically, the signed matter [m,d,f] is verifiable for [c,e] because each element is a Semigroup, and therefore multiplication is associative in accordance with equation <NUM> below.

Therefore, based on the results of block <NUM>, Charlie can verify whether Alice signed matter m.

One semigroup that may be used for the embodiments of the present disclosure is a plactic monoid. A monoid is any semigroup with an identity element. When clear from context, the identity element is written as <NUM>. Being an identity element means <NUM>a = a = a<NUM> for all a in the monoid.

A tableau consists of rows of symbols, sorted by length. For example, Alfred Young in <NUM> defined a set of boxes or cells arranged in a left-justified form, now known as a Young diagram. <FIG> shows a Young diagram <NUM> in which the cells are sorted by length from the top down, with the top being the longest. This is referred to as English notation. <FIG> shows a Young diagram <NUM> sorted by length with the bottom being the longest. This is referred to a French notation. The embodiments of the present disclosure could use either notation, but French notation is used as an example in the present disclosure.

A Young tableau comprises filling the boxes in the Young diagram with symbols from some ordered set. If the ordered set has no duplicates, this is referred to as a standard tableau. If the symbols in the ordered set are allowed to repeat, this may create a semistandard tableau. Specifically, in a semistandard tableau, each row is sorted from lowest to highest (left to right), with repeats allowed. Also, in a semistandard tableau, each is column is sorted, but with no repeats allowed. In French notation, the columns are sorted lowest-to-highest (bottom-to-up).

An example of a semistandard tableau with single-digit symbols is shown with regards to Table <NUM> below.

Knuth (<NUM>) defined an associative binary operation applicable to semistandard Young tableaus via algorithms of Robinson (<NUM>) and Schensted (<NUM>). Schutzenberger and Lascoux (<NUM>) studied the resulting algebra, naming it the plactic monoid.

Multiplication can occur on a symbol by symbol basis. An example is shown in Table <NUM> below, which shows the creation of a plactic monoid based on the string 'helloworld' using a single-symbol tableau. In Table <NUM>, each row must be sorted. In this case, when the next symbol can be added to the end of bottom row and leave the row sorted, it is added to this row. When the symbol cannot be added to the end of the bottom row, it replaces the symbol in that row which would leave the row sorted, and the replaced symbol is added to the row above in a similar fashion. If there is no row above a symbol, then a new row is created.

As seen from Table <NUM>, multiplication is achieved through repeated insertion of symbols in a semistandard tableau.

In some embodiments, a string equivalence may be created from the tableau of Table <NUM>. Specifically, instead of considering the plactic monoid as the set of semistandard tableaus, the set of all string forms may be considered up to equivalence relation. Two strings are equivalent if they generate the same semistandard tableaus. In this form, each tableau has alternate representation as strings,.

From the last row of Table <NUM> for the "helloworld" string, the row reading string is "wheodlllor" and the column reading string is "whedolllor". These two strings, and several others, will generate the same tableau as "helloworld" generates.

The "helloworld" and "wheodlllor" strings are equivalent because they both generate the same tableau. Thus they are both alternative representations of the same semistandard tableau.

A simplified example of a C program to implement such multiplication is shown with regards to Table <NUM> below.

In the example code in Table <NUM>, the program input is any text, with each ASCII character except nulls and newlines representing a generator element of the plactic monoid. The output of the program is a product of these, according to Knuth's version of the Robinson-Schensted algorithm, represented as a semistandard tableau, as shown for example in Table <NUM>.

While the examples of Tables <NUM> and <NUM> use ASCII characters as the ordered set for single-symbol tableaus, in practice any ordered set may be used as long as it is agreed to by both parties in a digital signature situation, as described below. Further, while single-symbol tableaus are provided in the examples, in practice multi-symbol tableaus could equally be used.

Each element of the plactic monoid is therefore a product of symbols (also called generators). Products of symbols are considered equivalent if they can be related by a sequence of Knuth transformations as described above.

Elements of the plactic monoid are therefore equivalence classes of such products of symbols.

As a plactic monoid is a semigroup, it can be used for multiplicative signatures, referred to herein as plactic signatures. Further, as the plactic monoid does not seem closely related to the RSA or ECDSA, the security of plactic signatures is independent of security these signature schemes. Plactic signatures therefore seem resistant to known quantum attacks.

Reference is now made to <FIG>. In the embodiment of <FIG>, Alice wishes to sign a message m through a plactic signature scheme. In particular, as with <FIG> and <FIG>, Alice and Charlie communicate with each other.

In accordance with the embodiment of <FIG>, Alice, at block <NUM>, creates the message digest a utilizing a hash function. The hash function used with <FIG> is not necessarily a standard hash function, since it needs to map messages into semigroup elements. Rather, a form of embedded hashing may be used.

Specifically, in the case of plactic signatures, a message may be mapped into a semistandard tableau. A traditional hash can be represented as a byte string. Standard techniques allow this byte string to be made as long as necessary. A simple way to turn a byte string into a semistandard tableau is to convert it into a string of characters or integers or any sortable entries, and then apply the Robinson-Schensted algorithm, which converts the string into a tableau.

At block <NUM>, Alice can choose private key b before choosing a public key [c,e] by computing the endpoint e from equation <NUM> above. This results in a viable public key.

Therefore, a semistandard tableau for e can be generated based on b and c. The generation can use the examples of Tables <NUM> and <NUM>, for example.

At block <NUM>, the raw signature d can be computed by Alice. Specifically, a semistandard tableau for d can be generated based on a and b. The generation can use the examples of Tables <NUM> and <NUM>, for example.

Further, if f is a keyed hash function having a key k, the hash function h may be fixed across the whole system. In this case, the key k is sufficient to specify function f. This allows f to have a short specification so that the signature [d,f] is not too long. For example, the signature could be [d,k].

Further, in some cases, the multiplicative signature scheme from the embodiment of <FIG> could be used in plactic signatures when the hash function is an identity function in the embodiment of <FIG>.

Charlie receives message <NUM> and at block <NUM> may generate the matter a=f(m). Specifically, at block <NUM>, Charlie creates the message digest a utilizing the same hash function as Alice used.

At block <NUM> Charlie may compute the endmatter ae by multiplying elements a and e using Knuth multiplication.

At block <NUM>, Charlie may compute the signcheck dc by multiplying elements d and c using Knuth multiplication.

At block <NUM>, a check can be made to determine whether the endmatter and the signcheck are the same. Specifically, the signed matter [m,d,f] is verifiable for [c,e] semistandard tableau.

In a simplified example, using the identify function as the hash function, a message "helloworld" as found in Table <NUM> above may be the a or m for the equations above.

Assume that Alice has a simple private key b as "privatekey". The checker c is the string "checker".

Based on these values, d=ab results in the following tableau:.

The endpoint is e=bc and is shown in Table <NUM> below:.

Therefore [a,d] and [c,e] can be obtained by Charlie, who can calculate the endmatter ae as:.

Further, Charlie can calculate the signcheck dc as:.

Based on the above, the endmatter and signcheck match, and therefore Alice signed the message.

While the Examples of Tables <NUM> to <NUM> are simplified for illustration, in practice stronger checker and private keys would be used. Further, the use of a hash other than a unitary hash is also possible.

In the above, security is provided by having a difficult division. In particular, private key b for public key [c,e] could be found using a division operator by computing b = e/c.

To ensure that signatures are secure, division must therefore be difficult, at least for any inputs e and c using in signatures, or else an attacker could find the private key and sign any message.

Further, for security, the semigroup should not have a fast cross-multiplier. In particular, a cross-multiplier is an operator, written as */ such that: <MAT>.

Equation <NUM> only holds if there are values u and v such that ux = vy.

Some semigroups have fast cross-multipliers. In a commutative semigroup x */y = x defines a cross multiplier. In a semigroup with a zero element, x */y = <NUM> defies a cross multiplier. In a group with efficient inversion, x*/ y = y-<NUM> defines a cross multiplier.

The plactic monoid is non-communitive, has no zero element and has no inverse and so therefore these three definitions fail to give cross multipliers for a plastic monoid.

A further attack vector is to forge unhashed signatures. To forge a matter a in an unhashed multiplicative, an attempt to factor a = a<NUM>a<NUM> is performed. A signer is then asked to sign matter a<NUM>. The signer returns a signature d<NUM>. the forger can then compute d=a<NUM>d<NUM>, which is a valid signature on matter a. in this case, the forgery is aided by the signer.

Based on the above, a hashed multiplicative signature may need to be used to implement plactic signatures in some cases.

Therefore, plactic signatures could be used for digital signatures. Plactic signatures have no known quantum vulnerabilities and could therefore be used to provide additional security to communications.

The above methods may be implemented using any computing device. One simplified diagram of a computing device is shown with regard to <FIG>. The computing device of <FIG> could be any fixed or mobile computing device.

In <FIG>, device <NUM> includes a processor <NUM> and a communications subsystem <NUM>, where the processor <NUM> and communications subsystem <NUM> cooperate to perform the methods of the embodiments described above. Communications subsystem <NUM> may, in some embodiments, comprise multiple subsystems, for example for different radio technologies.

Processor <NUM> is configured to execute programmable logic, which may be stored, along with data, on device <NUM>, and shown in the example of <FIG> as memory <NUM>. Memory <NUM> can be any tangible, non-transitory computer readable storage medium which stores instruction code that, when executed by processor <NUM> cause device <NUM> to perform the methods of the present disclosure. The computer readable storage medium may be a tangible or in transitory/non-transitory medium such as optical (e.g., CD, DVD, etc.), magnetic (e.g., tape), flash drive, hard drive, or other memory known in the art.

Alternatively, or in addition to memory <NUM>, device <NUM> may access data or programmable logic from an external storage medium, for example through communications subsystem <NUM>.

Communications subsystem <NUM> allows device <NUM> to communicate with other devices or network elements and may vary based on the type of communication being performed. Further, communications subsystem <NUM> may comprise a plurality of communications technologies, including any wired or wireless communications technology.

Communications between the various elements of device <NUM> may be through an internal bus <NUM> in one embodiment. However, other forms of communication are possible.

While operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be employed. Moreover, the separation of various system components in the implementation descried above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

While the above detailed description has shown, described, and pointed out the fundamental novel features of the disclosure as applied to various implementations, it will be understood that various omissions, substitutions, and changes in the form and details of the system illustrated may be made by those skilled in the art. In addition, the order of method steps are not implied by the order they appear in the claims.

Typically, storage mediums can include any or some combination of the following: a semiconductor memory device such as a dynamic or static random access memory (a DRAM or SRAM), an erasable and programmable read-only memory (EPROM), an electrically erasable and programmable read-only memory (EEPROM) and flash memory; a magnetic disk such as a fixed, floppy and removable disk; another magnetic medium including tape; an optical medium such as a compact disk (CD) or a digital video disk (DVD); or another type of storage device. Note that the instructions discussed above can be provided on one computer-readable or machine-readable storage medium, or alternatively, can be provided on multiple computer-readable or machine-readable storage media distributed in a large system having possibly a plurality of nodes. Such computer-readable or machine-readable storage medium or media is (are) considered to be part of an article (or article of manufacture). The storage medium or media can be located either in the machine running the machine-readable instructions, or located at a remote site from which machine-readable instructions can be downloaded over a network for execution.

Claim 1:
A method for verification at a computing device of a signed message received from a first party over a public communications channel, the method comprising:
extracting, by the computing device, a message digest "a" from the signed message, the message digest "a" belonging to a semigroup;
obtaining, by the computing device, a public key [c,e] for the first party, elements of the public key including a checker "c" and an endpoint "e", checker "c" and endpoint "e" belonging to the semigroup and the endpoint comprising a multiplication of a private key "b" for the first party and the checker "c";
wherein the semigroup is a plactic monoid , and wherein multiplication is Knuth multiplication;
multiplying the message digest "a" and the endpoint "e" to create an endmatter "ae" = "abc";
extracting, by the computing device, a signature "d" from the signed message, the signature "d" belonging to the semigroup and being a multiplication of message digest "a" and private key "b";
multiplying the signature "d" and the checker "c" to create a signcheck "dc" = "abc"; and
verifying that the endmatter "ae" matches the signcheck "dc",
wherein the checker "c" is a fixed value.